I. D. ChueshovI.D.Chueshov Dissipative Systems Infinite-Dimensional Introduction Theory I. D. Chueshov Introduction to the Theory of InfiniteDimensional Dissipative Systems 966–7021–64–5

I . D . C h u e s h o v

Dissipative

Systems

Infinite-Dimensional

Introduction Theory

I. D. Chueshov

Introduction to the Theory of Infinite�Dimensional Dissipative Systems

966–7021–64–5

O R D E R

www.acta.com.ua

I . D . C h u e s h o vC h u e s h o v

U n i v e r s i t y l e c t u r e s i n c o n t e m p o r a r y m a t h e m a t i c s

Dissipativeissipative

Systemsystems

of Infinite-DimensionalInfinite-Dimensional

Introductionntroduction TheoryTheoryto the

This book provides an exhau -stive introduction to the scope of main ideas and methods of the theory of infinite-dimensional dis -sipative dynamical systems which has been rapidly developing in re -cent years. In the examples sys tems generated by nonlinear partial differential equations arising in the different problems of modern mechanics of continua are considered. The main goal of the book is to help the reader to master the basic strategies used in the study of infinite-dimensional dissipative systems and to qualify him/her for an independent scien -tific research in the given branch. Experts in nonlinear dynamics will find many fundamental facts in the convenient and practical form in this book.

The core of the book is com -posed of the courses given by the author at the Department of Me chanics and Mathematics at Kharkov University during a number of years. This book con -tains a large number of exercises which make the main text more complete. It is sufficient to know the fundamentals of functional analysis and ordinary differential equations to read the book.

Translated by Constantin I. Chueshov

from the Russian edition («ACTA», 1999)

Translation edited by Maryna B. Khorolska

Author: I. D. ChueshovI. D. Chueshov

Title: Introduction to the Theory Introduction to the Theory of Infinite�Dimensional of Infinite�Dimensional Dissipative SystemsDissipative Systems

ISBN: 966–7021–64–5

You can O R D E R O R D E R this bookwhile visiting the website of «ACTA» Scientific Publishing House http://www.acta.com.uawww.acta.com.ua/en/«A

CTA

»20

02

I . D . C h u e s h o v

IntroductionIntroductionIntroductionIntroduction

to the Theory of Infinite-Dimensionalto the Theory of Infinite-Dimensionalto the Theory of Infinite-Dimensionalto the Theory of Infinite-Dimensional

Dissipative SystemsDissipative SystemsDissipative SystemsDissipative Systems

A C T A 2 0 0 2

UDC 517

2000 Mathematics Subject Classification:primary 37L05; secondary 37L30, 37L25.

This book provides an exhaustive introduction to the scopeof main ideas and methods of the theory of infinite-dimen-sional dissipative dynamical systems which has been rapidlydeveloping in recent years. In the examples systems genera-ted by nonlinear partial differential equations arising in thedifferent problems of modern mechanics of continua are con-sidered. The main goal of the book is to help the reader tomaster the basic strategies used in the study of infinite-di-mensional dissipative systems and to qualify him/her for anindependent scientific research in the given branch. Expertsin nonlinear dynamics will find many fundamental facts in theconvenient and practical form in this book.

The core of the book is composed of the courses given bythe author at the Department of Mechanics and Mathematicsat Kharkov University during a number of years. This bookcontains a large number of exercises which make the maintext more complete. It is sufficient to know the fundamentalsof functional analysis and ordinary differential equations toread the book.

Translated by Constantin I. Chueshov

from the Russian edition («ACTA», 1999)

Translation edited by Maryna B. Khorolska

ACTA Scientific Publishing HouseKharkiv, UkraineE-mail: [email protected]

© I. D. Chueshov, 1999, 2002© Series, «ACTA», 1999© Typography, layout, «ACTA», 2002

ISBN 966-7021-20-3 (series)

ISBN 966-7021-64-5

Свідоцтво ДК №179

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ContentsContentsContentsContents

. . . . Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

C h a p t e r 1 . Basic Concepts of Basic Concepts of Basic Concepts of Basic Concepts of ththththe Theorye Theorye Theorye Theory

of Infinite-Dimensional Dynamical Syof Infinite-Dimensional Dynamical Syof Infinite-Dimensional Dynamical Syof Infinite-Dimensional Dynamical Syststststemsemsemsems

. . . . § 1 Notion of Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . 11

. . . . § 2 Trajectories and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . 17

. . . . § 3 Definition of Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

. . . . § 4 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . . 24

. . . . § 5 Theorems on Existence of Global Attractor . . . . . . . . . . . . . . 28

. . . . § 6 On the Structure of Global Attractor . . . . . . . . . . . . . . . . . . . 34

. . . . § 7 Stability Properties of Attractor and Reduction Principle . . 45

. . . . § 8 Finite Dimensionality of Invariant Sets . . . . . . . . . . . . . . . . . 52

. . . . § 9 Existence and Properties of Attractors of a Class of Infinite-Dimensional Dissipative Systems . . . . . . . . . . . . . 61

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

C h a p t e r 2 . Long-Time Behaviour of SolutionsLong-Time Behaviour of SolutionsLong-Time Behaviour of SolutionsLong-Time Behaviour of Solutions

to a Class of Semilinear Parabolic Equationsto a Class of Semilinear Parabolic Equationsto a Class of Semilinear Parabolic Equationsto a Class of Semilinear Parabolic Equations

. . . . § 1 Positive Operators with Discrete Spectrum . . . . . . . . . . . . . . 77

. . . . § 2 Semilinear Parabolic Equations in Hilbert Space . . . . . . . . . . 85

. . . . § 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

. . . . § 4 Existence Conditions and Properties of Global Attractor . . 101

. . . . § 5 Systems with Lyapunov Function . . . . . . . . . . . . . . . . . . . . . 108

. . . . § 6 Explicitly Solvable Model of Nonlinear Diffusion . . . . . . . . . 118

. . . . § 7 Simplified Model of Appearance of Turbulence in Fluid . . . 130

. . . . § 8 On Retarded Semilinear Parabolic Equations . . . . . . . . . . . 138

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4 C o n t e n t s

C h a p t e r 3 . Inertial ManifoldsInertial ManifoldsInertial ManifoldsInertial Manifolds

. . . . § 1 Basic Equation and Concept of Inertial Manifold . . . . . . . . 149

. . . . § 2 Integral Equation for Determination of Inertial Manifold . . 155

. . . . § 3 Existence and Properties of Inertial Manifolds . . . . . . . . . . 161

. . . . § 4 Continuous Dependence of Inertial Manifold on Problem Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

. . . . § 5 Examples and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

. . . . § 6 Approximate Inertial Manifolds for Semilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . 182

. . . . § 7 Inertial Manifold for Second Order in Time Equations . . . . 189

. . . . § 8 Approximate Inertial Manifolds for Second Orderin Time Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

. . . . § 9 Idea of Nonlinear Galerkin Method . . . . . . . . . . . . . . . . . . . 209

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

C h a p t e r 4 . The Problem on NonlinearThe Problem on NonlinearThe Problem on NonlinearThe Problem on Nonlinear

Oscillations of a Plate in a Supersonic Gas FlowOscillations of a Plate in a Supersonic Gas FlowOscillations of a Plate in a Supersonic Gas FlowOscillations of a Plate in a Supersonic Gas Flow

. . . . § 1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

. . . . § 2 Auxiliary Linear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

. . . . § 3 Theorem on the Existence and Uniqueness of Solutions . . 232

. . . . § 4 Smoothness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

. . . . § 5 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . 246

. . . . § 6 Global Attractor and Inertial Sets . . . . . . . . . . . . . . . . . . . . 254

. . . . § 7 Conditions of Regularity of Attractor . . . . . . . . . . . . . . . . . . 261

. . . . § 8 On Singular Limit in the Problemof Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

. . . . § 9 On Inertial and Approximate Inertial Manifolds . . . . . . . . . 276

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Contents 5

C h a p t e r 5 . Theory of FunTheory of FunTheory of FunTheory of Functctctctionalsionalsionalsionals

ththththat Uniquely Determine Long-Time Dynamicsat Uniquely Determine Long-Time Dynamicsat Uniquely Determine Long-Time Dynamicsat Uniquely Determine Long-Time Dynamics

. . . . § 1 Concept of a Set of Determining Functionals . . . . . . . . . . . 285

. . . . § 2 Completeness Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

. . . . § 3 Estimates of Completeness Defect in Sobolev Spaces . . . . 306

. . . . § 4 Determining Functionals for AbstractSemilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . 317

. . . . § 5 Determining Functionals for Reaction-Diffusion Systems . . 328

. . . . § 6 Determining Functionals in the Problemof Nerve Impulse Transmission . . . . . . . . . . . . . . . . . . . . . . 339

. . . . § 7 Determining Functionalsfor Second Order in Time Equations . . . . . . . . . . . . . . . . . . 350

. . . . § 8 On Boundary Determining Functionals . . . . . . . . . . . . . . . . 358

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

C h a p t e r 6 . Homoclinic ChaosHomoclinic ChaosHomoclinic ChaosHomoclinic Chaos

in Infinite-Dimensional Syin Infinite-Dimensional Syin Infinite-Dimensional Syin Infinite-Dimensional Syststststemsemsemsems

. . . . § 1 Bernoulli Shift as a Model of Chaos . . . . . . . . . . . . . . . . . . . 365

. . . . § 2 Exponential Dichotomy and Difference Equations . . . . . . . 369

. . . . § 3 Hyperbolicity of Invariant Setsfor Differentiable Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 377

. . . . § 4 Anosov’s Lemma on -trajectories . . . . . . . . . . . . . . . . . . . 381

. . . . § 5 Birkhoff-Smale Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

. . . . § 6 Possibility of Chaos in the Problemof Nonlinear Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . 396

. . . . § 7 On the Existence of Transversal Homoclinic Trajectories . . 402

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

. . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

�

Палкой щупая дорогу,Бродит наугад слепой,Осторожно ставит ногуИ бормочет сам с собой.И на бельмах у слепогоПолный мир отображен:Дом, лужок, забор, корова,Клочья неба голубого —Все, чего не видит он.

Вл. Ходасевич«Слепой»

A blind man tramps at random touching the road with a stick.He places his foot carefully and mumbles to himself.The whole world is displayed in his dead eyes.There are a house, a lawn, a fence, a cowand scraps of the blue sky — everything he cannot see.

Vl. Khodasevich

«A Blind Man»

PrefacePrefacePrefacePreface

The recent years have been marked out by an evergrowing interest in theresearch of qualitative behaviour of solutions to nonlinear evolutionarypartial differential equations. Such equations mostly arise as mathematicalmodels of processes that take place in real (physical, chemical, biological,etc.) systems whose states can be characterized by an infinite number ofparameters in general. Dissipative systems form an important class of sys-tems observed in reality. Their main feature is the presence of mechanismsof energy reallocation and dissipation. Interaction of these two mecha-nisms can lead to appearance of complicated limit regimes and structuresin the system. Intense interest to the infinite-dimensional dissipative sys-tems was significantly stimulated by attempts to find adequate mathemati-cal models for the explanation of turbulence in liquids based on the notionof strange (irregular) attractor. By now significant progress in the study ofdynamics of infinite-dimensional dissipative systems have been made.Moreover, the latest mathematical studies offer a more or less common line(strategy), which when followed can help to answer a number of principalquestions about the properties of limit regimes arising in the system underconsideration. Although the methods, ideas and concepts from finite-di-mensional dynamical systems constitute the main source of this strategy,finite-dimensional approaches require serious revaluation and adaptation.

The book is devoted to a systematic introduction to the scope of mainideas, methods and problems of the mathematical theory of infinite-dimen-sional dissipative dynamical systems. Main attention is paid to the systemsthat are generated by nonlinear partial differential equations arising in themodern mechanics of continua. The main goal of the book is to help thereader to master the basic strategies of the theory and to qualify him/herfor an independent scientific research in the given branch. We also hopethat experts in nonlinear dynamics will find the form many fundamentalfacts are presented in convenient and practical.

The core of the book is composed of the courses given by the author atthe Department of Mechanics and Mathematics at Kharkov University dur-ing several years. The book consists of 6 chapters. Each chapter corre-sponds to a term course (34-36 hours) approximately. Its body can beinferred from the table of contents. Every chapter includes a separate listof references. The references do not claim to be full. The lists consist of thepublications referred to in this book and offer additional works recommen-

8 P r e f a c e

ded for further reading. There are a lot of exercises in the book. They playa double role. On the one hand, proofs of some statements are presented as(or contain) cycles of exercises. On the other hand, some exercises containan additional information on the object under consideration. We recom-mend that the exercises should be read at least. Formulae and statementshave double indexing in each chapter (the first digit is a section number).When formulae and statements from another chapter are referred to,the number of the corresponding chapter is placed first.

It is sufficient to know the basic concepts and facts from functionalanalysis and ordinary differential equations to read the book. It is quite un-derstandable for under-graduate students in Mathematics and Physics.

I.D. Chueshov

C h a p t e r 1

Basic Concepts of the TheoryBasic Concepts of the TheoryBasic Concepts of the TheoryBasic Concepts of the Theory

of Infinite-Dimensionalof Infinite-Dimensionalof Infinite-Dimensionalof Infinite-Dimensional Dynamical Systems Dynamical Systems Dynamical Systems Dynamical Systems

C o n t e n t s

. . . . § 1 Notion of Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . 11

. . . . § 2 Trajectories and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . 17

. . . . § 3 Definition of Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

. . . . § 4 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . . 24

. . . . § 5 Theorems on Existence of Global Attractor . . . . . . . . . . . . . . 28

. . . . § 6 On the Structure of Global Attractor . . . . . . . . . . . . . . . . . . . 34

. . . . § 7 Stability Properties of Attractor and Reduction Principle . . . 45

. . . . § 8 Finite Dimensionality of Invariant Sets . . . . . . . . . . . . . . . . . 52

. . . . § 9 Existence and Properties of Attractors of a Class ofInfinite-Dimensional Dissipative Systems . . . . . . . . . . . . . . . 61

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

The mathematical theory of dynamical systems is based on the qualitative theo-ry of ordinary differential equations the foundations of which were laid by HenriPoincaré (1854–1912). An essential role in its development was also played by theworks of A. M. Lyapunov (1857–1918) and A. A. Andronov (1901–1952). At presentthe theory of dynamical systems is an intensively developing branch of mathematicswhich is closely connected to the theory of differential equations.

In this chapter we present some ideas and approaches of the theory of dynami-cal systems which are of general-purpose use and applicable to the systems genera-ted by nonlinear partial differential equations.

§ 1 Notion of Dynamical System§ 1 Notion of Dynamical System§ 1 Notion of Dynamical System§ 1 Notion of Dynamical System

In this book dynamical system dynamical system dynamical system dynamical system is taken to mean the pair of objects con-sisting of a complete metric space and a family of continuous mappings of thespace into itself with the properties

, , (1.1)

where coincides with either a set of nonnegative real numbers or a set. If , we also assume that is a continuous

function with respect to for any . Therewith is called a phase space phase space phase space phase space, ora state space, the family is called an evolutionary operator evolutionary operator evolutionary operator evolutionary operator (or semigroup),parameter plays the role of time. If , then dynamical system iscalled discretediscretediscretediscrete (or a system with discrete time). If , then is fre-quently called to be dynamical system with continuouscontinuouscontinuouscontinuous time. If a notion of dimen-sion can be defined for the phase space (e. g., if is a lineal), the value iscalled a dimensiondimensiondimensiondimension of dynamical system.

Originally a dynamical system was understood as an isolated mechanical systemthe motion of which is described by the Newtonian differential equations and whichis characterized by a finite set of generalized coordinates and velocities. Now peopleassociate any time-dependent process with the notion of dynamical system. Theseprocesses can be of quite different origins. Dynamical systems naturally arise inphysics, chemistry, biology, economics and sociology. The notion of dynamical sys-tem is the key and uniting element in synergetics. Its usage enables us to covera rather wide spectrum of problems arising in particular sciences and to work outuniversal approaches to the description of qualitative picture of real phenomenain the universe.

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12 B a s i c C o n c e p t s o f t h e T h e o r y o f I n f i n i t e - D i m e n s i o n a l D y n a m i c a l S y s t e m s

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Let us look at the following examples of dynamical systems.

E x a m p l e 1.1

Let be a continuously differentiable function on the real axis posessing theproperty , where is a constant. Consider the Cauchyproblem for an ordinary differential equation

, , . (1.2)

For any problem (1.2) is uniquely solvable and determines a dynamicalsystem in . The evolutionary operator is given by the formula ,where is a solution to problem (1.2). Semigroup property (1.1) holdsby virtue of the theorem of uniqueness of solutions to problem (1.2). Equationsof the type (1.2) are often used in the modeling of some ecological processes.For example, if we take , , then we get a logistic equ-ation that describes a growth of a population with competition (the value is the population level; we should take for the phase space).

E x a m p l e 1.2

Let and be continuously differentiable functions such that

,

with some constant . Let us consider the Cauchy problem

(1.3)

For any , problem (1.3) is uniquely solvable. It generatesa two-dimensional dynamical system , provided the evolutionary ope-rator is defined by the formula

,

where is the solution to problem (1.3). It should be noted that equationsof the type (1.3) are known as Liénard equations in literature. The van der Polequation:

and the Duffing equation:

which often occur in applications, belong to this class of equations.

f x� �x f x� � C 1 x2�� C

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g x� � � , � 0 , f x� �� x3 a x�� b��

N o t i o n o f D y n a m i c a l S y s t e m 13

E x a m p l e 1.3

Let us now consider an autonomous system of ordinary differential equations

. (1.4)

Let the Cauchy problem for the system of equations (1.4) be uniquely solvableover an arbitrary time interval for any initial condition. Assume that a solutioncontinuously depends on the initial data. Then equations (1.4) generate an di-mensional dynamical system with the evolutionary operator actingin accordance with the formula

,

where is the solution to the system of equations (1.4) such that, . Generally, let be a linear space and be

a continuous mapping of into itself. Then the Cauchy problem

(1.5)

generates a dynamical system in a natural way provided this problem iswell-posed, i.e. theorems on existence, uniqueness and continuous dependenceof solutions on the initial conditions are valid for (1.5).

E x a m p l e 1.4

Let us consider an ordinary retarded differential equation

, , (1.6)

where is a continuous function on . Obviously an initial conditionfor (1.6) should be given in the form

. (1.7)

Assume that lies in the space of continuous functions on thesegment In this case the solution to problem (1.6) and (1.7) can beconstructed by step-by-step integration. For example, if the solu-tion is given by

,

and if , then the solution is expressed by the similar formula in termsof the values of the function for and so on. It is clear that the so-lution is uniquely determined by the initial function . If we now define anoperator in the space by the formula

,

where is the solution to problem (1.6) and (1.7), then we obtain an infi-nite-dimensional dynamical system .

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Now we give several examples of discrete dynamical systems. First of all it should benoted that any system with continuous time generates a discrete system ifwe take instead of Furthermore, the evolutionary operator ofa discrete dynamical system is a degree of the mapping i. e. .Thus, a dynamical system with discrete time is determined by a continuous mappingof the phase space into itself. Moreover, a discrete dynamical system is very oftendefined as a pair consisting of the metric space and the continuous map-ping

E x a m p l e 1.5

Let us consider a one-step difference scheme for problem (1.5):

, , .

There arises a discrete dynamical system , where is the continuousmapping of into itself defined by the formula .

E x a m p l e 1.6

Let us consider a nonautonomous ordinary differential equation

, , , (1.9)

where is a continuously differentiable function of its variables and is pe-riodic with respect to i. e. for some . It is as-sumed that the Cauchy problem for (1.9) is uniquely solvable on any timeinterval. We define a monodromymonodromymonodromymonodromy operator (a period mapping) by the formula

where is the solution to (1.9) satisfying the initial condition. It is obvious that this operator possesses the property

(1.10)

for any solution to equation (1.9) and any . The arising dynamicalsystem plays an important role in the study of the long-time proper-ties of solutions to problem (1.9).

E x a m p l e 1.7 (Bernoulli shift)

Let be a set of sequences consisting of zeroes andones. Let us make this set into a metric space by defining the distance by theformula

.

Let be the shift operator on , i. e. the mapping transforming the sequence into the element , where . As a result, a dynamical

system comes into being. It is used for describing complicated (qua-sirandom) behaviour in some quite realistic systems.

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N o t i o n o f D y n a m i c a l S y s t e m 15

In the example below we describe one of the approaches that enables us to connectdynamical systems to nonautonomous (and nonperiodic) ordinary differential equa-tions.

E x a m p l e 1.8

Let be a continuous bounded function on . Let us define the hull of the function as the closure of a set

with respect to the norm

.

Let be a continuous function. It is assumed that the Cauchy problem

(1.11)

is uniquely solvable over the interval for any . Let us definethe evolutionary operator on the space by the formula

,

where is the solution to problem (1.11) and . As a result,a dynamical system comes into being. A similar construction is of-ten used when is a compact set in the space of continuous bounded func-tions (for example, if is a quasiperiodic or almost periodic function).As the following example shows, this approach also enables us to use naturallythe notion of the dynamical system for the description of the evolution of ob-jects subjected to random influences.

E x a m p l e 1.9

Assume that and are continuous mappings from a metric space into it-self. Let be a state space of a system that evolves as follows: if is the state ofthe system at time , then its state at time is either or withprobability , where the choice of or does not depend on time and theprevious states. The state of the system can be defined after a number of stepsin time if we flip a coin and write down the sequence of events from the right tothe left using and . For example, let us assume that after 8 flips we get thefollowing set of outcomes:

.

Here corresponds to the head falling, whereas corresponds to the tail fall-ing. Therewith the state of the system at time will be written in the form:

h x t�� R2

Lh

h x t��

h� x t�� h x t �� , � R�� !

hC

h x t�� : x R t R�� !

sup�

g x� �

x� t� � g x� � h�

x t�� , x 0� �� x0� �

0 +"�� h�

Lh�S� X R1 Lh#�

S� x0 h�

�� x �� h��

x t� � h�� h

�x t ��

R Lh St�#� �Lh C

h x t��

f0 f1 Y

Y y

k k 1� f0 y� � f1 y� �1 2$ f0 f1

0 1

10 110010

1 0t 8�


1

C

h

a

p

t

e

r

.

This construction can be formalized as follows. Let be a set of two-sided se-quences consisting of zeroes and ones (as in Example 1.7), i.e. a collectionof elements of the type

,

where is equal to either or . Let us consider the space con-sisting of pairs , where , . Let us define the mapping

: by the formula:

,

where is the left-shift operator in (see Example 1.7). It is easy to see thatthe th degree of the mapping actcts according to the formula

and it generates a discrete dynamical system . This system is oftencalled a universal random (discrete) dynamical system.

Examples of dynamical systems generated by partial differential equations will be gi-ven in the chapters to follow.

Assume that operators have a continuous inverse for any .Show that the family of operators defined by the equa-lity for and for form a group, i.e. (1.1)holds for all .

Prove the unique solvability of problems (1.2) and (1.3) in-volved in Examples 1.1 and 1.2.

Ground formula (1.10) in Example 1.6.

Show that the mapping in Example 1.8 possesses semi-group property (1.1).

Show that the value involved in Example 1.7 is a met-ric. Prove its equivalence to the metric

.

W f1 � f0 � f1 � f1 � f0 � f0 � f1 � f0� � y� ��

�2

% % n� % 1� %0%1%n� ��

%i

1 0 X �2 Y#�x % y�� % �2� y Y�

F X X&

F x� � F % y�� S% f%0y� ��

S �2n - F

Fn % y�� Sn% f%n 1� � � f%1 �

f%0� � y� ��

�2 Y# Fn��

E x e r c i s e 1.1 St t

S�t : t R� �

S�

t St� t 0 S�t S t

1�� t 0�t �� R�

E x e r c i s e 1.2

E x e r c i s e 1.3

E x e r c i s e 1.4 St

E x e r c i s e 1.5 d x y��

d* x y�� 2 i� xi

yi

�i "��

"

'�

T r a j e c t o r i e s a n d I n v a r i a n t S e t s 17

§ 2 Trajectories and Invariant Sets§ 2 Trajectories and Invariant Sets§ 2 Trajectories and Invariant Sets§ 2 Trajectories and Invariant Sets

Let be a dynamical system with continuous or discrete time. Its trajectorytrajectorytrajectorytrajectory

(or orbitorbitorbitorbit) is defined as a set of the type

,

where is a continuous function with values in such that for all and . Positive (negative) semitrajectorysemitrajectorysemitrajectorysemitrajectory is defined as a set

, ( , respectively), where a continuous on ( , respectively) function possesses the property for any

, ( , respectively). It is clear that any positivesemitrajectory has the form , i.e. it is uniquely determined byits initial state . To emphasize this circumstance, we often write .In general, it is impossible to continue this semitrajectory to a full trajectorywithout imposing any additional conditions on the dynamical system.

Assume that an evolutionary operator is invertible for some. Then it is invertible for all and for any there

exists a negative semitrajectory ending at the point .

A trajectory is called a periodic trajectoryperiodic trajectoryperiodic trajectoryperiodic trajectory (or a cyclecyclecyclecycle) ifthere exists , such that . Therewith the minimalnumber possessing the property mentioned above is called a periodperiodperiodperiod of a tra-jectory. Here is either or depending on whether the system is a continuousor a discrete one. An element is called a fixed pointfixed pointfixed pointfixed point of a dynamical system

if for all (synonyms: equilibrium pointequilibrium pointequilibrium pointequilibrium point , stationary stationary stationary stationary

pointpointpointpoint).

Find all the fixed points of the dynamical system ge-nerated by equation (1.2) with . Does there exista periodic trajectory of this system?

Find all the fixed points and periodic trajectories of a dynami-cal system in generated by the equations

Consider the cases and . Hint: use polar coordinates.

Prove the existence of stationary points and periodic trajecto-ries of any period for the discrete dynamical system described

X St

��

( u t� � : t T� ��u t� � X S�u t� � u t �� T�� t T�

(+ u t� � : t 0 �� (– u t� � : t 0� �� T�T– u t� � S�u t� � u t �� 0� t 0 � 0 t 0 � t 0��

(� (� St v : t 0 ��v (+ (+ v� ��

(� v� �


t 0� t 0� v X�(– (– v� �� v

( u t� � : t T� ��T T�� T 0� u t T�� u t� ��

T 0�T R Z

u0 X�X St�� St u0 u0� t 0

E x e r c i s e 2.2 R St

�� f x� � x x 1��

E x e r c i s e 2.3

R2

x� �y� x x2 y2�� 2 4 x2 y2�� 1�� ,��

y� �x y x2 y2�� 2 4 x2 y2�� 1�� .��)�)�

� 0* � 0�

E x e r c i s e 2.4


1

C

h

a

p

t

e

r

in Example 1.7. Show that the set of all periodic trajectories is densein the phase space of this system. Make sure that there exists a tra-jectory that passes at a whatever small distance from any point of thephase space.

The notion of invariant set plays an important role in the theory of dynamical sys-tems. A subset of the phase space is said to be:

a) positively invariantpositively invariantpositively invariantpositively invariant, if for all ;b) negatively invariantnegatively invariantnegatively invariantnegatively invariant, if for all ;c) invariantinvariantinvariantinvariant, if it is both positively and negatively invariant, i.e. if

for all .The simplest examples of invariant sets are trajectories and semitrajectories.

Show that is positively invariant, is negatively invariantand is invariant.

Let us define the sets

and

for any subset of the phase space . Prove that is a positivelyinvariant set, and if the operator is invertible for some then is a negatively invariant set.

Other important example of invariant set is connected with the notions of -limitand -limit sets that play an essential role in the study of the long-time behaviourof dynamical systems.

Let . Then the -limit setlimit setlimit setlimit set for is defined by

,

where . Hereinafter is the closure of a set in thespace . The set

,

where , is called the -limit setlimit setlimit setlimit set for .

Y X

StY Y+ t 0

StY Y, t 0

StY Y� t 0

E x e r c i s e 2.5 (� (–

(

E x e r c i s e 2.6

(+ A� � St

A� �t 0 - v S

tu : u A��

t 0 -��

(– A� � St

1�

t 0 - A� � v : S

tv A� �

t 0 -��

A X (+ A� �S

tt 0 ,�

(– A� �

%�

A X. % A

% A� � St

t s - A� �

Xs 0 /�

St A� � v St u : u A�� Y� �X Y

X

� A� � St1� A� �

t s -

Xs 0 /�

St1�

A� � v : St v A� �� A

T r a j e c t o r i e s a n d I n v a r i a n t S e t s 19

Lemma 2.1

For an element to belong to an -limit set , it is necessary and

sufficient that there exist a sequence of elements and a se-

quence of numbers , the latter tending to infinity such that

,

where is the distance between the elements and in the

space .

Proof.

Let the sequences mentioned above exist. Then it is obvious that for any there exists such that

.

This implies that

for all . Hence, the element belongs to the intersection of these sets,i.e. .

On the contrary, if , then for all

.

Hence, for any there exists an element such that

.

Therewith it is obvious that , , . This proves thelemma.

It should be noted that this lemma gives us a description of an -limit set but doesnot guarantee its nonemptiness.

Show that is a positively invariant set. If for any there exists a continuous inverse to , then is invariant, i.e.

.

Let be an invertible mapping for every . Prove thecounterpart of Lemma 2.1 for an -limit set:

.

Establish the invariance of .

y % % A� �y

n � A.

tn

d Stn

yn

y�� n "&

lim ��

d x y�� x y

X

� 0� n0 0

Stn

yn

St

t � - A� � , n n0 �

y Stn

yn

n "&lim� S

t

t � - A� �

X

�

� 0� y

y % A� ��y % A� �� n 0 1 2 � � ��

y St

t n - A� �

X

�

n zn

zn S

t n - t A� � ,� d y zn�� 1

n��

zn

Stn

yn

� yn

A� tn

n

%

E x e r c i s e 2.7 % A� � t 0�St % A� �

St% A� � % A� ��


t 0��

y � A� � yn

� A tn

tn

+" ; d Stn

1�y

ny��

n "&lim&�0��0 0�

� �� !

1�

� A� �


1

C

h

a

p

t

e

r

Let be a periodic trajectory of a dy-namical system. Show that for any .

Let us consider the dynamical system constructed inExample 1.1. Let and be the roots of the function

, . Then the segment isan invariant set. Let be a primitive of the function ( ). Then the set is positively invariantfor any .

Assume that for a continuous dynamical system thereexists a continuous scalar function on such that the value

is differentiable with respect to for any and

, .

Then the set is positively invariant for any .

§ 3 Definition of A§ 3 Definition of A§ 3 Definition of A§ 3 Definition of Atttttractortractortractortractor

Attractor is a central object in the study of the limit regimes of dynamical systems.Several definitions of this notion are available. Some of them are given below. Fromthe point of view of infinite-dimensional systems the most convenient concept is thatof the global attractor.

A bounded closed set is called a global attractorglobal attractorglobal attractorglobal attractor for a dynamical sys-tem , if

1) is an invariant set, i.e. for any ;2) the set uniformly attracts all trajectories starting in bounded sets,

i.e. for any bounded set from

.

We remind that the distance between an element and a set is defined by theequality:

,

where is the distance between the elements and in .The notion of a weak global attractor is useful for the study of dynamical sys-

tems generated by partial differential equations.

E x e r c i s e 2.9 ( u t� � : "� t "� � ��( % u� � � u� �� u (�

E x e r c i s e 2.10 R St�� a b f x� � :

f a� � f b� � 0� � a b� I x : a x b� � ��F x� � f x� �

F� x� � f x� �� x : F x� � c� �c

E x e r c i s e 2.11 X St

�� V y� � X

V Sty� � t y X�

dtd

�� V St y� �� V St y� � 2�� 0 , 2 0 , y X��

y : V y� � R� � R 2 �$

A1 X.X S

t�� A1 S

tA1 A1� t 0�

A1B X

St y A1�� dist : y B�� !

supt "&lim 0�

z A

z A�� dist d z y�� : y A� �inf�d z y�� z y X

D e f i n i t i o n o f A t t r a c t o r 21

Let be a complete linear metric space. A bounded weakly closed set iscalled a global weak attractorglobal weak attractorglobal weak attractorglobal weak attractor if it is invariant and for anyweak vicinity of the set and for every bounded set there exists

such that for .We remind that an open set in weak topology of the space can be described

as finite intersection and subsequent arbitrary union of sets of the form

,

where is a real number and is a continuous linear functional on .It is clear that the concepts of global and global weak attractors coincide in the

finite-dimensional case. In general, a global attractor is also a global weak attrac-tor, provided the set is weakly closed.

Let be a global or global weak attractor of a dynamical sys-tem . Then it is uniquely determined and contains any boun-ded negatively invariant set. The attractor also contains the

limit set of any bounded .

Assume that a dynamical system with continuoustime possesses a global attractor . Let us consider a discrete sys-tem , where with some . Prove that is a glo-bal attractor for the system . Give an example which showsthat the converse assertion does not hold in general.

If the global attractor exists, then it contains a global minimal attractorglobal minimal attractorglobal minimal attractorglobal minimal attractor which is defined as a minimal closed positively invariant set possessing the property

for every .

By definition minimality means that has no proper subset possessing the proper-ties mentioned above. It should be noted that in contrast with the definition of theglobal attractor the uniform convergence of trajectories to is not expected here.

Show that , provided is a compact set.

Prove that for any . Therewith, if isa compact, then .

By definition the attractor contains limit regimes of each individual trajectory.It will be shown below that in general. Thus, a set of real limit regimes(states) originating in a dynamical system can appear to be narrower than the globalattractor. Moreover, in some cases some of the states that are unessential from thepoint of view of the frequency of their appearance can also be “removed” from ,for example, such states like absolutely unstable stationary points. The next twodefinitions take into account the fact mentioned above. Unfortunately, they require

X A2S

tA2 A2� t 0��

� A2 B X.t0 t0 � B�� S

tB �. t t0

X

Ul c� x X : l x� � c��

c l X

A

A

E x e r c i s e 3.1 A

X St

�� A

% - % B� � B X.


�� A1

X Tn�� T St0

� t0 0� A1X Tn��

A1 A3

Sty A3�� dist

t "&lim 0� y X�

A3

A3

E x e r c i s e 3.3 StA3 A3� A3

E x e r c i s e 3.4 % x� � A3� x X� A3A3 % x� � : x X� �-�

A3A3 A1*

A3


1

C

h

a

p

t

e

r

additional assumptions on the properties of the phase space. Therefore, these defini-tions are mostly used in the case of finite-dimensional dynamical systems.

Let a Borel measure such that be given on the phase space ofa dynamical system . A bounded set in is called a Milnor attractorMilnor attractorMilnor attractorMilnor attractor

(with respect to the measure ) for if is a minimal closed invariant setpossessing the property

for almost all elements with respect to the measure . The Milnor attractoris frequently called a probabilistic global minimal attractor.

At last let us introduce the notion of a statistically essential global minimal at-tractor suggested by Ilyashenko. Let be an open set in X and let be itscharacteristic function: , ; , . Let us define theaverage time which is spent by the semitrajectory emanating from in the set by the formula

.

A set is said to be unessential with respect to the measure if

.

The complement to the maximal unessential open set is called an IlyashenkoIlyashenkoIlyashenkoIlyashenko

aaaatttttractor tractor tractor tractor (with respect to the measure ).It should be noted that the attractors and are used in cases when the na-

tural Borel measure is given on the phase space (for example, if is a closed mea-surable set in and is the Lebesgue measure).

The relations between the notions introduced above can be illustrated by thefollowing example.

E x a m p l e 3.1

Let us consider a quasi-Hamiltonian system of equations in :

(3.1)

where and is a positive number. It is easyto ascertain that the phase portrait of the dynamical system generated by equa-tions (3.1) has the form represented on Fig. 1.

3 3 X� � "� X

X St

�� A4 X

3 X St

�� A4

St y A4�� distt "&lim 0�

y X� 3

U XU

x� �X

Ux� � 1� x U� X

Ux� � 0� x U4

� x U�� (+ x� � x

U

� x U�� T "&

lim 1T�� X

US

tx� � td

0

T

��

U 3

M U� � 3 x : � x U�� 0� �� 0�A5

3A4 A5

X

RN 3

R2

q�p5

5H 3Hq5

5H ,��

p�H55q�� 3H

p55H ,��

�))�))�

H p q�� 1 2$� � p2 q4 q2�� 3

D e f i n i t i o n o f A t t r a c t o r 23

A separatrix (“eight cur-ve”) separates the do-mains of the phase planewith the different quali-tative behaviour of thetrajectories. It is given bythe equation .The points insidethe separatrix are charac-terized by the equation

. Therewithit appears that

,

,

.

Finally, the simple calculations show that , i.e. the Ilyashenko at-tractor consists of a single point. Thus,

,

all inclusions being strict.

Display graphically the attractors of the system generatedby equations (3.1) on the phase plane.

Consider the dynamical system from Example 1.1 with . Prove that ,

, and .

Prove that and in general.

Show that all positive semitrajectories of a dynamical systemwhich possesses a global minimal attractor are bounded sets.

In particular, the result of the last exercise shows that the global attractor can existonly under additional conditions concerning the behaviour of trajectories of the sys-tem at infinity. The main condition to be met is the dissipativity discussed in the nextsection.

Fig. 1. Phase portrait of system (3.1)

H p q�� 0�p q��

H p q�� 0�

A1 A2 p q�� : H p q�� 0� ��

A3 p q�� : H p q�� 0�� !

p q�� :p55

H p q�� q55

H p q�� 0� �� !

-�

A4 p q�� : H p q�� 0� ��

A5 0 0� ��

A1 A2 A3 A4 A56 6 6�

E x e r c i s e 3.5 Aj

E x e r c i s e 3.6

f x� � x x2 1�� A1 x : 1 x 1� �� A3 x 0 x� 17� � �� A4 A5 x 17� ��

E x e r c i s e 3.7 A4 A3. A5 A3.

E x e r c i s e 3.8


1

C

h

a

p

t

e

r

§ 4 Dissipativity and Asymptotic§ 4 Dissipativity and Asymptotic§ 4 Dissipativity and Asymptotic§ 4 Dissipativity and Asymptotic

CompactnessCompactnessCompactnessCompactness

From the physical point of view dissipative systems are primarily connected with ir-reversible processes. They represent a rather wide and important class of the dy-namical systems that are intensively studied by modern natural sciences. Thesesystems (unlike the conservative systems) are characterized by the existence of theaccented direction of time as well as by the energy reallocation and dissipation.In particular, this means that limit regimes that are stationary in a certain sense canarise in the system when . Mathematically these features of the qualitativebehaviour of the trajectories are connected with the existence of a bounded absor-bing set in the phase space of the system.

A set is said to be absorbingabsorbingabsorbingabsorbing for a dynamical system if forany bounded set in there exists such that for every

. A dynamical system is said to be dissipativedissipativedissipativedissipative if it possesses a boun-ded absorbing set. In cases when the phase space of a dissipative system is a Banach space a ball of the form can be taken as an absor-bing set. Therewith the value is said to be a radius of dissipativityradius of dissipativityradius of dissipativityradius of dissipativity.

As a rule, dissipativity of a dynamical system can be derived from the existenceof a Lyapunov type function on the phase space. For example, we have the followingassertion.

Theorem 4.1.

Let the phase sLet the phase sLet the phase sLet the phase sppppace of a continuous dynamical system be a Ba-ace of a continuous dynamical system be a Ba-ace of a continuous dynamical system be a Ba-ace of a continuous dynamical system be a Ba-

nach space. Assume that:nach space. Assume that:nach space. Assume that:nach space. Assume that:

(a) there exists a continuous function on possessing the pro-there exists a continuous function on possessing the pro-there exists a continuous function on possessing the pro-there exists a continuous function on possessing the pro-

pertiespertiespertiesperties

,,,, (4.1)

where are continuous functions on and where are continuous functions on and where are continuous functions on and where are continuous functions on and

when ;when ;when ;when ;

(b) there exist a derivative for and positive numbers there exist a derivative for and positive numbers there exist a derivative for and positive numbers there exist a derivative for and positive numbers

and such that and such that and such that and such that

forforforfor .... (4.2)

Then the dynamical system is dissipative.Then the dynamical system is dissipative.Then the dynamical system is dissipative.Then the dynamical system is dissipative.

Proof.

Let us choose such that for . Let

and be such that for . Let us show that

t +"&

B0 X. X St

�� B X t0 t0 B� �� S

tB� � B0.

t t0 X St

�� X X S

t��

x X : xX

R�� R

X St

��

U x� � X

81 x� � U x� � 82 x� ��

8j

r� � R� 81 r� � +"&r "&

dtd��U S

ty� � t 0

� 2dtd��U St y� � �� St y 2�

X St��

R0 2 81 r� � 0� r R0

l 82 r� � : r 1 R0�� sup�

R1 R0 1�� 81 r� � l� r R1�

D i s s i p a t i v i t y a n d A s y m p t o t i c C o m p a c t n e s s 25

for all and . (4.3)

Assume the contrary, i.e. assume that for some such that thereexists a time possessing the property . Then the continuity of implies that there exists such that . Thus, equation(4.2) implies that

,

provided . It follows that for all . Hence, forall . This contradicts the assumption. Let us assume now that is an arbitrarybounded set in that does not lie inside the ball with the radius . Then equation(4.2) implies that

, , (4.4)

provided . Here

.

Let . If for a time the semitrajectory enters the ball withthe radius , then by (4.3) we have for all . If that does not takeplace, from equation (4.4) it follows that

for ,

i.e. for . Thus,

, .

This and (4.3) imply that the ball with the radius is an absorbing set for the dy-namical system . Thus, Theorem 4.1 is proved.

Show that hypothesis (4.2) of Theorem 4.1 can be replacedby the requirement

,

where and are positive constants.

Show that the dynamical system generated in by the diffe-rential equation (see Example 1.1) is dissipative, pro-vided the function possesses the property: ,where and are constants (Hint: ). Find an up-per estimate for the minimal radius of dissipativity.

Consider a discrete dynamical system , where isa continuous function on . Show that the system is dissi-pative, provided there exist and such that

for .

St y R1� t 0 y R0�

y X� y R0�t 0� S

ty R1� S

ty

0 t0 t� � 2 St0

y� R0 1��

U Sty� � U S

t0y� � , t t0 �

Sty 2� U S

ty� � l� t t0 S

ty R1�

t t0 B

X R0

U St y� � U y� � � t lB � t�� y B�

Sty 2�

lB U x� � : x B� �sup�

y B� t* lB

l�� $� Sty

2 Sty R1� t t*

81 St y� � U St y� �� l� tlB

l��

Sty R1� t � 1� l

Bl��

St B x : x R1� �. tlB

l��

R1X S

t��

E x e r c i s e 4.1

dtd��U S

ty� � ( U S

ty� �� C�

( C

E x e r c i s e 4.2 Rx� f x� �� 0�

f x� � x f x� � x29 C� 9 0� C U x� � x2�

E x e r c i s e 4.3 R f n�� f

R R f�� 2 0� 0 � 1� �

f x� � � x� x 2�


1

C

h

a

p

t

e

r

Consider a dynamical system generated (see Exam-ple 1.2) by the Duffing equation

,

where and are real numbers and . Using the propertiesof the function

show that the dynamical system is dissipative for small enough. Find an upper estimate for the minimal radius of dissi-pativity.

Prove the dissipativity of the dynamical system generatedby (1.4) (see Example 1.3), provided

, .

Show that the dynamical system of Example 1.4 is dissipativeif is a bounded function.

Consider a cylinder with coordinates , , and the mapping of this cylinder which is defined

by the formula , where

,

.

Here and are positive parameters. Prove that the discrete dyna-mical system is dissipative, provided . We notethat if , then the mapping is known as the Chirikov map-ping. It appears in some problems of physics of elementary parti-cles.

Using Theorem 4.1 prove that the dynamical system generated by equations (3.1) (see Example 3.1) is dissipative.(Hint: ).

In the proof of the existence of global attractors of infinite-dimensional dissipativedynamical systems a great role is played by the property of asymptotic compactness.For the sake of simplicity let us assume that is a closed subset of a Banach space.The dynamical system is said to be asymptotically compact asymptotically compact asymptotically compact asymptotically compact if for any

its evolutionary operator can be expressed by the form

, (4.5)

where the mappings and possess the properties:

E x e r c i s e 4.4 R2 St

��

x�� x� x3 a x�� b�a b � 0�

U x x�� 12�� x� 2

14�� x4 a

2�� x2 : x x�

�2�� x2�; <

= >��

R2 St

�� : 0�

E x e r c i s e 4.5

xk fk x1 x2 xN� � �� k 1�

N

' xk

2C�

k 1�

N

'9�� 9 0�

E x e r c i s e 4.6

f z� �

E x e r c i s e 4.7 Ц x 8�� x R�8 0 1� �� T

T x 8�� x ? 8 ?��

x ? �x k 2@8sin��

8? 8 x ? 1mod� ��

� k

Ц Tn�� 0 � 1� �� 1� T

E x e r c i s e 4.8 R2 St��

U x� � H p q�� 2�

X

X St

�� t 0� S

t

St St

1� �St

2� ��

St1� �

St2� �


a) for any bounded set in

, ;

b) for any bounded set in there exists such that the set

(4.6)

is compact in , where is the closure of the set .A dynamical system is said to be compact compact compact compact if it is asymptotically compact and

one can take in representation (4.5). It becomes clear that any finite-di-mensional dissipative system is compact.

Show that condition (4.6) is fulfilled if there exists a compactset in such that for any bounded set the inclusion ,

holds. In particular, a dissipative system is compact if itpossesses a compact absorbing set.

Lemma 4.1.

The dynamical system is asymptotically compact if there exists

a compact set such that

(4.7)

for any set bounded in .

Proof.

The distance to a compact set is reached on some element. Hence, for any and there exists an element such that

.

Therefore, if we take , it is easy to see that in this case de-composition (4.5) satisfies all the requirements of the definition of asymptoticcompactness.

Remark 4.1.

In most applications Lemma 4.1 plays a major role in the proof of the

property of asymptotic compactness. Moreover, in cases when the phase

space of the dynamical system does not possess the structure

of a linear space it is convenient to define the notion of the asymptotic

compactness using equation (4.7). Namely, the system is said

to be asymptotically compact if there exists a compact possessing

property (4.7) for any bounded set in . For one more approach

to the definition of this concept see Exercise 5.1 below.

B X

rB

t� � St1� �

yX

y B�sup 0&� t +"&

B X t0

(t0

2� �B� �� S

t

2� �

t t0 - B�

X (� � (

St1� � 0�

E x e r c i s e 4.9

K H B St2� �

B K.t t0 B� �

X St

�� K

Stu K�� dist : u B� �sup

t "&lim 0�

B X

t 0� u X� v St2� �

u� K�

St u K�� dist St u St2� �

u��

St1� �

u Stu S

t2� �

u��

X X St��

X St�� K

B X


1

C

h

a

p

t

e

r

Consider the infinite-dimensional dynamical system genera-ted by the retarded equation

,

where and is bounded (see Example 1.4). Show thatthis system is compact.

Consider the system of Lorentz equations arising as a three-mode Galerkin approximation in the problem of convection in a thinlayer of liquid:

Here , , and are positive numbers. Prove the dissipativity ofthe dynamical system generated by these equations in .Hint: Consider the function

on the trajectories of the system.

§ 5 Theorems on Existence§ 5 Theorems on Existence§ 5 Theorems on Existence§ 5 Theorems on Existence

of Global Aof Global Aof Global Aof Global Atttttractortractortractortractor

For the sake of simplicity it is assumed in this section that the phase space isa Banach space, although the main results are valid for a wider class of spaces(see, e. g., Exercise 5.8). The following assertion is the main result.

Theorem 5.1.

Assume that a dynamical system is dissipative and asymptoti-Assume that a dynamical system is dissipative and asymptoti-Assume that a dynamical system is dissipative and asymptoti-Assume that a dynamical system is dissipative and asymptoti-

cally compact. Let be a bounded absorbing set of the system cally compact. Let be a bounded absorbing set of the system cally compact. Let be a bounded absorbing set of the system cally compact. Let be a bounded absorbing set of the system .... Then Then Then Then

the set is a nonempty compact set and is a global attractor of thethe set is a nonempty compact set and is a global attractor of thethe set is a nonempty compact set and is a global attractor of thethe set is a nonempty compact set and is a global attractor of the

dynamical system The attractor is a connected set in dynamical system The attractor is a connected set in dynamical system The attractor is a connected set in dynamical system The attractor is a connected set in

In particular, this theorem is applicable to the dynamical systems from Exercises4.2–4.11. It should also be noted that Theorem 5.1 along with Lemma 4.1 gives thefollowing criterion: a dissipative dynamical system possesses a compact global at-tractor if and only if it is asymptotically compact.

The proof of the theorem is based on the following lemma.

E x e r c i s e 4.10

x� t� � �x t� �� f x t 1��

� 0� f z� �


x� Ax� Ay ,��

y� r x y� x z ,��

z� b z� x y .��)�)�

A r b

R3

V x y z� �� 12�� x2 y2 z r� A�� 2� ��

X

X St

�� B X S

t��

A % B� ��X S

t�� .... A X .

T h e o r e m s o n E x i s t e n c e o f G l o b a l A t t r a c t o r 29

Lemma 5.1.

Let a dynamical system be asymptotically compact. Then for

any bounded set of the -limit set is a nonempty compact

invariant set.

Proof.

Let . Then for any sequence tending to infinity the set is relatively compact, i.e. there exist a sequence and an ele-

ment such that tends to as . Hence, the asymptoticcompactness gives us that

as .

Thus, . Due to Lemma 2.1 this indicates that is non-empty.

Let us prove the invariance of -limit set. Let . Then accordingto Lemma 2.1 there exist sequences , and such that

. However, the mapping is continuous. Therefore,

, .

Lemma 2.1 implies that . Thus,

, .

Let us prove the reverse inclusion. Let . Then there exist sequences and such that . Let us consider the se-

quence , . The asymptotic compactness implies that thereexist a subsequence and an element such that

.

As stated above, this gives us that

.

Therefore, . Moreover,

.

Hence, . Thus, the invariance of the set is proved.Let us prove the compactness of the set . Assume that is a se-

quence in . Then Lemma 2.1 implies that for any we can find and such that . As said above, the property of asymp-

totic compactness enables us to find an element and a sequence suchthat

.

X St

�� B X % % B� �

yn

B� tn

� Stn

2� �yn ,

n 1 2 � �� nk

y X� Stnk

2� �ynk

y k "&

y Stn

k

yn

k� S

tn

k

1� �y

nk

y Stn

k

2� �y

nk

�� 0& k "&

y Stn

k

yn

kk "&lim� % B� �

% y % B� ��tn

�, tn

"& zn

� B.S

tn

zn

y& St

St t

n� z

nS

t �Stn

zn

Sty&� n "&

St y % B� ��

St% B� � % B� �. t 0�

y % B� ��v

n � B. t

n: t

n"& � Stn

vn y&y

nS

tn t� vn

� tn

t tn

kz X�

z Stnk

t�2� � yn

kk "&lim�

z Stnk

t� yn

kk "&lim�

z % B� ��

St z Stk "&lim �Stnk

t� vnk

Stnk

vnkk "&

lim y� � �

y St% B� �� % B� �

% B� � zn

�% B� � n t

nn

yn

B� zn Stnyn� 1 n$�

z nk

�

Stnky

nkz� 0 , k "&&


1

C

h

a

p

t

e

r

This implies that and . This means that is a closed andcompact set in . Lemma 5.1 is proved completely.

Now we establish Theorem 5.1. Let be a bounded absorbing set of the dynamicalsystem. Let us prove that is a global attractor. It is sufficient to verify that

uniformly attracts the absorbing set . Assume the contrary. Then the value does not tend to zero as . This means that

there exist and a sequence such that

.

Therefore, there exists an element such that

. (5.1)

As before, a convergent subsequence can be extracted from the sequence. Therewith Lemma 2.1 implies

which contradicts estimate (5.1). Thus, is a global attractor. Its compactnessfollows from the easily verifiable relation

.

Let us prove the connectedness of the attractor by reductio ad absurdum. Assumethat the attractor is not a connected set. Then there exists a pair of open sets and such that

, , , .

Let be a convex hull of the set , i.e.

.

It is clear that is a bounded connected set and . The continuity of themapping implies that the set is also connected. Therewith .Therefore, , . Hence, for any the pair , cannotcover . It follows that there exists a sequence of points such that . The asymptotic compactness of the dynamical systemenables us to extract a subsequence such that tends in to anelement as . It is clear that and . These equationscontradict one another since . Therefore, Theorem

5.1 is proved completely.

z % B� �� znkz& % B� �

H

B

% B� �% B� � B

Sty % B� �� :dist y B� �sup t "&9 0� t

n: t

n"& �

Stny % B� �� :dist y B�

� �� !

sup 29

yn

B�

Stny

n% B� �� dist 9 n� 1 2 � ��

Stnk

ynk

�Stn

yn

�

z Stnk

yn

kk "&lim� % B� ��

% B� �

A % B� �� St

2� �B

t � /

� 0�/�

A U1U2

Ui

A/ B* i 1 2�� A U1 U2-. U1 U2/ B�

Ac conv A� �� A

Ac Ci vi : vi

i 1�

N

' A , Ci 0 , Ci

i 1�

N

' � 1 N� 1 2 � �� !

�

Ac Ac A6S

tS

tAc A S

tA� S

tAc.

Ui

StAc/ B* i 1 2�� t 0� U1 U2

StAc x

nS

ny

nS

nAc��

xn

U1 U2-4n

k � xnk

Snkynk

� X

y k "& y U1 U2-4 y % Ac� ��% Ac� � % B� �. A U1 U2-.�


It should be noted that the connectedness of the global attractor can also be provedwithout using the linear structure of the phase space (do it yourself).

Show that the assumption of asymptotic compactness in Theo-rem 5.1 can be replaced by the Ladyzhenskaya assumption: the se-quence contains a convergent subsequence for anybounded sequence and for any increasing sequence

such that . Moreover, the Ladyzhenskaya as-sumption is equivalent to the condition of asymptotic compactness.

Assume that a dynamical system possesses a compactglobal attractor . Let be a minimal closed set with the property

for every .

Then and , i.e. coincides with theglobal minimal attractor (cf. Exercise 3.4).

Assume that equation (4.7) holds. Prove that the global at-tractor possesses the property .

Assume that a dissipative dynamical system possesses a glo-bal attractor . Show that for any bounded absorbing set

of the system.

The fact that the global attractor has the form , where is an absorb-ing set of the system, enables us to state that the set not only tends to the at-tractor , but is also uniformly distributed over it as . Namely, the followingassertion holds.

Theorem 5.2.

Assume that a dissipative dynamical system possesses a com-Assume that a dissipative dynamical system possesses a com-Assume that a dissipative dynamical system possesses a com-Assume that a dissipative dynamical system possesses a com-

pact global attractor pact global attractor pact global attractor pact global attractor .... Let Let Let Let be a bounded absorbing set for be a bounded absorbing set for be a bounded absorbing set for be a bounded absorbing set for .... Then Then Then Then

.... (5.2)

Proof.

Assume that equation (5.2) does not hold. Then there exist sequences and such that

for some . (5.3)

The compactness of enables us to suppose that converges to an element. Therewith (see Exercise 5.4)

,

E x e r c i s e 5.1

Stnu

n �

un

� X.tn

� T+. tn

+"&


�� A A*

Sty A*�� dist

t "&lim 0� y X�

A* A. A* % x� � : x X� �-� A*

E x e r c i s e 5.3

A A % K� � K.�

E x e r c i s e 5.4

A A % B� ��B

A A % B� �� B

StB

A t "&

X St�� A B X St��

a St B�� :dist a A� �supt "&lim 0�

an

� .A. t

n: t

n"& �

an StnB�� 9 dist 9 0�

A an �a A�

a S�m

ym

m "&lim y

m � B.��


1

C

h

a

p

t

e

r

where is a sequence such that . Let us choose a subsequence such that for every . Here is chosen such that

for all . Let . Then it is clear that and

.

Equation (5.3) implies that

.

This contradicts the previous equation. Theorem 5.2 is proved.

For a description of convergence of the trajectories to the global attractor it is con-venient to use the Hausdorff metric Hausdorff metric Hausdorff metric Hausdorff metric that is defined on subsets of the phase spaceby the formula

, (5.4)

where and

. (5.5)

Theorems 5.1 and 5.2 give us the following assertion.

Corollary 5.1.

Let be an asymptotically compact dissipative system. Then its

global attractor possesses the property for any

bounded absorbing set of the system .

In particular, this corollary means that for any there exists such thatfor every the set gets into the -vicinity of the global attractor ;and vice versa, the attractor lies in the -vicinity of the set . Here isa bounded absorbing set.

The following theorem shows that in some cases we can get rid of the require-ment of asymptotic compactness if we use the notion of the global weak attractor.

Theorem 5.3.

Let the phase space of a dynamical system be a separableLet the phase space of a dynamical system be a separableLet the phase space of a dynamical system be a separableLet the phase space of a dynamical system be a separable

Hilbert space. Assume that the system is dissipative and its evolu-Hilbert space. Assume that the system is dissipative and its evolu-Hilbert space. Assume that the system is dissipative and its evolu-Hilbert space. Assume that the system is dissipative and its evolu-

tionary operator tionary operator tionary operator tionary operator is weakly closed, i.e. for all is weakly closed, i.e. for all is weakly closed, i.e. for all is weakly closed, i.e. for all the weak convergence the weak convergence the weak convergence the weak convergence

and imply that and imply that and imply that and imply that .... Then the dynamical system Then the dynamical system Then the dynamical system Then the dynamical system

possesses a global weak attractor possesses a global weak attractor possesses a global weak attractor possesses a global weak attractor....

The proof of this theorem basically repeats the reasonings used in the proof of Theo-rem 5.1. The weak compactness of bounded sets in a separable Hilbert space playsthe main role instead of the asymptotic compactness.

�m � �m "& mn

��mn

tn

tB

� n 1 2 � �� tB

StB .

B. t tB

zn

S�mntn

� ym

n� z

n � B.

a S�mn

ymnn "&

lim Stnzn

n "&lim� �

an

Stnz

n�� dist a

nStn

B�� dist 9

2 C D�� h C D�� h D C�� max�

C D� X�

h C D�� c D�� :dist c C� �sup�

X St

�� A 2 S

tB A��

t "&lim 0�

B X St

��

� 0� t� 0�t t�� S

tB � A

A � StB B

H H St

�� H S

t��

St

t 0�y

ny& S

ty

nz& z S

ty�

H St

��


Lemma 5.2.

Assume that the hypotheses of Theorem 5.3 hold. For we define

the weak -limit set by the formula

, (5.6)

where is the weak closure of the set . Then for any bounded set

the set is a nonempty weakly closed bounded invariant

set.

Proof.

The dissipativity implies that each of the sets isbounded and therefore weakly compact. Then the Cantor theorem on the col-lection of nested compact sets gives us that is a non-empty weakly closed bounded set. Let us prove its invariance. Let .Then there exists a sequence such that weakly. Thedissipativity property implies that the set is bounded when is largeenough. Therefore, there exist a subsequence and an element suchthat and weakly. The weak closedness of implies that

. Since for , we have that for all .Hence, . Therefore, . The proof of the reverseinclusion is left to the reader as an exercise.

For the proof of Theorem 5.3 it is sufficient to show that the set

, (5.7)

where is a bounded absorbing set of the system , is a global weak attractorfor the system. To do that it is sufficient to verify that the set is uniformly attract-ed to in the weak topology of the space . Assume the contrary. Thenthere exist a weak vicinity of the set and sequences and

such that . However, the set is weakly compact. There-fore, there exist an element and a sequence such that

.

However, for . Thus, for all and , which is impossible. Theorem 5.3 is proved.

Assume that the hypotheses of Theorem 5.3 hold. Show thatthe global weak attractor is a connected set in the weak topologyof the phase space .

Show that the global weak minimal attractor is a strictly invariant set.

B H.% %

wB� �

%w B� � St B� �t s -

ws 0 /�

Y� �w Y

B H. %w B� �

(ws

B� � St

B� �t s -� �

w�

%w

B� � (w

sB� �

s 0 /�y %

wB� ��

yn

St

B� �t n -� y

ny&

Sty

n � t

ynk

� z

ynky& S

ty

nkz& S

t

z Sty� S

tynk

(ws

B� �� nk

s z (ws

B� �� s

z %w

B� �� St%

wB� � %

wB� �.

Aw

%w

B� ��

B H St

�� B

Aw

%w

B� �� H

� Aw

yn

� B. tn

: tn&

"& � Stny

n�4 Stn

yn

�z �4 n

k �

z w Stnk

ynkk "&

lim��

Stnkyn

k(

ws

B� �� tnks z (

w

sB� �� s 0 z �

%w

B� ��

E x e r c i s e 5.5

Aw

H

E x e r c i s e 5.6 Aw* %

wx� � :-�

x H� �


1

C

h

a

p

t

e

r

Prove the existence and describe the structure of global andglobal minimal attractors for the dynamical system generated bythe equations

for every real .

Assume that is a metric space and is an asymptoti-cally compact (in the sense of the definition given in Remark 4.1)dynamical system. Assume also that the attracting compact iscontained in some bounded connected set. Prove the validity of theassertions of Theorem 5.1 in this case.

In conclusion to this section, we give one more assertion on the existence of the globalattractor in the form of exercises. This assertion uses the notion of the asymptoticsmoothness (see [3] and [9]). The dynamical system is said to be asympto- asympto- asympto- asympto-

tically smooth tically smooth tically smooth tically smooth if for any bounded positively invariant set there exists a compact such that as , where the

value is defined by formula (5.5).

Prove that every asymptotically compact system is asymptoti-cally smooth.

Let be an asymptotically smooth dynamical system.Assume that for any bounded set the set

is bounded. Show that the system posses-ses a global attractor of the form

.

In addition to the assumptions of Exercise 5.10 assume that is pointwise dissipative, i.e. there exists a bounded set such that as for every point

. Prove that the global attractor is compact.

§ 6 On the Structure of Global Attractor§ 6 On the Structure of Global Attractor§ 6 On the Structure of Global Attractor§ 6 On the Structure of Global Attractor

The study of the structure of global attractor of a dynamical system is an importantproblem from the point of view of applications. There are no universal approaches tothis problem. Even in finite-dimensional cases the attractor can be of complicatedstructure. However, some sets that undoubtedly belong to the attractor can be poin-

E x e r c i s e 5.7

x� 3x y� x x2 y2�� ,��

y� x 3y y x2 y2��

3

E x e r c i s e 5.8 X X St

��

K

X St�� St B B. , t 0 � �

B X. K h St B K�� 0& t "&h . .��

E x e r c i s e 5.9

E x e r c i s e 5.10 X St�� B X. (+ B� � �

St B� �t 0 -� X St�� A

A w B� � : B X. , B is bounded �-�


X St

�� B0 X. dist X S

ty B0�� 0& t "&

y X� A

O n t h e S t r u c t u r e o f G l o b a l A t t r a c t o r 35

ted out. It should be first noted that every stationary point of the semigroup be-longs to the attractor of the system. We also have the following assertion.

Lemma 6.1.

Assume that an element lies in the global attractor of a dynamical

system . Then the point belongs to some trajectory that lies

in wholly.

Proof.

Since and , then there exists a sequence suchthat , , . Therewith for discrete time the re-quired trajectory is , where for and

for . For continuous time let us consider the value

Then it is clear that for all and for ,. Therewith . Thus, the required trajectory is also built in the

continuous case.

Show that an element belongs to a global attractor if andonly if there exists a bounded trajectory such that .

Unstable sets also belong to the global attractor. Let be a subset of the phasespace of the dynamical system . Then the unstable set emanatingunstable set emanatingunstable set emanatingunstable set emanating

from from from from is defined as the set of points for every of which there existsa trajectory such that

.

Prove that is invariant, i.e. for all.

Lemma 6.2.

Let be a set of stationary points of the dynamical system

possessing a global attractor . Then .

Proof.

It is obvious that the set lies in the attractorof the system and thus it is bounded. Let . Then there exists a tra-jectory such that and

.

St

z A

X St

�� z (A

St A A� z A� zn � A.z0 z� S1zn zn 1�� n 1 2 � ��

( un : n Z� �� un Sn z� n 0 un �z n�� n 0�

u t� �S

tz t 0 , �

St n� zn n t n� 1 n�� 1 2 � ��

�

u t� � A� t R� S� u t� � u t �� 0 t R� u 0� � z�

E x e r c i s e 6.1 z

( u t� � : "� t "� � ��u 0� � z�

Y

X X St

�� Y M+ Y� � z X�

( u t� � : t T� ��

u 0� � z u t� � Y�� distt "�&

lim� 0� �

E x e r c i s e 6.2 M+ Y� � StM+ Y� � M+ Y� ��

t 0�

� X St

�� A M+ �� A.

� z : St z z� t 0�� z M+ ��

(z u t� � t T�� u 0� � z�

u �� dist 0 � "�&�&


1

C

h

a

p

t

e

r

Therefore, the set is bounded when is largeenough. Hence, the set tends to the attractor of the system as .However, for . Therefore,

.

This implies that . The lemma is proved.

Assume that the set of stationary points is finite. Showthat

,

where are the stationary points of (the set is calledan unstable manifold emanating from the stationary point ).

Thus, the global attractor includes the unstable set . It turns out that un-der certain conditions the attractor includes nothing else. We give the following defi-nition. Let be a positively invariant set of a semigroup , . Thecontinuous functional defined on is called the Lyapunov functionLyapunov functionLyapunov functionLyapunov function of thedynamical system on if the following conditions hold:

a) for any the function is a nonincreasing function with re-spect to ;

b) if for some and the equation holds, then for all , i.e. is a stationary point of the semigroup .

Theorem 6.1.

Let a dynamical system possess a compact attractor Let a dynamical system possess a compact attractor Let a dynamical system possess a compact attractor Let a dynamical system possess a compact attractor .... Assume Assume Assume Assume

also that the Lyapunov function exists on also that the Lyapunov function exists on also that the Lyapunov function exists on also that the Lyapunov function exists on .... Then , where Then , where Then , where Then , where

is the set of stationary points of the dynamical system. is the set of stationary points of the dynamical system. is the set of stationary points of the dynamical system. is the set of stationary points of the dynamical system.

Proof.

Let . Let us consider a trajectory passing through (its existence fol-lows from Lemma 6.1). Let

and .

Since , the closure is a compact set in . This implies that the -limitset

of the trajectory is nonempty. It is easy to verify that the set is invariant:. Let us show that the Lyapunov function is constant on .

Indeed, if , then there exists a sequence tending to such that

Bs

u �� : � s�� s 0�S

tB

st +"&

z StB

s� t s

z A�� dist Sty A�� :dist y B

s� � 0&sup t +"&��

z A�

E x e r c i s e 6.3 �

M+ �� M+ zk

� �k 1�

l

-�

zk

St

M+ zk

� �z

k

A M+ ��

Y St : StY Y. t 0�D y� � Y

X St

�� Y

y Y� D Sty� �

t 0 t0 0� y X� D y� � D St0

y� ��y S

ty� t 0 y S

t

X St�� A

D y� � A A M+ ��

y A� ( y

( u t� � : t T� �� (�– u t� � : t ��

(�– A. (�

–� � X �

� (� � (�–� �

� 0�/�

( � (� �St � (� � � (� �� D y� � � (� �

u � (� �� tn � "�


.

Consequently,

.

By virtue of monotonicity of the function along the trajectory we have

.

Therefore, the function is constant on . Hence, the invariance of the set gives us that , for all . This means that

lies in the set of stationary points. Therewith (verify it yourself)

.

Hence, . Theorem 6.1 is proved.

Assume that the hypotheses of Theorem 6.1 hold. Then forany element its -limit set consists of stationary pointsof the system.

Thus, the global attractor coincides with the set of all full trajectories connecting thestationary points.

Assume that the system possesses a compact globalattractor and there exists a Lyapunov function on . Assume thatthe Lyapunov function is bounded below. Show that any semitrajec-tory of the system tends to the set of stationary points of the sys-tem as , i.e. the global minimal attractor coincides with theset .

In particular, this exercise confirms the fact realized by many investigators that theglobal attractor is a too wide object for description of actually observed limit regimesof a dynamical system.

Assume that is a dynamical system generated by thelogistic equation (see Example 1.1): .Show that is a Lyapunov function for this sys-tem.

Show that the total energy

is a Lyapunov function for the dynamical system generated (seeExample 1.2) by the Duffing equation

, .

u tn� �tn

"�&lim u�

D u� � D u tn

� �� n "&

lim�

D u� �

D u� � D u �� : � 0� �sup�

D u� � � (� �� (� � D S

tu� � D u� �� t 0� u � (� �� (� �

�

u t� � � (� �� distt "�&

lim 0�

y M+ ��

E x e r c i s e 6.4

y A� % % y� �

E x e r c i s e 6.5 X St�� X

�

t +"&�

E x e r c i s e 6.6 R St

�� x� �x x 1�� 0 � 0��

V x� � x3 3$ x2 2$��

E x e r c i s e 6.7

E x x�� 12��x� 2

14�� x4 a

2��x2 b x��

x�� x� x3 a x�� b� � 0�


1

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h

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If in the definition of a Lyapunov functional we omit the second requirement, thena minor modification of the proof of Theorem 6.1 enables us to get the following as-sertion.

Theorem 6.2.

Assume that a dynamical system possesses a compact global at-Assume that a dynamical system possesses a compact global at-Assume that a dynamical system possesses a compact global at-Assume that a dynamical system possesses a compact global at-

tractor tractor tractor tractor and there exists a continuous function on and there exists a continuous function on and there exists a continuous function on and there exists a continuous function on such that such that such that such that

does not increase with respect to for any does not increase with respect to for any does not increase with respect to for any does not increase with respect to for any .... Let Let Let Let be a set ofbe a set ofbe a set ofbe a set of

elements such that for all elements such that for all elements such that for all elements such that for all .... Here Here Here Here isisisisa trajectory of the system passing through a trajectory of the system passing through a trajectory of the system passing through a trajectory of the system passing through ( ).... Then Then Then Then

and contains the global minimal attractor and contains the global minimal attractor and contains the global minimal attractor and contains the global minimal attractor ....

Proof.

In fact, the property was established in the proof of Theorem 6.1.As to the property , it follows from the constancy of the function onthe -limit set of any element .

Apply Theorem 6.2 to justify the results of Example 3.1 (seealso Exercise 4.8).

If the set of stationary points of a dynamical system is finite, then Theo-rem 6.1 can be extended a little. This extension is described below in Exercises 6.9–6.12. In these exercises it is assumed that the dynamical system is continu-ous and possesses the following properties:

(a) there exists a compact global attractor ;(b) there exists a Lyapunov function on ;(c) the set of stationary points is finite, therewith

for and the indexing of possesses the property

. (6.1)

We denote

, , .

Show that for all .

Assume that . Then

. (6.2)

Assume that the function is defined on the whole . Then(6.2) holds for any bounded set ,where is a positive number.

X St�� A E y� � X

E St y� � t y X� �

u A� E u t� �� E u� �� " t "� �� u t� � �u u 0� � u� M+ �� A�

� A* % x� �x X�-�

M+ �� A�A* �. E u� �

% % x� � x X�

E x e r c i s e 6.8

� X St

��

X St

��

A

D x� � A

� z1 zN

� �� D zi

� � *D z

j� �* i j* z

j

D z1� � D z2� � D zN

� ��

Aj

M+ zk

� �k 1�

j

-� j 1 2 N� � �� A0 B�


Aj

Aj

� j 1 2 N� ��

E x e r c i s e 6.10 B Aj\ zj �.

Sty A

j 1�� :dist y B� �supt "&lim 0�

E x e r c i s e 6.11 D X

B x : D x� � D zj� � 9�� .9


Assume that is the closure of the set and is its boundary. Show that

and

, .

It can also be shown (see the book by A. V. Babin and M. I. Vishik [1]) that undersome additional conditions on the evolutionary operator the unstable manifolds

are surfaces of the class , therewith the facts given in Exercises 6.9–6.12remain true if strict inequalities are substituted by nonstrict ones in (6.1). It shouldbe noted that a global attractor possessing the properties mentioned above is fre-quently called regularregularregularregular.

Let us give without proof one more result on the attractor of a system with a fi-nite number of stationary points and a Lyapunov function. This result is importantfor applications.

At first let us remind several definitions. Let be an operator acting in a Ba-nach space . The operator is called Frechét differentiable at a pointFrechét differentiable at a pointFrechét differentiable at a pointFrechét differentiable at a point

provided that there exists a linear bounded operator suchthat

for all from some vicinity of the point x, where as . Therewith,the operator is said to belong to the class , on a set if it isdifferentiable at every pointdifferentiable at every pointdifferentiable at every pointdifferentiable at every point and

for all from some vicinity of the point . A stationary point of the mapping is called hyperbolichyperbolichyperbolichyperbolic if in some vicinity of the point , the spectrum

of the linear operator does not cross the unit circle and the spec-tral subspace of the operator corresponding to the set is finite-dimen-sional.

Theorem 6.3.

Let Let Let Let be a Banach space and let a continuous dynamical systembe a Banach space and let a continuous dynamical systembe a Banach space and let a continuous dynamical systembe a Banach space and let a continuous dynamical system

possess the properties: possess the properties: possess the properties: possess the properties:

1) there exists a global attractor ;there exists a global attractor ;there exists a global attractor ;there exists a global attractor ;

2) there exists a vicinity of the attractor such thatthere exists a vicinity of the attractor such thatthere exists a vicinity of the attractor such thatthere exists a vicinity of the attractor such that

for all , provided and belong to for all ;for all , provided and belong to for all ;for all , provided and belong to for all ;for all , provided and belong to for all ;

3) there exists a Lyapunov function continuous on ;there exists a Lyapunov function continuous on ;there exists a Lyapunov function continuous on ;there exists a Lyapunov function continuous on ;

4) the set of stationary points is finite and all thethe set of stationary points is finite and all thethe set of stationary points is finite and all thethe set of stationary points is finite and all the

points are hyperbolic;points are hyperbolic;points are hyperbolic;points are hyperbolic;

E x e r c i s e 6.12 M+ zj

� �� M+ zj

� �M+ z

j� �5 M+ z

j� �� \M+ z

j� �� M+z

j5 .

Aj 1�.

St M+ zj� �� M+ zj� �� St M+ zj� �5 M+ zj� �5�

St

M+ zj

� � C1

S

X S

x X� S ? x� � : X X&

S y� � S x� �� S ? x� � y x�� ( x y�� x y��

y ( �� 0& � 0&S C1 �� 0 � 1� �� Y

x Y�

S ? x� � S ? y� �� L X X�� C x y� ��

y x Y� z

S S C1 �� z

S ? z� � C : C 1� �C : C 1� �

X

X St�� A

F A

Stx S

ty� C e� t �� S� x S� y��

t � 0 Stx S

ty F t 0

X

� z1 zN

� � ��


1

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h

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p

t

e

r

5) the mapping is continuous.the mapping is continuous.the mapping is continuous.the mapping is continuous.

Then for any compact set in the estimateThen for any compact set in the estimateThen for any compact set in the estimateThen for any compact set in the estimate

(6.3)

holds for all , where does not depend on holds for all , where does not depend on holds for all , where does not depend on holds for all , where does not depend on ....

The proof of this theorem as well as other interesting results on the asymptotic be-haviour of a dynamical system possessing a Lyapunov function can be found in thebook by A. V. Babin and M. I. Vishik [1].

To conclude this section, we consider a finite-dimensional example that showshow the Lyapunov function method can be used to prove the existence of periodictrajectories in the attractor.

E x a m p l e 6.1 (on the theme by E. Hopf)

Studying Galerkin approximations in a model suggested by E. Hopf for the de-scription of possible mechanisms of turbulence appearence, we obtain the fol-lowing system of ordinary differential equations

Here is a positive parameter, and are real parameters. It is clear that theCauchy problem for (6.4)–(6.6) is solvable, at least locally for any initial condi-tion. Let us show that the dynamical system generated by equations (6.4)–(6.6)is dissipative. It will also be sufficient for the proof of global solvability. Let usintroduce a new unknown function . Then equations (6.4)–(6.6) can be rewritten in the form

These equations imply that

on any interval of existence of solutions. Hence,

.

t u�� Stu&

B X

Sty A�� :dist y B�

� �� !

sup CB

e G t��

t 0 G 0� B

(6.4)

(6.5)

(6.6)

u� 3u v2 w2� � � 0 ,�

v� : v v u� Hw�� 0 ,�

w� :w w u� Hv� � 0 .��)�)�

3 : H

u* u 3 2$ :��

u� * 3u* v2 w2� � � 332�� :�; <= > ,�

v�12�� 3 v v u*� Hw�� 0 ,�

w�12�� 3w w u*� Hv� � 0 .�

�)))�)))�

12��

dtd

�� u* 2v

2w

2� �� 3 u* 2 32�� v

2w

2�� 332�� :�; <= > u*�

dtd

�� u* 2v

2w

2� �� ; <= > 3 u* 2

v2

w2� �� 3

32�� :�; <= >2

�


Thus,

Firstly, this equation enables us to prove the global solvability of problem (6.4)–(6.6) for any initial condition and, secondly, it means that the set

is absorbing for the dynamical system generated by the Cauchy prob-lem for equations (6.4)–(6.6). Thus, Theorem 5.1 guarantees the existence ofa global attractor . It is a connected compact set in .

Verify that is a positively invariant set for .

In order to describe the structure of the global attractor we introduce the polarcoordinates

,

on the plane of the variables . As a result, equations (6.4)–(6.6) are trans-formed into the system

therewith, . System (6.7) and (6.8) has a stationary point for all and . If , then one more stationary point

occurs in system (6.7) and (6.8). It corresponds to a periodic trajectoryof the original problem (6.4)–(6.6).

Show that the point is a stable node of system (6.7)and (6.8) when and it is a saddle when .

Show that the stationary point is stable( ) being a node if and a focus if .

If , then (6.7) and (6.8) imply that

.

Therefore,

.

u* t� � 2v t� � 2

w t� � 2

u* 0� � 2v 0� � 2

w 0� � 2� �; <= > e 3 t� 3

2�� :�; <= >2

1 e 3� t�� .

�

�

� �

B0 u v w� �� : u32�� :��; <

= >2v2 w2� � 1

32�� :�; <= >2

�� !

�

R3 St

��

A R3

E x e r c i s e 6.13 B0 R3 St

��

A

v t� � r t� � 8 t� �cos� w t� � r t� � 8 t� �sin�

v w� �

(6.7)

(6.8)

u� 3u r2� � 0 ,�

r� : r u r�� 0 ,��

8 t� � H t� 80�� u 0��r 0� � 3 0� : R� : 0� u :��r 3:��

E x e r c i s e 6.14 0 0�� : 0� : 0�

E x e r c i s e 6.15 u : r� 3 :�� : 0� 3 8$� : 0� � : 3

8��

: 0�

12��

dtd

�� u2 r2�� 3 :�� u2 r2�� min� 0�

u t� � 2r t� � 2� u 0� � 2

r 0� � 2e 2 min 3 :�� t��


1

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h

a

p

t

e

r

Hence, for the global attractor of the system consists of the singlestationary exponentially attracting point

.

Prove that for the global attractor of problem (6.4)–(6.6) consists of the single stationary point .Show that it is not exponentially attracting.

Now we consider the case . Let us again refer to problem (6.7) and (6.8). It isclear that the line is a stable manifold of the stationary point .Moreover, it is obvious that if , then the value remains positive for all

. Therefore, the function

(6.9)

is defined on all the trajectories, the initial point of which does not lie on the line. Simple calculations show that

(6.10)

and

; (6.11)

therewith, . Equation (6.10) implies thatthe function does not increase along the trajectories. Therefore, any semi-trajectory emanating from the point ofthe system generated by equations (6.7) and (6.8) possesses theproperty for . Therewith, equation (6.9) impliesthat this semitrajectory can not approach the line at a distance less then

. Hence, this semitrajectory tends to . Moreover, for any the set

is uniformly attracted to , i.e. for any there exists such that

.

Indeed, if it is not true, then there exist , a sequence , and such that . The monotonicity of and property (6.11) imply that

for all . Let be a limit point of the sequence . Then after passingto the limit we find out that

: 0� A R3 St

��

u 0 v� 0 w� 0� � � �

E x e r c i s e 6.16 : 0�u 0 , � v 0 , w 0� � �

: 0�r 0� u 0 r� 0� � �

r t0� � 0� r t� �t t0�

V u r�� 12�� u :�� 2 1

2�� r2 3: rln� ��

r 0� �

dtd�� V u t� � r t� �� 3 u t� � :�� 2� 0�

V u r�� V : 3:�� 12�� u :� 2

r 3:�� 2�; <= >�

V : 3:�� 1 2$� � 3 : e 3 :� �$� �ln�V u r�� u t� � r t� �� t R�� u0 r0� r0 0*� �R R�# S

t��

V u t� � r t� �� V u0 r0�� t 0 r 0� �

1 3:� �$� � V u0 r0�� exp y u :��r 3:�� R�

B� y u r�� : V u r��

y � 0� t0 t0 � ��

St B� y : y y� �� .

�0 0� tn

+"& zn

B��Stn

zn

y� �0� V y� �

V Stz

n� � V S

tn

zn

� � V : 3:�� 12�� 0

2�

0 t tn

� � z zn

�


with . Thus, the last inequality is impossible since . Hence

. (6.12)

The qualitative behaviour of solutions to problem (6.7) and (6.8) on the semiplaneis shown on Fig. 2.

In particular, the observations above mean that the global minimal attractor of the dynamical system generated by equations (6.4)–(6.6) consists

of the saddle point and the stable limit cycle

(6.13)

for . Therewith, equation (6.12) implies that the cycle uniformly attractsall bounded sets in possessing the property

, (6.14)

i.e. which lie at a positive distance from the line .

Using the structure of equations (6.7) and (6.8) near the sta-tionary point , prove that a bounded set pos-sessing property (6.14) is uniformly and exponentially attracted tothe cycle , i.e.

for , where is a positive constant.

V St z� � V : 3:�� 12�� 0

2� t 0 �

z r 0� �4 St z y& u :� ��r 3:��

Sty y�� :dist y B�� sup

t "&lim 0�

Fig. 2. Qualitative behaviour of solutions to problem (6.7), (6.8): a) , b) 3 8$� : 0� � : 3 8$��

Amin R3 St�� u 0 v� 0 w� 0� � � �

C: u : v2 w2�� 3:��

: 0� C:B R3

d v2 w2� : u v w� �� B� �inf 0��v 0 w� 0� � �


u : r 3:�� B

C:

Sty C:�� dist y B�� sup C e

( t tB

��

t tB

(


1

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h

a

p

t

e

r

Now let lie in the global attractor of the system .Assume that and . Then (see Lemma 6.1) there existsa trajectory lying in such that .The analysis given above shows that as . Let us show that

when . Indeed, the function is monotonely nonde-creasing as . If we argue by contradiction and use the fact that isbounded we can easily find out that

and therefore

as . (6.16)

Equation (6.7) gives us that

. (6.17)

Since is bounded for all , we can get the equation

by tending in (6.17). Therefore, by virtue of (6.16) we find that as . Thus, as . Hence, for the global attractor

y0 u0 v0 w0� �� A R3 St

�� r0 0* r0

2 v02 w0

2�� 3:�*( y t� � u t� � v t� � w t� �� t R�� A y 0� � y0�

y t� � C:& t +"&y t� � 0& t "�& V u t� � r t� ��

t "�& y t� �

Fig. 3. Attractor of the system (6.4)–(6.6);a) , b) 3 8$� : 0� � : 3 8$��

V u t� � r t� �� t "�&

lim "�

r t� � v t� � 2w t� � 2�; <

= >1 2� 0& t "�&

u t� � e 3 t �� u s� � e 3 t �� r �� 2 �d

s

t

��

u s� � s R�

u t� � e 3 t �� r �� 2 �d

"�

t

��

s –"& u t� � 0&t –"& y t� � 0& t –"& : 0� A

S t a b i l i t y P r o p e r t i e s o f A t t r a c t o r a n d R e d u c t i o n P r i n c i p l e 45

of the system coincides with the union of the unstable manifold emanating from the point and the limit cycle (6.13). The at-tractor is shown on Fig. 3.

§ 7 Stability Properties of Attractor§ 7 Stability Properties of Attractor§ 7 Stability Properties of Attractor§ 7 Stability Properties of Attractor

and Reduction Principleand Reduction Principleand Reduction Principleand Reduction Principle

A positively invariant set in the phase space of a dynamical system is saidto be stable (in Lyapunov’s sense)stable (in Lyapunov’s sense)stable (in Lyapunov’s sense)stable (in Lyapunov’s sense) in if its every vicinity contains somevicinity such that for all . Therewith, is said to be asymp-asymp-asymp-asymp-

totically stable totically stable totically stable totically stable if it is stable and as for every . A setis called uniformly asymptotically stableuniformly asymptotically stableuniformly asymptotically stableuniformly asymptotically stable if it is stable and

(7.1)

The following simple assertion takes place.

Theorem 7.1.

Let Let Let Let be the compact global attractor of a continuous dynamical sys-be the compact global attractor of a continuous dynamical sys-be the compact global attractor of a continuous dynamical sys-be the compact global attractor of a continuous dynamical sys-

tem tem tem tem .... Assume that there exists its bounded vicinity such that the Assume that there exists its bounded vicinity such that the Assume that there exists its bounded vicinity such that the Assume that there exists its bounded vicinity such that the

mapping is continuous on mapping is continuous on mapping is continuous on mapping is continuous on .... Then Then Then Then is a stable set.is a stable set.is a stable set.is a stable set.

Proof.

Assume that is a vicinity of . Then there exists such that for . Let us show that there exists a vicinity of the attractor such that

for all . Assume the contrary. Then there exist sequences and such that , and . The set beingcompact, we can choose a subsequence such that as

. Therefore, the continuity property of the function gives us that . This contradicts the equation . Thus,there exists such that for . We can choose such that

for all . Therefore, the attractor is stable. Theorem 7.1

is proved.

It is clear that the stability of the global attractor implies its uniform asymptoticstability.

Assume that is a positively invariant set of a system. Prove that if there exists an element such that its

-limit set possesses the property , then is not stable.

R3 St

�� M+ 0� �u 0 v� 0 w� 0� � � �

M X St�� X �

� ? St � ?� � �. t 0 M

St y M& t "& y � ?�M

St y M�� dist : y � ?� �supt "&lim 0 .�

A

X St�� U

t u�� St u& R+ U# A

� A T 0� StU �.

t T � ? A

St� ? �. t 0 T�� u

n �

tn

� un

A�� dist 0& tn

� 0 T�� Stnu

n�4 A

nk

� unku& A� tnk

&t& 0 T�� t u�� S

tu&

Stnkun

kS

tu& A� Stn

un

�4� ? S

t� ? �. t 0 T�� T

St� ? U/� � �. t 0 A

E x e r c i s e 7.1 M

X St�� y M4� � y� � � y� � M/ B* M


1

C

h

a

p

t

e

r

In particular, the result of this exercise shows that the global minimal attractor canappear to be an unstable set.

Let us return to Example 3.1 (see also Exercises 4.8 and 6.8).Show that:

(a) the global attractor and the Milnor attractor are stable;

(b) the global minimal attractor and the Ilyashenko at-tractor are unstable.

Now let us consider the question concerning the stability of the attractor with re-spect to perturbations of a dynamical system. Assume that we have a family of dy-namical systems with the same phase space and with an evolutionaryoperator depending on a parameter which varies in a complete metric space

. The following assertion was proved by L. V. Kapitansky and I. N. Kostin [6].

Theorem 7.2.

Assume that a dynamical system possesses a compact global at-Assume that a dynamical system possesses a compact global at-Assume that a dynamical system possesses a compact global at-Assume that a dynamical system possesses a compact global at-

tractor for every . Assume that the following conditions hold:tractor for every . Assume that the following conditions hold:tractor for every . Assume that the following conditions hold:tractor for every . Assume that the following conditions hold:

(a) there exists a compact such that for all ;there exists a compact such that for all ;there exists a compact such that for all ;there exists a compact such that for all ;

(b) if if if if ,,,, and and and and ,,,, then for some then for some then for some then for some

....

Then the family of aThen the family of aThen the family of aThen the family of attttttttractors is upper semicontinuous at the point ,ractors is upper semicontinuous at the point ,ractors is upper semicontinuous at the point ,ractors is upper semicontinuous at the point ,

i.e.i.e.i.e.i.e.

(7.2)

as as as as ....

Proof.

Assume that equation (7.2) does not hold. Then there exist a sequence and a sequence such that for some . But

the sequence lies in the compact . Therefore, without loss of generality we canassume that for some and . Let us show that this re-sult leads to contradiction. Let be a trajectory of the dy-namical system passing through the element ( ). Using thestandard diagonal process it is easy to find that there exist a subsequence and a sequence of elements such that

for all ,

where . Here is a fixed number. Sequential application of condition(b) gives us that

E x e r c i s e 7.2

A1 A4

A3A5

X StC�� X

StC C

I

X StC��

AC C I�K X. AC K. C I�

Ck C0& xk AC

k� xk x0& St0

Ck xk St0

x0&t0 0�

AC C0

h AC

k AC0�� y A

C0�� :dist y AC

k� �sup� 0&

Ck

C0&

Ck &C0& xk A

Ck� xk A

C0�� dist 9 9 0�xk K

xk x0& K� x0 K� x0 AC04

(k uk t� � : –" t "� � ��X St

Ck�� xk uk 0� � xk�

k n� � �um � K.

uk n� � m t0��

n "&lim u

m� m 0 1 2 � � ��

u0 x0� t0 0�


for all and . It follows that the function

gives a full trajectory passing through the point . It is obvious that the trajectory is bounded. Therefore (see Exercise 6.1), it wholly belongs to , but that con-

tradicts the equation . Theorem 7.2 is proved.

Following L. V. Kapitansky and I. N. Kostin [6], for define the upper limit of the attractors along bythe equality

,

where denotes the closure operation. Prove that if the hypothe-ses of Theorem 7.2 hold, then is a nonempty compact in-variant set lying in the attractor .

Theorem 7.2 embraces only the upper semicontinuity of the family of attractors. In order to prove their continuity (in the Hausdorff metric defined by equation

(5.4)), additional conditions should be imposed on the family of dynamical systems. For example, the following assertion proved by A. V. Babin and M. I. Vishik

concerning the power estimate of the deviation of the attractors and in theHausdorff metric holds.

Theorem 7.3.

Assume that a dynamical system possesses a global attractorAssume that a dynamical system possesses a global attractorAssume that a dynamical system possesses a global attractorAssume that a dynamical system possesses a global attractor

for every for every for every for every .... Let the following conditions hold: Let the following conditions hold: Let the following conditions hold: Let the following conditions hold:

(a) there exists a bounded set such that for all there exists a bounded set such that for all there exists a bounded set such that for all there exists a bounded set such that for all

andandandand

,,,, (7.3)

with constants and independent of and withwith constants and independent of and withwith constants and independent of and withwith constants and independent of and with

;;;;

(b) for any and the estimate for any and the estimate for any and the estimate for any and the estimate

(7.4)

um l� u

k n� �n "&

lim m l�� t0�� Sl t0

Ck n� �

uk n� �

n "&lim m t0�� S

l t0

C0u

m� � �

m 1 2 � �� l 1 2 m� � ��

u t� �S

t

C0u0 t 0 , �

St t0 m�C0 u

m t0m� t� t0 m 1�� m�� 1 2 � ��

�)�)�

�

( x0( A

C0

x0 AC04

E x e r c i s e 7.3 C C0&A C0 ; I� � AC I

A C0 I�� AC : C I 0 C C0�� dist 9� �� -9 0�/�

.� �A C0 I��

AC0

AC �

X StC��

AC AC0

X StC��

AC C I�B0 X. AC B0. C I�

h StC

B0 AC�� C0 e G t� � C� I�

C0 0� G 0� C

h B A�� b A�� :dist b B� �sup�C1 C2� I� u1 u2� B0�

St

C1 u1 St

C2 u2�� dist C1e� t u1 u2�� dist C1 C2�� dist��


1

C

h

a

p

t

e

r

holds, with constants and independent of holds, with constants and independent of holds, with constants and independent of holds, with constants and independent of ....

Then there exists such thatThen there exists such thatThen there exists such thatThen there exists such that

.... (7.5)

Here Here Here Here is the Hausdorff metric defined by the formulais the Hausdorff metric defined by the formulais the Hausdorff metric defined by the formulais the Hausdorff metric defined by the formula

....

Proof.

By virtue of the symmetry of (7.5) it is sufficient to find out that

. (7.6)

Equation (7.3) implies that for any

for all (7.7)

when . Here is an -vicinity of the set. It follows from equation (7.4) that

(7.8)

Since , we have . Therefore, with , equa-tion (7.7) gives us that

. (7.9)

For any the estimate

holds. Hence, we can find that

for all and . Consequently, equation (7.9) implies that

for . It means that

Thus, equation (7.8) gives us that for any

C1 � C

C2 0�

2 AC1 A

C2�� C2 C1 C2�� dist� �qq,,,,�

GG ��

2 . .��

2 B A�� h B A�� h A B�� ;;;; �max�

h AC1 A

C2�� C2 C1 C2�� dist� �q

�

� 0�

StC

B0 � � AC� �. C I�

t t* � C0�� G� 1� 1 �$ln C0ln�� AC� � �AC

h St

C1B0 St

C2B0�� St

C1x St

C2y�� dist

y B0�inf

x B0�sup

St

C1x S

t

C2x�� dist

x B0�sup C1 e� t C1 C2�� .dist

�

� �

�

AC B0. AC StC

AC StC

B0.� t t* � C0��

AC StC

B0 � � AC� �. .

x z� X�

x AC�� dist x z�� dist z AC�� dist��

x AC�� dist x z�� dist ��

x X� z �� AC� ��

x AC�� dist x StC

B0�� dist �� x� X�

t t* � C0��

h AC1 A

C2�� x AC2�� dist

x AC1�

sup

x St

C2B0�� dist

x AC1�

sup � h St

C1B0 St

C2B0�� .

�

�

�

� 0�


for . By taking , and in this formula we find estimate (7.6). Theorem 7.3 is proved.

It should be noted that condition (7.3) in Theorem 7.3 is quite strong. It can be veri-fied only for a definite class of systems possessing the Lyapunov function (see Theo-rem 6.3).

In the theory of dynamical systems an important role is also played by the no-tion of the Poisson stability. A trajectory of a dynamicalsystem is said to be Poisson stable Poisson stable Poisson stable Poisson stable if it belongs to its -limit set .It is clear that stationary points and periodic trajectories of the system are Poissonstable.

Show that any Poisson stable trajectory is contained in theglobal minimal attractor if the latter exists.

A trajectory is Poisson stable if and only if any point of this trajectory is recurrent, i.e. for any vicinity there exists

such that .

The following exercise testifies to the fact that not only periodic (and stationary) tra-jectories can be Poisson stable.

Let be a Banach space of continuous functions boun-ded on the real axis. Let us consider a dynamical system with the evolutionary operator defined by the formula

.

Show that the element is recurrent forany real and (in particular, when is an irrationalnumber). Therewith the trajectory is Poisson stable.

In conclusion to this section we consider a theorem that is traditionally associatedwith the stability theory. Sometimes this theorem enables us to significantly decreasethe dimension of the phase space, this fact being very important for the study of infi-nite-dimensional systems.

Theorem 7.4. (reduction principle).

Assume that in a dissipative dynamical system there existsAssume that in a dissipative dynamical system there existsAssume that in a dissipative dynamical system there existsAssume that in a dissipative dynamical system there exists

a positively invariant locally compact set a positively invariant locally compact set a positively invariant locally compact set a positively invariant locally compact set possessing the property of possessing the property of possessing the property of possessing the property of

uniform attraction, i.e. for any bounded set the equationuniform attraction, i.e. for any bounded set the equationuniform attraction, i.e. for any bounded set the equationuniform attraction, i.e. for any bounded set the equation

h AC1 A

C2�� C1e� t C1 C2�� dist ��

t t* � C0�� C1 C2�� dist� �q

� qG

G �� t t* � C0�� G� 1� 1 �$ln C0ln��

( u t� � : "� t "� � ��X S

t�� % % (� �

E x e r c i s e 7.4

E x e r c i s e 7.5 ( x

� xJt 0� S

tx ��

E x e r c i s e 7.6 Cb R� �Cb R� � St��

St

f� � x� � f x t�� f x� � CbR� ��

f0 x� � %1sin x %2sin x��%1 %2 %1 %2$

( f0 x t�� : " t "� ��

X St

�� M

B X.


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(7.10)

holds. Let holds. Let holds. Let holds. Let be a global attractor of the dynamical system . Then be a global attractor of the dynamical system . Then be a global attractor of the dynamical system . Then be a global attractor of the dynamical system . Then

is also a global attractor of is also a global attractor of is also a global attractor of is also a global attractor of ....

Proof.

It is sufficient to verify that

(7.11)

for any bounded set . Assume that there exists a set such that (7.11) doesnot hold. Then there exist sequences and such that

(7.12)

for some . Let be a bounded absorbing set of . We choose a moment such that

. (7.13)

This choice is possible because is a global attractor of . Equation (7.10)implies that

.

The dissipativity property of gives us that when is largeenough. Therefore, local compactness of the set guarantees the existence of anelement and a subsequence such that

.

This implies that . Therefore, equation (7.12) gives us that. By virtue of the fact that this contradicts equation

(7.13). Theorem 7.4 is proved.

E x a m p l e 7.1

We consider a system of ordinary differential equations

(7.14)

It is obvious that for any initial condition problem (7.14) is uniquelysolvable over some interval . If we multiply the first equation by

and the second equation by and if we sum the results obtained, then we getthat

St y M�� disty B�sup

t "&lim 0�

A M St

�� A

X St

��

St y A�� disty B�sup

t "&lim 0�

B X. B

yn � B. tn : tn "& �

Stnyn A�� dist 9

9 0� B0 X St�� t0

St0

y A�� :dist y M B0/�� !

sup 92��

A M St

��

Stn t0� yn M�� dist 0& tn "&�

X St

�� Stn t0� y

nB0� n

M

z M B0/� nk

�

z Stnk

t0� yn

kk "&lim�

Stnky

nk

St0z&

St0z A�� dist 9 z M B0/�

y� y3 y�� y z2� yt 0�� y0� �

z� z 1 y2�� 0 zt 0�� z0 . � �

�)�)�

y0 z0�� 0 t* y0 z0��

y z


, .

This implies that the function possesses the property

, .

Therefore,

, .

This implies that any solution to problem (7.14) can be extended to the wholesemiaxis and the dynamical system generated by equation (7.14)is dissipative. Obviously, the set is positively invariant.Therewith the second equation in (7.14) implies that

, .

Hence, . Thus, the set exponentially attracts all the bound-ed sets in . Consequently, Theorem 7.4 gives us that the global attractor ofthe dynamical system is also the attractor of the system . Buton the set system of equations (7.14) is reduced to the differential equation

. (7.15)

Thus, the global attractors of the dynamical systems generated by equations(7.14) and (7.15) coincide. Therewith the study of dynamics on the plane is re-duced to the investigation of the properties of the one-dimensional dynamicalsystem.

Show that the global attractor of the dynamical system generated by equations (7.14) has the form

.

Figure the qualitative behaviour of the trajectories on the plane.

Consider the system of ordinary differential equations

(7.16)

Show that these equations generate a dissipative dynamical systemin . Verify that the set , is invariantand exponentially attracting. Using Theorem 7.4, prove that the glo-bal attractor of problem (7.16) has the form

.

Hint: Consider the variable instead of the variable .

12��

dtd

�� y2 z2�� y4 y2 z2�� 0� t 0 t* y0 z0��

V y z�� y2 z2��

dtd

�� V y t� � z t� �� 2 V y t� � z t� �� 2� t 0� t* y0 z0��

V y t� � z t� �� V y0 z0�� e 2 t� 1�� t 0� t* y0 z0��

R+ R2 St

�� M y 0�� : y R� ��

12��

dtd

�� z2 z2� 0� t 0�

z t� � 2z0

2e 2 t�� M

R2

M St

�� R2 St

�� M

y� y3 y�� 0� yt 0�� y0�


R2 St��

A y z�� : 1 y 1 z�� 0� ��

E x e r c i s e 7.8

y� y5 y3 1 2 z�� y 1 z2�� 0�

z� z 1 4 y4�� 2 y2 z2 y4 y2 3 2$�� 0 .��)�)�

R2 M y z�� : z y2�� y R� �

A

A y z�� : z y2� 1 y 1� ��

w z y2�� z


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§ 8 Finite Dimensionality§ 8 Finite Dimensionality§ 8 Finite Dimensionality§ 8 Finite Dimensionality

of Invariant Setsof Invariant Setsof Invariant Setsof Invariant Sets

Finite dimensionality is an important property of the global attractor which can beestablished in many situations interesting for applications. There are several ap-proaches to the proof of this property. The simplest of them seems to be the onebased on Ladyzhenskaya’s theorem on the finite dimensionality of the invariant set.However, it should be kept in mind that the estimates of dimension based on La-dyzhenskaya’s theorem usually turn out to be too overstated. Stronger estimates canbe obtained on the basis of the approaches developed in the books by A. V. Babinand M. I. Vishik, and by R. Temam (see the references at the end of the chapter).

Let be a compact set in a metric space . Then its fractal dimensionfractal dimensionfractal dimensionfractal dimension

is defined by

,

where is the minimal number of closed balls of the radius which coverthe set .

Let us illustrate this definition with the following examples.

E x a m p l e 8.1

Let be a segment of the length . It is evident that

.

Therefore,

.

Hence, , i.e. the fractal dimension coincides with the value of thestandard geometric dimension.

E x a m p l e 8.2

Let be the Cantor set obtained from the segment by the sequentualremoval of the centre thirds. First we remove all the points between and

. Then the centre thirds and of the two remainingsegments and are deleted. After that the centre parts

, , and of the fourremaining segments , , and , respec-tively, are deleted. If we continue this process to infinity, we obtain the Cantorset . Let us calculate its fractal dimension. First of all it should be noted that

M X

Mdimf

n M �� ln1 �$� �ln

�� 0&lim�

n M �� M

M l

l

2�� 1� n M �� l

2 �� 1��

1��ln l 2 ��

2��ln� n M �� ln 1

��ln l 2 ��2

��ln��

Mdimf 1�

M 0 1�� 1 3$

2 3$ 1 9$ 2 9$�� 7 9$ 8 9$�� 0 1 3$�� 2 3$ 1��

1 27$ 2 27$�� 7 27$ 8 27$�� 19 27$ 20 27$�� 25 27$ 26 27$�� 0 1 9$�� 2 9$ 1 3$�� 2 3$ 7 9$�� 8 9$ 1��

M

F i n i t e D i m e n s i o n a l i t y o f I n v a r i a n t S e t s 53

,

,

and so on. Each set can be considered as a union of segments of thelength . In particular, the cardinality of the covering of the set with thesegment of the length equals to . Therefore,

.

Thus, the fractal dimension of the Cantor set is not an integer (if a set possessesthis property, it is called fractal).

It should be noted that the fractal dimension is often referred to as the metric orderof a compact. This notion was first introduced by L. S. Pontryagin and L. G. Shnirel-man in 1932. It can be shown that any compact set with the finite fractal dimensionis homeomorphic to a subset of the space when is large enough.

To obtain the estimates of the fractal dimension the following simple assertionis useful.

Lemma 8.1.

The following equality holds:

,

where is the cardinality of the minimal covering of the com-

pact with closed sets diameter of which does not exceed (the dia-

meter of a set is defined by the value ).

Proof.

It is evident that . Since any set of the diameter liesin a ball of the radius , we have that . These two inequa-lities provide us with the assertion of the lemma.

All the sets are expected to be compact in Exercises 8.1–8.4 given below.

Prove that if , then .

Verify that .

Assume that is a direct product of two sets. Then

.

M Jk

k 0�

"

/�

J0 0 1�� , J1 0 1 3$�� 2 3$ 1�� -�

J2 0 1 9$�� 2 9$ 1 3$�� 2 3$ 7 9$�� 8 9$ 1�� ---�

Jk

2k

3 k� M

3 k� 2k

Mdimf 2kln2 3� k� �ln

��k "&lim 2ln

3ln��

Rd d 0�

Mdimf N M �� ln

1 �$� �ln��

� 0&lim�

N M �� M 2 �

X d X� � x y� : x y� X� �sup�

N M �� n M �� d

d n M 2 �� N M ��

E x e r c i s e 8.1 M1 M2+ M1dimf M2dimf�

E x e r c i s e 8.2 M1 M2-� �dimf M1dimf M2dimf� �max�

E x e r c i s e 8.3 M1 M2#

M1 M2#� �dimf M1dimf M2dimf��


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Let be a Lipschitzian mapping of one metric space intoanother. Then .

The notion of the dimension by Hausdorff is frequently used in the theory of dynami-cal systems along with the fractal dimension. This notion can be defined as follows.Let be a compact set in . For positive and we introduce the value

,

where the infimum is taken over all the coverings of the set with the balls of theradius . It is evident that is a monotone function with respectto . Therefore, there exists

.

The Hausdorff dimensionHausdorff dimensionHausdorff dimensionHausdorff dimension of the set is defined by the value

.

Show that the Hausdorff dimension does not exceed the frac-tal one.

Show that the fractal dimension coincides with the Hausdorffone in Example 8.1, the same is true for Example 8.2.

Assume that , where monotonicallytends to zero. Prove that (Hint:

when ).

Let . Show that .

Hint: when

.

Let . Prove that .

Find the fractal and Hausdorff dimensions of the global mini-mal attractor of the dynamical system in generated by the diffe-rential equation

.

The facts presented in Exercises 8.7–8.9 show that the notions of the fractal andHausdorff dimensions do not coincide. The result of Exercise 8.5 enables us to re-strict ourselves to the estimates of the fractal dimension when proving the finite di-mensionality of a set.

E x e r c i s e 8.4 g

g M� �dimf Mdimf�

M X d �

3 M d �� rj

� �d'inf�

M

rj

�� 3 M d ��

3 M d�� 3 M d �� 0&lim 3 M d ��

� 0�sup� �

M

MdimH d : 3 M d�� 0� �inf�

E x e r c i s e 8.5

E x e r c i s e 8.6

E x e r c i s e 8.7 M an

�n 1�" R.� a

n

MdimH 0� 3 M d �� a

n 1�d

n 2 d n�� an 1� � a

n� �

E x e r c i s e 8.8 M 1 n$ �n 1�" R.� Mdimf 1 2$�

;= n n M �� n 1 1

n 1�� 1

n 1�� n 2��

1n n 1�� <

>

E x e r c i s e 8.9 M1nln

�� !

n 2�

"R.� Mdimf 1�


R

y� y1y��sin� 0�


The main assertion of this section is the following variant of Ladyzhenskaya’stheorem. It will be used below in the proof of the finite dimensionality of global at-tractors of a number of infinite-dimensional systems generated by partial differentialequations.

Theorem 8.1.

Assume that Assume that Assume that Assume that is a compact set in a Hilbert space is a compact set in a Hilbert space is a compact set in a Hilbert space is a compact set in a Hilbert space .... Let Let Let Let bebebebe a contin- a contin- a contin- a contin-

uous mapping in such that uous mapping in such that uous mapping in such that uous mapping in such that .... Assume that there exists a finite- Assume that there exists a finite- Assume that there exists a finite- Assume that there exists a finite-

dimensional projector in the space dimensional projector in the space dimensional projector in the space dimensional projector in the space such thatsuch thatsuch thatsuch that

,,,, ,,,, (8.1)

,,,, ,,,, (8.2)

where where where where .... We also assume that We also assume that We also assume that We also assume that .... Then the compact Then the compact Then the compact Then the compact possesses possesses possesses possesses

a finite fractal dimension anda finite fractal dimension anda finite fractal dimension anda finite fractal dimension and

.... (8.3)

We remind that a projector in a space is defined as a bounded operator with theproperty . A projector is said to be finite-dimensional if the image isa finite-dimensional subspace. The dimension of a projector is defined as a num-ber .

The following lemmata are used in the proof of Theorem 8.1.

Lemma 8.2.

Let be a ball of the radius in . Then

. (8.4)

Proof.

Estimate (8.4) is self-evident when . Assume that . Let be a maximal set in with the property , .

By virtue of its maximality for every there exists such that. Hence, . It is clear that

, , .

Here is a ball of the radius centred at . Therefore,

.

This implies the assertion of the lemma.

Show that

, .

M H V

H V M� � M6P H

P Vv1 V v2�� l v1 v2�� v1 v2� M�

1 P�� Vv1 Vv2�� 9 v1 v2�� v1 v2� M�

9 1� l 1 9� M

Mdimf Pdim 9 l

1 9��ln 2

1 9��ln

1��

H P

P2 P� P PH

P

Pdim PHdim�

BR R Rd

N BR

�� n BR

�� 1 2 R��; <

= >d

� �

� R � R��1 �

l� � � B

R�

i�

j� �� i j*

x BR

� �i

x �i

� �� n BR

�� l�

B� 2$ �i

� � BR �� 2$. B� 2$ �

i� � B� 2$ �

j� �� / B� i j*

Br�� r �

l B� 2$� �Vol B� 2$ �i� �� Voli 1�

l

' BR � 2$�� Vol��


n BR

�� R

��; <= >d

BR

dimf d�


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Lemma 8.3.

Let be a closed subset in such that equations (8.1) and (8.2) hold

for all its elements. Then for any and the following esti-

mate holds:

, (8.5)

where is the dimension of the projector .

Proof.

Let be a minimal covering of the set with its closed subsets the di-ameter of which does not exceed . Equation (8.1) implies that in thereexist balls with radius such that . By virtue of Lemma 8.1there exists a covering of the set with the balls of the diameter

, where . Therefore, the collection

is a covering of the set . Here the sum of two sets and is defined by theequality

.

It is evident that

.

Equation (8.2) implies that . Therefore, . Hence, estimate (8.5) is valid. Lemma 8.3 is proved.

Let us return to the proof of Theorem 8.1. Since , Lemma 8.3 gives usthat

.

It follows that

.

We choose and such that

,

where . Then

.

Consequently,

.

� H

q 0� � 0�

N V � � q 9�� 1 4 lq��; <

= >n

N � ��

n Pdim� P

�i

� �

2 � PH

Bi

2 l � P V�i

Bi

.B

i j �

j 1�N

i P V�i

2 q � Ni

1 4 l q$� �� n�

Gi j

Bi j

1 P�� V �i

: i� 1 2 N � �� j� � � � 1 2 Ni

� � �� !

V � A B

A B� a b : a A� b B��

Gi j

diam Bi j

diam 1 P�� V�i

diam��

1 P�� V�i

diam 29 �� Gi j

diam �2 q 9��

M VM.

N M � q 9�� N M �� 1 4 lq��; <

= >n

��

N M q 9�� m�� N M 1�� 1 4 lq��; <

= >n m

� m�� 1 2 � ��

q m m ��

9 q� 1 �� 9 q�� m ��0 � 1� �

N M �� N M 9 q�� m�� N M 1�� 1 4 lq��; <

= >n m ��

Mdimf N M �� ln

1 �$� �ln��

� 0&lim n 1 4 l

q��; <

= >ln m ��

1 �$� �ln��

� 0&lim� ��


Obviously, the choice of can be made to fulfil the condition

.

Thus,

.

By taking we obtain estimate (8.3). Theorem 8.1 is proved.

Assume that the hypotheses of Theorem 8.1 hold and . Prove that .

Of course, in the proof of Theorem 8.1 a principal role is played by equations (8.1)and (8.2). Roughly speaking, they mean that the mapping squeezes sets along thespace while it does not stretch them too much along . Negative invari-ance of gives us that for all . Therefore, the set shouldbe initially squeezed. This property is expressed by the assertion of its finite dimen-sionality. As to positively invariant sets, their finite dimensionality is not guaranteedby conditions (8.1) and (8.2). However, as the next theorem states, they are attract-ed to finite-dimensional compacts at an exponential velocity.

Theorem 8.2.

Let be a continuous mapping defined on a compact set Let be a continuous mapping defined on a compact set Let be a continuous mapping defined on a compact set Let be a continuous mapping defined on a compact set in a Hi in a Hi in a Hi in a Hil-l-l-l-

bert space such that bert space such that bert space such that bert space such that .... Assume that there exists a finite-dimensi- Assume that there exists a finite-dimensi- Assume that there exists a finite-dimensi- Assume that there exists a finite-dimensi-

onal projector such that equations onal projector such that equations onal projector such that equations onal projector such that equations (8.1) and and and and (8.2) hold with hold with hold with hold with

and and and and .... Then for any there exists a positively invariant Then for any there exists a positively invariant Then for any there exists a positively invariant Then for any there exists a positively invariant

closed set such thatclosed set such thatclosed set such thatclosed set such that

(8.6)

andandandand

,,,, (8.7)

where where where where is an arbitrary number from the intervalis an arbitrary number from the intervalis an arbitrary number from the intervalis an arbitrary number from the interval ....

Proof.

The pair is a discrete dynamical system. Since is compact, Theo-rem 5.1 gives us that there exists a global attractor with the pro-perties and

. (8.8)

m ��

m �� lnq 9�� ln

�� 1��

Mdimf n 1 4 lq��; <

= > 1q 9��ln

1�ln�

q 1 2$ 1 9��


l 1 9�� dimf M 0�

V

1 P�� H PH

M M VkM. k 1 2 � �� M

V M

H VM M.P 0 9 1 2$� �

l 9� 1 K 9 1�� AK M.

Vky AK�� :dist y M� �sup Kk k,,,,� 1 2 � ��

AKdimf Pdim1 4 l

K 9��; <

= >ln

1K��ln

�� 1 4 l

q��; <

= >ln

12 q 9�� ln

��

� �) )� ) )� !

max��

q 0 1 2$ 9��

M Vk�� M

M0 VkMk 0 /�

V M0 M0�

h VkM M0�� Vky M0�� :dist y M� �sup 0&�


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We construct a set as an extension of . Let be a maximal set in pos-sessing the property for , . The existence of sucha set follows from the compactness of . It is obvious that

.

Lemma 8.3 with , and gives us that

with . Hereinafter . Therefore,

, . (8.9)

Let us prove that the set

(8.10)

possesses the properties required. It is evident that . Since , by virtue of (8.8) all the limit points of the set

lie in . Thus, is a closed subset in . The evident inequality

(8.11)

implies (8.6). Here and below . Let us prove(8.7). It is clear that

. (8.12)

Let be a minimal covering of the set with the closed sets the diameterof which is not greater than . By virtue of Lemma 8.3 there exists a covering of the set with closed subsets of the diameter . The cardinality of thiscovering can be estimated as follows

. (8.13)

Using the covering we can construct a covering of the same cardinality of theset with the balls of the radius centered at thepoints We increase the radius of every ball up to thevalue . The parameter will be chosen below. Thus, we considerthe covering

of the set . It is evident that every point belongs to this covering to-gether with the ball . If , the inequalities

AK M0 Ej

V jM

a b�� dist K j a b Ej

�� a b*V j M

Lj Card Ej� N Ej13��K j�; <

= > N V jM13��K j�; <

= >��

� M q� K 9�� 1 3$� � K j 1��

N V jM13��K j�; <

= > 1 4 l

K 9��; <

= >nN V j 1� M

13��K j 1��; <

= >�

K 9� n Pdim�

Lj Card Ej 1 4 l

K 9��; <

= >n j

N M13��; <

= >�� K 9�

AK M0 Vk Ej : j 1 2 � � � k 0 1 2 � � �� - �-�

VAK AK. VkEj .Vk j� M.

VkEj: j 1 2 � � � k 0 1 2 � � �� -

M0 AK M

h VkM AK�� h VkM Ek

�� Kk� �

h X Y�� x Y�� :dist x X� �sup�

AK V AK Ej : j 1 2 � �� - �-�

Fi

� AK2 � G

i �

VAK 2 � q 9��

N � q 9� �� N VAK � q 9�� 1 4 lq��; <

= >n

N AK ��

Gi �,VAK B xi 2 � q 9�� 2 � q 9��

xi i� 1 2 N � q 9� �� .� � ��2 � q 9 (� �� ( 0�

B xi 2 � q 9 (� �� i� 1 2 N � q 9� ��

VAK x VAK�B x 2 ( �� j 2


hold. By virtue of equation (8.11) with the help of (8.1) and (8.2) we have that

.

Therefore, , provided , i.e. if

.

Here is an integer part of the number . Consequently,

.

Therefore, equation (8.12) gives us that

,

where . Using (8.9) and (8.13) we find that

for . Here and further . Since

,

it is easy to find that

for ,

where the constant does not depend on (its value is unessential further).Therefore,

.

If we take , then after iterations we get

h Ej

VAK�� h V j M VAK�� h V jM VEj 1��

h V j M VEj 1�� l 9�� h V j 1� M Ej 1�� l 9�� K j 1��

h Ej

VAK�� 2 ( �� 2 ( � l 9�� K j 1��

j j0 2

l 9�2 ( ��ln

1K��ln

��

z� � z

VAK Ej

j j0 -� �

� � !

B xi

2 � q 9 (� �� i 1�

N � q 9� ��

-.-

N AK � �� N � q 9� �� Card Ej

j 0�

j0 1�

'��

� 2 q 9 (� ��

N AK � �� G N AK �� N M13��; <

= > 1 4 l

K 9��; <

= >n j

j 1�

j0 1�

'��

K 9� G 1 4 lq��; <

= >n

�

j0

1��ln

1K��ln

�� C l 9 ( K� � ��

1 4 l

K 9��; <

= >n j

j 0�

j0 1�

' H 1��; <

= >�� n

1 K$� �ln�� 1 4 l

K 9��; <

= >ln��

H 0� �

N AK � �� GN AK � �� H � ��

� �m 1��

N AK �m�� GN AK �m 1�� H � m 1��

Gm N AK 1�� HGm 1� ��G� � i� .

i 0�

m 1�

'�

�

�

��


1

C

h

a

p

t

e

r

Let us fix , and and choose suchthat and . Then summarizing the geometric progres-sion we obtain

(8.14)

Let be small enough and

,

where, as mentioned above, is an integer part of the number . Since ,equation (8.14) gives us that

,

where and are positive numbers which do not depend on . Therefore,

Simple calculations give us that

.

This easily implies estimate (8.7). Thus, Theorem 8.2 is proved.

Show that for formula (8.7) for the dimensionof the set can be rewritten in the form

. (8.15)

If the hypotheses of Theorem 8.2 hold, then the discrete dynamical system possesses a finite-dimensional global attractor . This attractor uniformly attractsall the trajectories of the system. Unfortunately, the speed of its convergence to theattractor cannot be estimated in general. This speed can appear to be small. However,Theorem 8.2 implies that the global attractor is contained in a finite-dimensional po-

9 0 1 2$�� K 9 1�� q 0 1 2$ 9�� ( 0�� 2 q 9 (� �� 1�� G 1*

N AK �m�� Gm N AK 1�� H� �m� Gm�� G�

��

Gm N AK 1�� H� �� G��; <

= > H� �� G�� m� .�

� �

�

� 0�

m �� 11 �$� �ln1 �$� �ln

��

z� � z � �m ��

N AK �� N AK �m�� a1 1 4 lq��; <

= >n m �� a2

1��; <

= >m ��

a1 a2 �

AKdimf

Nln AK �� 1��ln

�� 0&lim

m �� 1��ln

�� 0&lim 1

m��

m "&lim a1 1 4 e

q��

n ma2

1

��; <

= >m�

� �� !

.

�

�

�

AKdimf11��ln

�� 1 4 eq��; <

= >n

� ��; <= >max

� �� !

ln��

E x e r c i s e 8.13 9 K 1 2$� �AK

AK = Pdim1 4 l

K 9��; <

= >ln

12K��ln

��dimf

M Vk�� M0

E x i s t e n c e a n d P r o p e r t i e s o f A t t r a c t o r s … 61

sitively invariant set possessing the property of uniform exponential attraction. Fromthe applied point of view the most interesting corollary of this fact is that the dyna-mics of a system becomes finite-dimensional finite-dimensional finite-dimensional finite-dimensional exponentially fast independent ofthe speed of convergence of the trajectories to the global attractor. Moreover, the re-duction principle (see Theorem 7.4) is applicable in this case. Thus, finite-dimen-sional invariant exponentially attracting sets can be used to describe the qualitativebehaviour of infinite-dimensional systems. These sets are frequently referred to asinertial setsinertial setsinertial setsinertial sets , or fractal exponential attractorsfractal exponential attractorsfractal exponential attractorsfractal exponential attractors . In some cases they turn outto be surfaces in the phase space. In contrast with the global attractor, the inertialset of a dynamical system can not be uniquely determined. The construction in theproof of Theorem 8.2 shows it.

§ 9 Existence and Properties of Attractors§ 9 Existence and Properties of Attractors§ 9 Existence and Properties of Attractors§ 9 Existence and Properties of Attractors

of a Class of Infinite-Dimensionalof a Class of Infinite-Dimensionalof a Class of Infinite-Dimensionalof a Class of Infinite-Dimensional

Dissipative SystemsDissipative SystemsDissipative SystemsDissipative Systems

The considerations given in the previous sections are mainly of general character.They are related to a dissipative dynamical system of the generic structure. There-with, we inevitably make additional assumptions on the behaviour of trajectories ofthese systems (e.g., the asymptotic compactness, the existence of a Lyapunov func-tion, the squeezing property along a subspace, etc.). Thereby it is natural to askwhat properties of the original objects of a particular dynamical system guaranteethe fulfilment of the assumptions mentioned above. In this section we discuss thisquestion in terms of the dynamical system generated by a differential equation ofthe form

(9.1)

in a separable Hilbert space , where is a linear operator and is a nonlinearmapping which is coordinated with in some sense. Our main goal is to demon-strate the generic line of arguments as well as to describe those properties of theoperators and which provide the applicability of general theorems proved inthe previous sections. The main attention is paid to the questions of existence and fi-nite dimensionality of a global attractor. Nowadays the presented line of arguments(or a modification of it) is one of the main components of a great number of workson global attractors.

It is assumed below that the following conditions are fulfilled.

dtd�� y A y� B y� � y

t 0�� y0 � �

� A B

A

A B


1

C

h

a

p

t

e

r

(A) There exists a strongly continuous semigroup of continuous map-pings in such that is a solution to problem (9.1) in thesense that the following identity holds:

, (9.2)

where (see condition (B) below). The semigroup isdissipative, i.e. there exists such that for any from the collec-tion of all bounded subsets of the space the estimate

holds when and . We also assume that the set is bounded for any .

(B) The linear closed operator generates a semigroup which admits the estimate ( and are someconstants). There exists a sequence of finite-dimensional projectors

which strongly converges to the identity operator such that1) commutes with , i.e. for any ;2) there exists such that for ,

where ;3) as .

(C) For any the nonlinear operator possesses the properties:1) if ;2) for , and for some the fol-

lowing equations hold:

(the existence of the operator follows from (B2)).

It should be noted that although conditions (A)–(C) seem a little too lengthy, they arevalid for a class of problems of the theory of nonlinear oscillations as well as fora number of systems generated by parabolic partial differential equations.

The following assertion should be mainly interpreted as a principal result whichtestifies to the fact that the asymptotic behaviour of the system is determined bya finite set of parameters.

Theorem 9.1.

If conditions If conditions If conditions If conditions (A)–(C) are fulfilled, then the semigroup possesses a are fulfilled, then the semigroup possesses a are fulfilled, then the semigroup possesses a are fulfilled, then the semigroup possesses a

compact global attractor compact global attractor compact global attractor compact global attractor .... The attractor has a finite fractal dimension The attractor has a finite fractal dimension The attractor has a finite fractal dimension The attractor has a finite fractal dimension

which can be estimated as follows:which can be estimated as follows:which can be estimated as follows:which can be estimated as follows:

,,,, (9.3)

St

� y t� � Sty0�

Sty0 T

ty0 G t y0�� T

ty0 T

t �� B S� y0� � �d

0

t

��

Tt

A t�� exp� St

R 0� B

� �� Sty �

R� y B� t t0 B� � (+ B� � S

tBt 0 -� B � ��

A Tt

A t�� exp�T

tL1 % t� �exp� L1 %

Pn

�A P

nPn

A A Pn

. n

n0 Tt

1 Pn

�� L2 � t�� exp� n n0�� , L 2 0�

rn

A 1� 1 Pn

�� 0&� n "&

R ? 0� B u� �B u1� � B u2� �� C1 R ?� � u1 u2�� u

iR ?,� i 1 2��

n n0 , ui

R ? i�� 1 2�� A 0�

AA 1 Pn

�� B u1� � C2 R ?� � , �

AA 1 Pn

�� B u1� � B u2� �� C3 R ?� � u1 u2��

AA 1 Pn

��

St

�

�dimf a1 1 Pn

L1 L2ln�� 1 D R� � � 1�� Pn

dim�


where and is determined from the conditionwhere and is determined from the conditionwhere and is determined from the conditionwhere and is determined from the condition

.... (9.4)

Here Here Here Here and are some absolute constants and are some absolute constants and are some absolute constants and are some absolute constants....

When proving the theorem, we mainly rely on decomposition (9.2) and the lemmatabelow.

Lemma 9.1.

Let be a set of elements which for some have the form

, where , with the constant de-

termined by the condition for . Here the value

is the same as in (9.2) with the element being such that

for all . Then the set is precompact in for

.

Proof.

Properties (B2) and (C2) imply that

when for . Therefore, the set

, (9.5)

where for all , is bounded in the space

with the norm . The symbol denotes the restriction of an operator on

a subspace. However, property (B3) implies that

.

Therefore, the operator is compact. Hence, is compactly embedded into . It means that the set (9.5) is precom-pact in . This implies the precompactness of .

Lemma 9.2.

There exists a compact set in the space such that

(9.6)

for any bounded set and .

Proof.

Let , where is a bounded set in . Then for . By virtue of (9.2) we have that

D R� � % L1C1 R� �� n n0

rnA

a2 D R� � L1 L2 C3 R� �� 1�a3 D R� � 1 L2ln�� 1�� exp��

a1 a2� � a3

Kn

t 0 v u0 ��1 P

n�� G t u�� u0 P

n�� u0 c0 R� c0

Pn

c0� n 1 2 � ��G t u�� u ��S

tu R� t 0 K

n�

n n0

AA 1 Pn

�� G t , u� � L2 e � t �� AA 1 Pn

�� B S� u� � �C2 R� � L2

�� d

0

t

��

S�u R� � 0

v : v 1 Pn�� G t , u� � , t 0��

Stu R� t 0 D AA 1 P

n��

AA .

Pm A 1� 1 Pn�� A 1� 1 Pn�� m "&

lim 0�

A 1� 1 Pn�� D AA 1 Pn�� 1 P

n��

1 Pn

�� Kn

K �

h St B , K� � St y , K� � :dist y B� � L2 R e� t t0�� sup�

B �. t t0 t0 B� ��

u B� B � Stu R� t t0 =

= t0 B� �


1

C

h

a

p

t

e

r

,

where

.

It is evident that for . Therefore,

.

This implies (9.6) with , where is the closure in of the set described in Lemma 9.1.

Show that lies in the set

(9.7)

where and are some constants.

In particular, Lemma 9.2 means that the system is asymptotically compact.Therefore, we can use Theorem 5.1 (see also Exercise 5.3) to guarantee the exis-tence of the global attractor lying in .

Let us use Theorem 8.1 to prove the finite dimensionality of the attractor. Veri-fication of the hypotheses of the theorem is based on the following assertion.

Lemma 9.3.

Let . Then

(9.8)

and

(9.9)

for and .

Proof.

Decomposition (9.2) and condition (C1) imply that

.

With the help of Gronwall’s lemma we obtain (9.8).To prove (9.9) it should be kept in mind that decomposition (9.2) and equa-

tions (B2) and (C2) imply that for

St u 1 Pn�� Tt t0� St0u Wn t , t0 , u� ��

Wn t , t0 , u� � Pn St u 1 Pn�� G t t0 , St0u��

Wn

t , t0 , u� � Kn

� t t0

St u , Kn� �dist 1 Pn�� Tt t0� St0u L2 R e

� t t0��

K Kn

� �� Kn

� � �

Kn

E x e r c i s e 9.1 K Kn

� ��

Kn A� v1 v2 : v1 P

n� , v2 1 P

n��

v1 C1 , AAv2 C2� ,��

�

C1 C2

� , St

� �

� K Kn

� ��

St ui R , t 0 , i � 1 2��

Stu1 S

tu2� L1 D R� � t� � u1 u2�exp�

1 Pn�� St u1 St u2�� L2 e � t� 1 C0 rnA

L1C3 R� �D R� �

�� e� t�; <= > u1 u2��

n n0 � � D R� ��

Stu1 S

tu2� L1 u1 u2� C1 R� � e w �� S� u1 S� u2� �d

0

t

��; <L ML M= >

ew t�

n n0


(9.10)

Here the inequality is used . If we put (9.8) in the right-hand side of formula (9.10), we obtain estimate (9.9).

The following simple argument completes the proof of Theorem 9.1. Let us fixan arbitrary number and choose and such that

and .

Then the hypotheses of Theorem 8.1 with , , and hold for the attractor . Hence, it is finite-dimensional with

estimate (9.3) holding for its fractal dimension. Theorem 9.1 is proved.

Prove that the global attractor of problem (9.1) is stable(Hint: verify that the hypotheses of Theorem 7.1 hold).

Properties (A)–(C) also enable us to prove that the system generated by equation(9.1) possesses an inertial set. A compact set in the phase space is said tobe an inertial setinertial setinertial setinertial set (or a fractal exponential attractor) if it is positively invariant

, its fractal dimension is finite and it possessesthe property

(9.11)

for any bounded set and for , where and are positivenumbers. (The importance of this notion for the theory of infinite-dimensional dy-namical systems has been discussed at the end of Section 8).

Lemma 9.4.

Assume that properties (A)–(C) hold. Then the dynamical system

generated by equation (9.1) possesses the following properties:

1) there exist a compact positively invariant set and constants

such that

(9.12)

for any bounded set in and for ;

1 Pn

�� Stu1 S

tu2�� L2 e � t� u1 u2�

C0 rnA

C3 R� � e� � S�u1 S�u2� �d

0

t

�<MM>

.

�

�

;LL=

�

A A� 1 Pn

�� C0 rnA�

0 9 1� � t0 n

L2 e�� t0 9

2�� C0 rn

AL1

C3 R� �D R� �� e

� t0 1�

M � , V St0� � P Pn� l �

L Pn D R� � t0� �exp� �

E x e r c i s e 9.2 �

A exp �

StAexp Aexp.� � Aexpdimf "��

h St B A exp�� St y , Aexp� � :dist y B� � CB e( t t0�� sup�

B �. t t0 t0 B� � CB

(

� , St

� �K

C , ( 0�

St y K�� :dist y B� � C e( t t

B�� sup

B � t tB

0�


1

C

h

a

p

t

e

r

2) there exist a vicinity of the compact and numbers and

such that

, (9.13)

provided that for all the semitrajectories lie in the clo-

sure of the set ;

3) there exist a sequence of finite-dimensional projectors in the

space , constants , and a sequence of positive

numbers tending to zero as such that

(9.14)

for any .

Proof.

Let be a compact set from Lemma 9.2. Let

.

It is clear that and equation (9.12) holds for with and . Let us prove that is a compact set. Let be a sequence ofelements of . Then for some and .If there exists an infinitely increasing subsequence , then equation (9.6)gives us that

.

Therefore, the sequence possesses a limit point in . If isa bounded sequence, then by virtue of the compactness of there exist a num-ber , an element and a sequence such that and

. Therewith

The first term in the right-hand side of this inequality evidently tends to zero.As for the second term, our argument is the same as in the proof of formula(9.8). We use the boundedness of the set (see property (A)) and proper-ties (B) and (C2) to obtain the estimate

, . (9.15)

It follows that

.

� K 1N�1 0�

Sty1 S

ty2� 1N e

�1 ty1 y2��

t 0 Sty

i

��

Pn

�� 2 ,N �2 , H 0�

2n

� n "&

1 Pn

�� Sty1 S

ty2�� 2N e H t� 1 2

ne�2 t�� y1 y2��

y1, y2 K�

K

K* (� K� � StK

t 0 -��

StK* K*. K K*� C L2 R�

( �� K* zn

�K* (� K� �� z

nStn

yn

� tn

0� yn

K�tnk �

Stnky

nk

K�; <= >dist

k "&lim 0�

zn

� K K *. tn

�K

t0 0� y K� nk

� ynky&

tnkt&

Stnk

yn

kS

t0y� Stn

y St0

y� Stnk

ynk

Stnk

y� .��

(+ K� �

St y1 St y2� C eC

Kt

y1 y2�� y1, y2 K�

Stnk

ynk

Stnk

y�k "&lim 0�


Therefore,

.

The closedness of the set can be established with the help of similar argu-ments. Thus, property (9.12) is proved for . Now we suppose that

, where is chosen such that for all . It is obviousthat is a compact positively invariant set. As it is proved above, it is easy tofind the estimate of form (9.15) for all and from an arbitrary bounded set

. Here an important role is played by the boundedness of the set (seeproperty (A)). Therefore, for any there exists a constant

such that

.

Hence, for we have that

for . This implies estimate (9.12) with the constant depending on and . However, if we change the moment in equation (9.12), we can pre-sume that, for example, . Therewith . Thus, the first assertion of thelemma is proved.

Since the set lies in the ball of dissipativity , estimates(9.13) and (9.14) follow from Lemma 9.3. Moreover,

,

. (9.16)

Thus, Lemma 9.4 is proved.

Lemma 9.4 along with the theorem given below enables us to verify the existence ofan inertial set for the dynamical system generated by equation (9.1).

Theorem 9.2.

Let the phase space of a dynamical system be a HilbertLet the phase space of a dynamical system be a HilbertLet the phase space of a dynamical system be a HilbertLet the phase space of a dynamical system be a Hilbert

space. Assume that in there exists a compact positively invariant set space. Assume that in there exists a compact positively invariant set space. Assume that in there exists a compact positively invariant set space. Assume that in there exists a compact positively invariant set

possessing properties possessing properties possessing properties possessing properties (9.12)––––(9.14). Then for any there exists an. Then for any there exists an. Then for any there exists an. Then for any there exists an

inertial set of the dynamical system such thatinertial set of the dynamical system such thatinertial set of the dynamical system such thatinertial set of the dynamical system such that

(9.17)

for any bounded set and for any bounded set and for any bounded set and for any bounded set and .... Here, as above, Here, as above, Here, as above, Here, as above,

. Moreover,. Moreover,. Moreover,. Moreover,

,,,, (9.18)

Stnk

ynkSt0

y K*�& (+ K� ��

K*

K K*�K St0

K*� t0 y R� y K�K

y1 y2B (� B� �

B � �� C0 �C B , K*, t0� �� 0�

St0

y1 St0

y2� C0 y1 y2� , y1, y2 B K*-��

y B�

Sty , K� �dist S

t0S

t t0� y , St0

K*�� C0 St t0� y, K*� �dist�dist�

t t0� C K

B tBC 1� ( ��

K z � : z R��

N1 L1 , �1 D R� � , N2 L2 , �2 � D R� ��

H ( � , 2n C0 rnA

C3 R� � D R� � 1��

� � , St

� �� K

: 2ln�Aexp: � , S

t� �

h StB , Aexp

:� � C B, :� � (� 1( �� 1

: ( �1� ��; <= > t t

B��

� �� !

exp��

B t tB h X Y�� x Y�� :distsup�x X� �

Aexp:

C0 1 Pnln�� Pndim��dimf


1

C

h

a

p

t

e

r

where the number is determined from the conditionwhere the number is determined from the conditionwhere the number is determined from the conditionwhere the number is determined from the condition

(9.19)

and constant does not depend on and and constant does not depend on and and constant does not depend on and and constant does not depend on and ....

The proof of the theorem is based on the following preliminary assertions.

Lemma 9.5.

Let be a dynamical system, its phase space being a compact in

a Hilbert space . Assume that for all equations (9.13)and (9.14) are valid. Then for any there exists an inertial set

of the system such that

. (9.20)

Moreover, estimate (9.18) holds for the value .

Proof.

We use Theorem 8.2 with , , and , where and are chosen to fulfil

and .

In this case conditions (8.1) and (8.2) are valid for with and Therefore, there exists a bounded closed positively invariant

set with such that (see (8.6) and (8.15))

(9.21)

and

(9.22)

Assume that and consider the set

.

Here . It is easy to see that

.

Therefore, equations (9.20) and (9.18) follow from (9.21) and (9.20) after somesimple calculations.

Lemma 9.6.

Assume that in the phase space of a dynamical system

there exist compact sets and such that (a) ; (b) properties

n

2n

4N2� ��2H��

exp :�2

H��

� �� !

��

C0 : n

K , St� �� y1 y2�� K�

: 2ln�Aexp:

K , St� �

h StK , Aexp

:� � Sty Aexp

:�� :dist y K� � C: e : t��sup�

Aexp:dimf

M K� V St0

� 9 12�� e :�� t0

n

2 eH t0�N 9

2�� 1

4�� e :�� 2n e

�2 t0 1�

V St0� 9 1

2�� e :��

l Pn 1 e�1 t0 .N�

AK 9 K 1 2$� �

V m y , AK� � :dist y K� � Km , m�sup 1 2 � ��

AK 1 4 l

K 9��; <

= > 12K��ln

1�ln P

ndim��dimf

K 29 e :��

Aexp:

St AK : 0 0 t0� � �-�

: 1K�� 2ln�ln�

Aexp: 1 Aexp

:dimf��dimf

� � , St

� �K K0 K0 K.


(9.12) and (9.13) are valid for ; and (c) the set possesses the pro-

perty

, (9.23)

where . Then for any bounded set

and the following inequality holds

. (9.24)

Proof.

By virtue of (9.12) every bounded set reaches the vicinity in finitetime and stays in it. Therefore, it is sufficient to prove the lemma for a set

such that for , where denotes the closure of .Let and . Evidently,

for any and . With the help of (9.13) we have that

.

Therefore, for any and we have that

If we take an infimum over and a supremum over , we find that

for all . Hence, equations (9.12) and (9.23) give us that

for . If we choose , we obtain (9.24). Lemma 9.6is proved.

If we now use Lemma 9.6 with and estimate (9.20), we get equation(9.17). This completes the proof of Theorem 9.2.

Thus, by virtue of Lemma 9.4 and Theorem 9.2 the dynamical system gene-rated by equation (9.1) possesses an inertial set for which equations (9.17)–(9.19) hold with relations (9.16).

It should be noted that a slightly different approach to the construction of iner-tial sets is developed in the book by A. Eden, C. Foias, B. Nicolaenko, and R. Temam(see the list of references). This book contains further developments and applica-tions of the theory of inertial sets.

K K0

h StK , K0� � C e

(0 t��

h X Y�� x Y�� : dist x X� �sup� B �.t t

B

h StB , K0� � C

B

( (0( (0 �1� �� t�

� �� !

exp�

B �

B � �� StB �� . t 0 ��

k0 K0� y B�

St y k0� SO t S 1 O�� t y SO t k� SO t k k0��

0 O 1� � k K�

Sty k0� 1 e

�1O tS 1 O�� t

y k� SO tk k0��N�

0 O 1� � k K�

Sty , K0� �dist 1 e

�1O tS 1 O�� t

y k� SO tk , K0� �dist�N

1 e�1O t

S 1 O�� ty k� h SO t

K , K0� � . �N

� �

�

k K� y B�

h StB , K0� � 1 e

�1O th S 1 O�� t

B , K� �N h SO tK , K0� ��

0 O 1� �

h St B , K0� � CBBBBe

�1O ( 1 O�� tCK e

O (0 t��

t tB

O ( ( (0 �1� �� 1��

K0 Aexp:�

� , St

� �Aexp:


1

C

h

a

p

t

e

r

To conclude this section, we outline the results on the behaviour of the projec-tion onto the finite-dimensional subspace of the trajectories of the system

generated by equation (9.1).Assume that an element belongs to the global attractor of a dynamical

system . Lemma 6.1 implies that there exists a trajectory lying in wholly such that . Therewith the following assertion is valid.

Lemma 9.7.

Assume that properties (A)–(C) are fulfilled and let . Then the

following equation holds:

, , (9.25)

where is a trajectory passing through , the number can be

found from (B2) and the integral in (9.25) converges in the norm of the

space .

Proof.

Since , equation (9.2) gives us that

. (9.26)

A trajectory in the attractor possesses the property , .Therefore, property (B2) implies that

and .

These estimates enable us to pass to the limit in (9.26) as . Thereuponwe obtain (9.25).

The following assertion is valid under the hypotheses of Theorem 9.1.

Theorem 9.3.

There exists such that for all the following assertionsThere exists such that for all the following assertionsThere exists such that for all the following assertionsThere exists such that for all the following assertions

are valid:are valid:are valid:are valid:

1) for any two trajectories and lying in the attractor of thefor any two trajectories and lying in the attractor of thefor any two trajectories and lying in the attractor of thefor any two trajectories and lying in the attractor of the

system generated by equation system generated by equation system generated by equation system generated by equation (9.1) the equality the equality the equality the equality

for all implies that ;for all implies that ;for all implies that ;for all implies that ;

2) for any two solutions and of the system for any two solutions and of the system for any two solutions and of the system for any two solutions and of the system (9.1) the equa- the equa- the equa- the equa-

tiontiontiontion

(9.27)

implies that as implies that as implies that as implies that as ....

Pn�

� , St

� �y0 �

� St

�� ( y t� � , t R� �� y 0� � y0�

y0 ��

1 Pn�� y0 1 Pn�� T ��

"�

0

�� B y �� d n n0

y �� y0 n0

�

y0 Sty t��

1 Pn�� y0 1 Pn�� Tt y t�� 1 Pn�� T ��

t�

0

� B y �� d�; <L ML M= >

�

y t� � R� t " , "��

1 Pn�� Tty t�� L2 R e � t�� 1 Pn�� T �� B y �� L2 C

Re � ��

t "�&

N0 n0 N N0

y1 t� � y2 t� �P

Ny1 t� � P

Ny2 t� ��

t R� y1 t� � y2 t� ��u1 t� � u2 t� �

PN u1 t� � u2 t� �� t �"&

lim 0�

u1 t� � u2 t� �� 0& t "&


We can also obtain an upper estimate of the number from the inequali-We can also obtain an upper estimate of the number from the inequali-We can also obtain an upper estimate of the number from the inequali-We can also obtain an upper estimate of the number from the inequali-

ty ty ty ty ....

Proof.

Equation (9.25) implies that for any trajectory lying in the attractorof system (9.1) the equation

,

holds. Therefore, if , then properties (B2), (B3), and (C2) give usthat

It follows that the estimate

holds for , where . If we tend , we obtain thefirst assertion, provided .

Now let us prove the second assertion of the theorem. Let

.

Then

.

Therefore, equation (9.10) for the function gives us that

.

This and Gronwall’s lemma imply that

Therefore, if , then equation (9.27) gives us that . Thus,the second assertion of Theorem 9.3 is proved.

N0r

N0

Aa0 � L2 C3 R� �� 1��

yi

t� �

1 PN

�� yi

t� � 1 PN

�� Tt �� B y

i�� , id

"�

t

� 1 2 ��

PN y1 t� � PN y2 t� ��

y1 t� � y2 t� �� C0 rN

AL2 C3 R� � e � t �� y1 �� y2 �� . d

"�

t

��

y1 t� � y2 t� �� AN

� t� AN

t t0�� e� � y1 �� y2 �� d

"�

t0

�exp�

t t0 AN C0 rN

AL2 C3 R� �� t0 "�&

AN ��

�N

t� � PN

u1 t� � u2 t� ��

u1 t� � u2 t� �� N t� �� 1 PN�� u1 t� � u2 t� ��

P t� � u1 u2�� t� �exp

P t� � �N t� � e� t L2 u1 0� � u2 0� �� AN P �� d

0

t

��

u1 t� � u2 t� �� N t� � L2 � AN�� t�� u1 0� � u2 0� ��

AN

�N

�� AN

�� t �� . dexp

0

t

�

�

�

exp��

AN

�� u1 t� � u2 t� �� 0&


1

C

h

a

p

t

e

r

Theorem 9.3 can be presented in another form. Let be a basisin the space . Let us define linear functionals on

. Theorem 9.3 implies that the asymptotic behaviour of trajectories ofthe system is uniquely determined by its values on the functionals .Therefore, it is natural that the family of functionals is said to be the determin-ing collection. At present some general approaches have been worked out whichenable us to define whether a particular set of functionals is determining. Chapter 5is devoted to the exposition of these approaches. It should be noted that for the firsttime Theorem 9.3 was proved for the two-dimensional Navier-Stokes system byC. Foias and D. Prodi (the second assertion) and by O. A. Ladyzhenskaya (the firstassertion).

Concluding the chapter, we would like to note that the list of references givenbelow does not claim to be full. It contains only references to some monographs andreviews devoted to the developments of the questions touched on here and compris-ing intensive bibliography.

ek

: k 1 dN

� �� P

N� l

ju� � u , e

j� �� , j �

1 dN

�� , S

t� � l

j

lj

�

ReferencesReferencesReferencesReferences

1. BABIN A. V., VISHIK M. I. Attractors of evolution equations. –Amsterdam: North-Holland, 1992.

2. TEMAM R. Infinite-dimensional dynamical systems in Mechanics and

Physics. –New York: Springer, 1988.3. HALE J. K. Asymptotic behavior of dissipative systems. –Providence: Amer.

Math. Soc., 1988.4. EDEN A., FOIAS C., NICOLAENKO B., TEMAM R. Exponential attractors for dissipa-

tive evolution equations. –New York: Willey, 1994.5. LADYZHENSKAYA O. A. On the determination of minimal global attractors for the

Navier-Stokes equations and other partial differential equations // Russian Math. Surveys. –1987. –Vol. 42. –# 6. –P. 27–73.

6. KAPITANSKY L. V., KOSTIN I. N. Attractors of nonlinear evolution equations

and their approximations // Leningrad Math. J. –1991. –Vol. 2. –P. 97–117.7. CHUESHOV I. D. Global attractors for nonlinear problems of mathematical

physics // Russian Math. Surveys. –1993. –Vol. 48. –# 3. –P. 133–161.8. PLISS V. A. Integral sets of periodic systems of differential equations.

–Moskow: Nauka, 1977 (in Russian).9. HALE J. K. Theory of functional differential equations. –Berlin; Heidelberg;

New York: Springer, 1977.

C h a p t e r 2

Long-Time Behaviour of SolutionsLong-Time Behaviour of SolutionsLong-Time Behaviour of SolutionsLong-Time Behaviour of Solutions

to a Class of Semilinear Parabolic Equationsto a Class of Semilinear Parabolic Equationsto a Class of Semilinear Parabolic Equationsto a Class of Semilinear Parabolic Equations

C o n t e n t s

. . . . § 1 Positive Operators with Discrete Spectrum . . . . . . . . . . . . . . 77

. . . . § 2 Semilinear Parabolic Equations in Hilbert Space . . . . . . . . . . 85

. . . . § 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

. . . . § 4 Existence Conditions and Properties of Global Attractor . . 101

. . . . § 5 Systems with Lyapunov Function . . . . . . . . . . . . . . . . . . . . . 108

. . . . § 6 Explicitly Solvable Model of Nonlinear Diffusion . . . . . . . . . 118

. . . . § 7 Simplified Model of Appearance of Turbulence in Fluid . . . 130

. . . . § 8 On Retarded Semilinear Parabolic Equations. . . . . . . . . . . . 138

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

In this chapter we study well-posedness and the asymptotic behaviour of solu-tions to a class of abstract nonlinear parabolic equations. A typical representative ofthis class is the nonlinear heat equation

considered in a bounded domain of with appropriate boundary conditions onthe border . However, the class also contains a number of nonlinear partial dif-ferential equations arising in Mechanics and Physics that are interesting from theapplied point of view. The main feature of this class of equations lies in the fact thatthe corresponding dynamical systems possess a compact absorbing set.

The first three sections of this chapter are devoted to the questions of existenceand uniqueness of solutions and a brief description of examples. They are indepen-dent of the results of Chapter 1. In the other sections containing the discussion ofasymptotic properties of solutions we use general results on the existence and pro-perties of global attractors proved in Chapter 1. In Sections 6 and 7 we present twoquite simple infinite-dimensional systems for which the asymptotic behaviour of thetrajectories can be explicitly described. In Section 8 we consider a class of systemsgenerated by infinite-dimensional retarded equations.

The list of references at the end of the chapter consists only of the books re-commended for further reading.

§ 1 Positive Operators§ 1 Positive Operators§ 1 Positive Operators§ 1 Positive Operators

with Discrete Spectrumwith Discrete Spectrumwith Discrete Spectrumwith Discrete Spectrum

This section contains some auxiliary facts that play an important role in the subse-quent considerations related to the study of the asymptotic properties of solutionsto abstract semilinear parabolic equations.

Assume that is a separable Hilbert space with the inner product andthe norm . Let be a selfadjoint positive linear operator with the domain .An operator is said to have a discrete spectrumdiscrete spectrumdiscrete spectrumdiscrete spectrum if in the space there existsan orthonormal basis of the eigenvectors:

, , , (1.1)

such that

, . (1.2)

The following exercise contains a simple example of an operator with discrete spec-trum.

�u

� t�� 2 u

�xi2

�� f x u�� i 1�

d

��

Rd

�

H . .�� . A D A� �

A H

ek

�

ek

ej

�� k j

� A ek

k

ek

� k j� 1 2 ��

0 1 2 �� k

k ��lim ��

78 L o n g - T i me B e h a v i o u r o f S o l u t i o n s t o a C l a s s o f S e mi l i n e a r P a r a b o l i c Eq u a t i o n s

2

C

h

a

p

t

e

r

Let and let be an operator defined by theequation with the domain which consists of conti-nuously differentiable functions such that (a) ,(b) is absolutely continuous and (c) . Showthat is a positive operator with discrete spectrum. Find its eigen-vectors and eigenvalues.

The above-mentioned structure of the operator enables us to define an operator for a wide class of functions defined on the positive semiaxis. It can be

done by supposing that

,

, . (1.3)

In particular, one can define operators with . For these ope-rators are bounded. However, in this case it is also convenient to introduce the line-als if we regard as a completion of the space with respect to thenorm .

Show that the space with can be identi-fied with the space of formal series such that

.

Show that for any the operator can be defined onevery space as a bounded mapping from into

such that

, . (1.4)

Show that for all the space is a sepa-rable Hilbert space with the inner product and the norm .

The operator with the domain is a positive operatorwith discrete spectrum in each space .

Prove the continuity of the embedding of the space into for , i.e. verify that and .

Prove that is dense in for any .

E x e r c i s e 1.1 H L2 0 1�� A

Au u�� D A� �u x� � u 0� � u 1� � 0� �

u � x� � u � L2 0 1�� A

A

f A� � f � �

D f A� �� h ck

ek

H : ck2

f k

� �� 2 ��k 1�

�

��k 1�

�

��

�

f A� � h ck f k� � ek

k 1�

�

�� h D f A� ��

A� � R� � 0��

D A�� D A �� H

A � .

E x e r c i s e 1.2 � � D A �� 0!c

ke

k�

ck2 k

2 �

k 1�

�

� ��

E x e r c i s e 1.3 R� A

D A�� D A�� D A� ��

A D A�� D A� �� A 1 2�

A 1 A

2"�

E x e r c i s e 1.4 � R� �� D A�� #u v�� A�u A�v��

u � A�u�

E x e r c i s e 1.5 A �1 $��$

E x e r c i s e 1.6 �� ! �� % u C u ��

E x e r c i s e 1.7 �� !

P o s i t i v e O p e r a t o r s w i t h D i s c r e t e S p e c t r u m 79

Let for . Show that the linear functional can be continuously extended from the space to

and for any and .

Show that any continuous linear functional on has theform: , where . Thus, is the spaceof continuous linear functionals on .

The collection of Hilbert spaces with the properties mentioned in Exercises 1.7–1.9is frequently called a scalescalescalescale of Hilbert spaces. The following assertion on the com-pactness of embedding is valid for the scale of spaces .

Theorem 1.1.

Let Let Let Let .... The The The Thennnn the space is compactly embedded into the space is compactly embedded into the space is compactly embedded into the space is compactly embedded into ,,,, i.e. i.e. i.e. i.e.

every sequence bounded in is compact in every sequence bounded in is compact in every sequence bounded in is compact in every sequence bounded in is compact in ....

Proof.

It is well known that every bounded set in a separable Hilbert space is weaklycompact, i.e. it contains a weakly convergent sequence. Therefore, it is sufficient toprove that any sequence weakly tending to zero in converges to zero with re-spect to the norm of the space . We remind that a sequence in weaklyconverges to an element if for all

.

Let the sequence be weakly convergent to zero in and let

, . (1.5)

Then for any we have

. (1.6)

Here we applied the fact that for

.

Equations (1.5) and (1.6) imply that

.

We fix and choose such that

. (1.7)

E x e r c i s e 1.8 f �$� $ 0! F g� � #f g�� # H � $�

f g�� f $ g $�"� f �$� g � $��

E x e r c i s e 1.9 F �$F f� � f g�� g � $�� $�

�$

�$ �

$1 $2! �$1�$2

�$1�$2

�$1�$2

fn

� �$f �$� g �$�

fn

g�� $n ��lim f g�� $�

fn � �$1

fn $1

C� n 1 2 ��

N

fn $2

2 k

2$2 fn

ek

�� 2 1

N2 $1 $2��

�� k

2$1 fn

ek

�� 2

k N�

�

��k 1�

N 1�

��

k N&

k

2$2 1

N

2 $1 $2��

k

2$1�

fn $2

2 k

2$2 fn

ek

�� 2 C N

2 $1 $2��

k 1�

N 1�

��

' 0! N

fn $2

2 k

2$2 fn

ek

�� 2 '�k 1�

N 1�

��


2

C

h

a

p

t

e

r

Let us fix the number . The weak convergence of to zero gives us

, .

Therefore, it follows from (1.7) that

.

By virtue of the arbitrariness of we have

.

Thus, Theorem 1.1 is proved.

Show that the resolvent , , isa compact operator in each space .

We point out several properties of the scale of spaces that are important forfurther considerations.

Show that in each space the equation

, ,

defines an orthoprojector onto the finite-dimensional subspace ge-nerated by the set of elements . Moreover,for each we have

.

Using the Hölder inequality

, , ,

prove the interpolation inequality

, , .

Relying on the result of the previous exercise verify thatfor any the following interpolation estimate holds:

,

where , , and .Prove the inequality

,

where and is a positive number.

N fn

fn

ek

�� n ��

lim 0� k 1 2 � N 1��

fn $2

'�n ��

lim

'

fn $2n ��lim 0�

E x e r c i s e 1.10 R A� � A �� 1�� k

(�$

�$ �

E x e r c i s e 1.11 �$

Pl u u ek�� ek

k 1�

l

�� u �$� �� $ ��

ek

k 1 2 � l� � �� $

Plu u� $l ��

lim 0�


ak

bk

ak

p

k

�) *+ ,- .1 p�

bk

q

k

�) *+ ,- .1 q�

�k

� 1p��

1q�� 1� a

kb

k0!�

A/u Au/

u1 /�"� 0 / 1� � u D A� ��


$1 $2 R��

u $ /� � u $1

/u $2

1 /��

$ /� � /$1 1 /�� $2�� 0 / 1� � u �max $1 $2��

u $ /� �2 ' u $1

2C/ '

/1 /��

u $2

2��

0 / 1� � '


Equations (1.3) enable us to define an exponential operator , , inthe scale . Some of its properties are given in exercises 1.14–1.17.

For any and the linear operator maps into and possesses the property

.

The following semigroup property holds:

, .

For any and the following equation is valid:

. (1.8)

For any the exponential operator defines a dissi-pative compact dynamical system . What can you sayabout its global attractor?

Let us introduce the following notations. Let be the space of stronglycontinuous functions on the segment with the values in , i.e. they are con-tinuous with respect to the norm . In particular, Exercise 1.16 meansthat if . By we denote the subspace of

that consists of the functions which possess strong (in ) deri-vatives lying in . The space is defined similarlyfor any natural . We remind that the strong derivative (in ) of a function ata point is defined as an element such that

.

Let for some . Show that

for all , , and . Moreover,

, .

Let be the space of functions on the segment with the valuesin for which the integral

exists. Let be the space of essentially bounded functions on with the values in and the norm

t A�� exp t 0&�$ �

E x e r c i s e 1.14 � R� t 0! t A�� exp�� $$ 0&0 e t A� u � �

et 1�

u ��


t1 A�� t2 A�� exp"exp t1 t2�� A�� exp� t1 t2 0&�

E x e r c i s e 1.16 u �$� $ R�

e t A� u e 1A� u� $t 1�lim 0�

E x e r c i s e 1.17 $ R� e t A�

�$ e t A��

C a b ��2�� a b��

.� A� .�

e t A� u C R� �� u �� C1 a b ��2�� C a b ��2�� f t� � ��

f � t� � C a b ��2�� Ck a b ��2�� k �� f t� �

t t0� v ��

1h�� f t0 h�� f t0� �� v�

�h 0�lim 0�

E x e r c i s e 1.18 u0 �$� $

e t A� u0 Ck � �� 2��

� 0! � R� k 1 2 ��

dk

tkd�� e t A� u0 A�� k e t A� u0� k 1 2 ��

L2 a b ��2�� a b��

uL2 a b ��2�� 2

u t� � �2

td

a

b

3�L� a b ��2�� a b��

��


2

C

h

a

p

t

e

r

.

We consider the Cauchy problem

, ; , (1.9)

where and . The weak solutionweak solutionweak solutionweak solution (in ) to thisproblem on the segment is defined as a function

(1.10)

such that and equalities (1.9) hold. Here the derivative is considered in the generalized sense, i.e. it is defined by the equality

, ,

where is the space of infinitely differentiable scalar functions on vanishing near the points and .

Show that every weak solution to problem (1.9) possesses theproperty

. (1.11)

(Hint: first prove the analogue of formula (1.11) for ,then use Exercise 1.11).

Prove the theorem on the existence and uniqueness of weaksolutions to problem (1.9). Show that a weak solution to thisproblem can be represented in the form

. (1.12)

Let and let a function possess the property

for some . Then formula (1.12) gives us a solution to pro-blem (1.9) belonging to the class

.

Such a solution is said to be strong in .

uL� a b D A�� 2��

ess A�u t� �t a b��

sup�

ydtd

�� A y� f t� �� t a b�� y a� � y0�

y �� f t� � L2 a b F� 1 24�2�� a b��

y t� � C a b ��2�� L2 a b �� 1 24�2�� 0�

yd td4 L2 a b F� 1 24�2�� y � t� � yd td4#

5 t� � y � t� � td

a

b

3 5 � t� � y t� � td

a

b

3�� 5 C0�

a b��

C0�

a b�� a b�� a b


y t� � �2

y 1� �� 1

2��

2 1d

a

t

3� y0 � f 1� �� 1

2��

2 1d

a

t

3��

yn

t� � Pn

y t� ��


y t� �

y t� � e t a�� A� y0 e t 1�� A� f 1� � 1d

a

t

3��

E x e r c i s e 1.21 y0 �� f t� �

f t1� � f t2� �� C t1 t2� /�

0 / 1��

C a b�� 2� � C1 �a b� ��2�� C �a b � �� 1�2�� 00

��


The following properties of the exponential operator play an important partin the further considerations.

Lemma 1.1.

Let be the orthoprojector onto the closure of the span of elements

in and let , . Then

1) for all , and the following inequality holds:

; (1.13)

2) for all , and the following estimate is valid:

, (1.14)

in the case we supose that in (1.14).

Proof.

Estimate (1.13) follows from the equation

.

In the proof of (1.14) we similarly have that

.

This gives us the inequality

.

Since is attained when , we have that

Therefore,

.

This implies estimate (1.14). Lemma 1.1 is proved.

In particular, we note that it follows from (1.14) that

, . (1.15)

e t A�

QN

ek

k N 1�&� � H PN

I QN

�� N 0 1 2 �� h H� 0& t R�

A PN

e t A� h N

e

Nt

h�

h D A � �� t 0! � &

A�QN

e t A� h� �

t��) *- .� �

N 1�� e

t N 1��

A h�

� � 0� 00 0�

A PN e t A� h2 k

2 e

2 t k

�h ek�� 2

k 1�

N

��

A�QN

e t A� h 2 � �e t �� 2

k2

h ek

�� 2

k N 1��

�

� N 1�&

max�

A�QN

e t A� h1

t� �� 6� � e 6�� A h6 t

N 1�&max�

67 e 6� 6 0&2 �max 6 7�

67 e 6�� 6

N 1� t&max

N 1� t� �7 e

N 1� t� if N 1� t 7 ;&�

77 e 7� if N 1� t 7 .��

8�8�

�

67 e 6�� 77 N 1� t� �7�� e

N 1� t��

6 N 1� t&

max

A� e t A� h� �

t��) *- .� �

1� �� e

t 1�A h"� � &


2

C

h

a

p

t

e

r

Using estimate (1.15) and the equation

, , ,

prove that

; , (1.16)

provided , where a constant does not dependon and (cf. Exercise 1.16).

Show that

, , . (1.17)

Lemma 1.2.

Let for . Then there exists a unique solu-

tion to the nonhomogeneous equation

, , (1.18)

that is bounded in on the whole axis. This solution can be

represented in the form

. (1.19)

We understand the solution to equation (1.18) on the whole axis as a function such that for any the function is a weak solution

(in ) to problem (1.9) on the segment with .

Proof.

If there exist two bounded solutions to problem (1.18), then their diffe-rence is a solution to the homogeneous equation. Therefore,

for and for any . Hence,

.

If we tend here, then we obtain that . Thus, the bounded so-lution to problem (1.18) is unique. Let us prove that the function definedby formula (1.19) is the required solution. Equation (1.15) implies that

for and . Therefore, integral (1.19) exists and it can be uniform-ly estimated with respect to as follows:


e t A� u e s A� u� A e 1A� u 1d

s

t

3�� t s& u � �

e t A� u e s A� u� / C/ $� t s� $ /�u $� t s� 0!

/ $ 1 /�� C/ $�t s


A� e t A� �t��) *- .� e �� t 0! � 0!

f t� � L� R �� 7�� 0 7 1��v t� � C R ��

vdtd

�� A v� f t� �� t R�

��

v t� � et 1�� A

f 1� � 1d

��

t

3�

v t� � �C R �� a b� v t� �� 1 24� a b�� y0 v a� ��

w t� � w t� � �t t0�� A� �w t0� �exp� t t0& t0

A�w t� � et t0�� 1�

A�w t0� � C et t0�� 1��

t0 �� w t� � 0�v t� �

A�et 1�� A� 7

t 1��) *- .

7 17� e

t 1�� 1� ess A� 7� f 1� �

1 R�sup�

t 1! 0 7 1� �t

S e m i l i n e a r P a r a b o l i c E q u a t i o n s i n H i l b e r t S p a c e 85

,

where for and

for .

The continuity of the function in follows from the following equationthat can be easily verified:

.

This also implies (see Exercise 1.18) that is a solution to equation (1.18).Lemma 1.2 is proved.

§ 2 Semilinear Parabolic § 2 Semilinear Parabolic § 2 Semilinear Parabolic § 2 Semilinear Parabolic EquationsEquationsEquationsEquations

in Hilbert Spacein Hilbert Spacein Hilbert Spacein Hilbert Space

In this section we prove theorems on the existence and uniqueness of solutions toan evolutionary differential equation in a separable Hilbert space of the form

, , (2.1)

where is a positive operator with discrete spectrum and is a nonlinearcontinuous mapping from into , , possessing the property

(2.2)

for all and from the domain of the operator and such that. Here is a nondecreasing function of the parameter that does

not depend on and is a norm in the space .A function is said to be a mild solutionmild solutionmild solutionmild solution (in ) to problem (2.1) on the

half-interval if it lies in for every and for all satisfies the integral equation

. (2.3)

The fixed point method enables us to prove the following assertion on the localexistence of mild solutions.

A�v t� � 1 k� 1

1 7�� ess A� 7� f 1� �

1 R�sup�

k 0� 7 0�

k 77 s 7� e s� sd

0

�

3� 0 7 1� �

v t� � ��

v t� � et t0�� A�

v t0� � et 1�� A�

g 1� � 1d

t0

t

3��

v t� �

H

udtd

�� A u� B u t�� ut s� u0�

A B . .�� D A/� � R9 H 0 / 1��

B u1 t�� B u2 t�� M :� � A/ u1 u2��

u1 u2 �/ D A/� �� A/

A/uj :� M :� � :t . H

u t� � �/s s T �� C s s T � �/2�� T � T�

t s s T ��

u t� � et s�� A�

u0 et 1�� A�

B u 1� � 1�� 1d

s

t

3��


2

C

h

a

p

t

e

r

Theorem 2.1.

Let Let Let Let .... Then there exists depending on and such that Then there exists depending on and such that Then there exists depending on and such that Then there exists depending on and such that

problem problem problem problem (2.1) possesses a unique mild solution on the half-interval possesses a unique mild solution on the half-interval possesses a unique mild solution on the half-interval possesses a unique mild solution on the half-interval

.... Moreover, either or the solution cannot be continued Moreover, either or the solution cannot be continued Moreover, either or the solution cannot be continued Moreover, either or the solution cannot be continued

in up to the moment in up to the moment in up to the moment in up to the moment ....

Proof.

On the space we define the mapping

.

Let us prove that for any . Assume that and . It is evident that

. (2.4)

By virtue of (1.8) we have that if then

.

Therefore, it is sufficient to estimate the second term in (2.4). Equation (1.15) im-plies that

(2.5)

(if , then the coefficient in the braces should be taken to be equal to 1). Thus, maps into itself. Let . In

we consider a ball of the form

.

Let us show that for small enough the operator maps into itself and is con-tractive. Since for , equation (2.2) gives

u0 �/� T * / u0 /

s s T* �� T* ��/ t s T*��

Cs /� C s s T �/2�� #

G u� � t� � et s�� A�

u0 et 1�� A�

B u 1� � 1�� 1d

s

t

3��

G u� � t� � C s s T �/2�� T 0! t1 t2� �s s T�� t1 t2�

G u� � t2� � et2 t1�� A�

G u� � t1� � et2 1�� A�

B u 1� � 1�� 1d

t1

t2

3��

t2 t1 ,�

G u� � t1� � et2 t1�� A�

G u� � t1� �� / 0�

et2 1�� A�

B u 1� � 1�� 1d

t1

t2

3/

�

/t2 1��) *- ./ 1

/� 1d

t1

t2

3 B u 1� � 1�� 1 s s T��

max

t2 t1� 1 /� //1 /�� 1

/t2 t1� /�

� ��

B u 1� � 1�� 1 s s T��

max

�

�

"�

/ 0�G C

s /� C s s T �/2�� v0 t� � e t s�� A� u0� Cs /�

U u t� � Cs /� : u v0�

Cs /�

u t� � v0 t� �� / 1�s s T�� max#�

� ��

�

T G U

u Cs /�

1 u0 /�� u U�


for all , where is a fixed number. Therefore, with the help of (2.5) we findthat

.

Similarly we have

for . Consequently, if we choose such that

and ,

we obtain that is a contractive mapping of into itself. Therefore, possessesa unique fixed point in . Thus, we have constructed a solution on the seg-ment . Taking as an initial moment, we can construct a solutionon the segment with the initial condition .If we continue our reasoning, then we can construct a solution on some maximal half-interval . Moreover, it is possible that . Theorem 2.1 is proved.

Let and let be such that isthe maximal half-interval of the existence of the mild solution to problem (2.1). Then we have either and

or .

Using equations (1.16) and (2.5), prove that for any mild solu-tion to problem (2.1) on the estimate

, , (2.6)

is valid, provided , , and .

Let and let be a mild solution to problem (2.1)on the half-interval . Then

for any , , and . Moreover, equations (2.1)are valid if they are understood as the equalities in and , re-spectively.

It is frequently convenient to use the Galerkin method in the study of properties ofmild solutions to the problem of the type (2.1). Let be the orthoprojector in

B u 1� � 1�� 1 s s T��

max B 0 1��

1 u0 /�� M 1 u0 /�� C T0 u0 /�� #

�

�

1 s s T�� max�

T T0� T0

G u� � v0�C

s /�T1 /� C1 T0 / u0 /� �� "�

G u� � G v� ��C

s /�T1 /� C2 T0 /�� M 1 u /�� u v�

Cs /�

"�

u v� U� T1

T11 /�

C1 T0 / u0 /� �� 1� T11 /�

C2 T0 /�� M 1 u0 /�� 1�

G U G

U Cs /�%s s T1�� s T1�

s T1� s T1 T2� �� u0 u s T1��

s s T* �� T* ��

E x e r c i s e 2.1 u0 �/� T* T* / u0�� s s T*��u t� �

T* �� u t� � /t s T*��

lim �� ,� T* ��

E x e r c i s e 2.2

u t� � s s T*��

u t� � u 1� �� C / � T� �� t 1� / �� t 1� s s T��

u0 �/� 0 � /� � T T*�

E x e r c i s e 2.3 u0 �/� u t� �s s T*��

u t� � C s s T� �/� �� C s � s T�� 1 $��

C1 s � s T�� $��

0 0

0

�

$ 0! 0 � T� � T T*�� $� �/

Pm H


2

C

h

a

p

t

e

r

onto the span of elements . Galerkin approximate solutionGalerkin approximate solutionGalerkin approximate solutionGalerkin approximate solution

of the order of the order of the order of the order with respect to the basis is defined as a continuously diffe-rentiable function

(2.7)

with the values in the finite-dimensional space that satisfies the equations

, , . (2.8)

It is clear that (2.8) can be rewritten as a system of ordinary differential equa-tions for the functions .

Show that problem (2.8) is equivalent to the problem of find-ing a continuous function with the values in that satis-fies the integral equation

. (2.9)

Using the method of the proof of Theorem 2.1, prove the localsolvability of problem (2.9) on a segment , where the pa-rameter can be chosen to be independent of . Moreover,the following uniform estimate is valid:

, , (2.10)

where is a constant.

The following assertion on the convergence of approximate functions to exact onesholds.

Theorem 2.2.

Let Let Let Let .... Assume that there exists a sequence of approximate solu- Assume that there exists a sequence of approximate solu- Assume that there exists a sequence of approximate solu- Assume that there exists a sequence of approximate solu-

tions on a segment for which estimate tions on a segment for which estimate tions on a segment for which estimate tions on a segment for which estimate (2.10) is valid. Then is valid. Then is valid. Then is valid. Then

there exists a mild solution to problem there exists a mild solution to problem there exists a mild solution to problem there exists a mild solution to problem (2.1) on the segment on the segment on the segment on the segment

andandandand

,,,, (2.11)

where where where where is a is a is a is a positive constant independent of positive constant independent of positive constant independent of positive constant independent of ....

e1 e2 � em

� � � �m e

k �

um t� � gk t� � ek

k 1�

m

��

Pm H

dtd

��um

t� � A um

t� �� Pm

B um

t� � t�� t s! um

t s�Pm

u0�

gk t� �

E x e r c i s e 2.4

um

t� � Pm

H

um

t� � et s�� A�

Pm

u0 et 1�� A�

Pm

B um

1� � 1�� 1d

s

t

3��

E x e r c i s e 2.5

s s T�� T 0! m

um t� � /s s T�� max R� m 1 2 3 ��

R 0!

u0 �/�u

mt� � s s T��

u t� � s s T��

u t� � um

t� �� / C 1 Pm

�� u0 /1

m 1�1 /�

��) *- .�

s s T�� max

C C / R T� �� s


Proof.

Let . We use (2.9), (1.14) and (1.17) to find that for we have

Therefore, equations (2.2) and (2.10) give us that

(2.12)

where

.

It is evident that . By changing the variable in the integral, we obtain

.

Thus, equation (2.12) implies

Hence, if we use Lemma 2.1 which is given below, we can find that

(2.13)

n m! / 0!

un t� � um t� �� / Pn Pm�� u0 /

/t 1��) *- . /

m 1�/� e

m 1� t 1��

B un1� � 1�� 1d

s

t

3

e /� /t 1��) *- ./ B u

n1� � 1�� B� u

m1� � 1�� 1 .d

s

t

3

�

� �

�

�

un t� � um t� �� / Pn Pm�� u0 /�

B 0 1�� s s T�� max M R� �R�) *

- . Jm t s��

M R� � // e /� t 1�� /�un 1� � um 1� �� / 1 ,d

s

t

3

�

� �

�

Jm t s�� /t 1��) *- . / m 1�

/� e

m 1� t 1�� 1d

s

t

3�J

mt s�� J

mt ��

; m 1� t 1��

Jm

t �� m 1�

1� /� 1 // ; /�e

;� ;d

0

�

3�) *+ ,+ ,- .

m 1�

1� /� 1 k�� #�

un

t� � um

t� �� / Pn

Pm

�� u0 /a1 / R��

m 1�1 /��

a2 / R�� t 1�� /�u

n1� � u

m1� �� / 1 .d

s

t

3

� �

�

�

un

t� � um

t� �� / C Pn

Pm

�� u0 /1

m 1�1 /�

��) *- .�


2

C

h

a

p

t

e

r

for all , where is a constant depending on , , and . It is alsoevident that estimate (2.13) remains true for . It means that the sequence ofapproximate solutions is a Cauchy sequence in the space .Therefore, there exists an element such that equation(2.11) holds. Estimates (2.10) and (2.11) enable us to pass to the limit in (2.9) andto obtain equation (2.3) for . Theorem 2.2 is proved.

Show that if the hypotheses of Theorem 2.2 hold, then the es-timate

is valid with . Here and are constants independentof , , and .

The following assertion provides a simple sufficient condition of the global solvabilityof problem (2.1).

Theorem 2.3.

Assume that the constant in Assume that the constant in Assume that the constant in Assume that the constant in (2.2) does not depend on does not depend on does not depend on does not depend on ,,,, i.e. the i.e. the i.e. the i.e. the

mapping satisfies the global Lipschitz conditionmapping satisfies the global Lipschitz conditionmapping satisfies the global Lipschitz conditionmapping satisfies the global Lipschitz condition

(2.14)

for all with some constant for all with some constant for all with some constant for all with some constant .... Then problem Then problem Then problem Then problem (2.1) has a unique has a unique has a unique has a unique

mild solution on the half-interval mild solution on the half-interval mild solution on the half-interval mild solution on the half-interval ,,,, provided provided provided provided .... Moreover, Moreover, Moreover, Moreover,

for any two solutions and the estimate for any two solutions and the estimate for any two solutions and the estimate for any two solutions and the estimate

,,,, ,,,, (2.15)

holds, where and holds, where and holds, where and holds, where and are constants that depend on are constants that depend on are constants that depend on are constants that depend on ,,,, ,,,, and only and only and only and only....

The proof of this theorem is based on the following lemma (see the book by Henry [3],Chapter 7).

Lemma 2.1.

Assume that is a continuous nonnegative function on the interval

such that

, , (2.16)

where and . Then there exists a constant

such that

. (2.17)

t s s T�� C 0! / R T

/ 0�u

mt� � � C s s T �/2��

u t� � C s s T �/2��

u t� �

E x e r c i s e 2.6

u t� � um

t� �� s s T�� max

a1

m 1�/ ��

�� 1 Pm

�� u0 /

a2

m 1�1 ��

0 � /� � a1 a2/ R T

M :� � :B u t��

B u1 t�� B u2 t�� M u1 u2� /�

uj

�/� M 0!s �� u0 �/�

u1 u2

u1 t� � u2 t� �� / a1 ea2 t s��

u1 s� � u2 s� �� /� t s&

a1 a2 / 1 M

5 t� �0 T��

5 t� � c0 t70�

c1 t 1�� 71�5 1� � 1d

0

t

3�� t 0 T��

c0 c1� 0& 0 70 71 1�� K �K 71 c1 T� ��

5 t� �c0

1 70�� t70�

K 71 c1 T� ��


Proof of Theorem 2.3.

Let be a solution to problem (2.1) on the maximal half-interval of its exis-tence . Assume that . Condition (2.14) gives us that

for all and . Therefore, from (2.3) and (1.15) we find that

.

Hence, for we have that

.

Therefore, Lemma 2.1 implies that the value is bounded on whichis impossible (see Exercise 2.1). Thus, the solution exists for any half-interval

. For the proof of estimate (2.15) we note that, as above, inequalities (2.3)and (1.15) for the function give us that

.

If we apply Lemma 2.1, we find that

for .

Therefore, the estimate

holds for , where is natural and . Thus, Theorem 2.3

is proved.

Using estimate (1.14), prove that if the hypotheses of Theo-rem 2.3 hold, then the inequality

(2.18)

is valid for any two solutions and . Here and is the orthoprojector onto the span of , the num-ber is the same as in (2.15) and depends on , , and .

Let us consider one more case in which we can guarantee the global solvabilityof problem (2.1). Assume that condition (2.2) holds for and

u t� �s s T�� T ��

B u t�� B 0 t�� M u /� M0 T� � M u /��

u �/� t 0 T��

u t� � / u0 //

t 1��) *- . / 1

/� M0 T� � M u 1� � /�� 1d

s

t

3��

t s s T��

u t� � / C0 u0 / T /� �� C1 T /�� t 1�� /�u 1� � / 1d

s

t

3��

u t� � / s s T��

s s T�� w t� � u1 t� � u2 t� ��

w t� � / w s� � //

t 1��) *- . / 1

/� M w 1� � / 1d

s

t

3��

w t� � / C / 1 M� �� w s� �� s t s 1��

w t� � / C n 1� w s� � / C t s�� Cln �exp w s� � /� �

t s n $� �� n 0 $ 1� �

E x e r c i s e 2.7

QN

u1 t� � u2 t� �� /

e

N 1� t s�� a3

N 1�1 /�

�� ea2 t s��

� ��

u1 s� � u2 s� �� /

�

�

u1 t� � u2 t� � QN I PN��PN e1 � eN� �a2 a3 1 / M

/ 1 24�


2

C

h

a

p

t

e

r

, (2.19)

where satisfies the global Lipschitz condition (2.14) with andis a potential operator on the space . This means that there exists

a Frechét differentiable functional on such that , i.e.

.

Theorem 2.4.

Let Let Let Let (2.2) be valid with be valid with be valid with be valid with and let decomposition and let decomposition and let decomposition and let decomposition (2.19) take place. take place. take place. take place.

Assume that the functional is bounded below on Assume that the functional is bounded below on Assume that the functional is bounded below on Assume that the functional is bounded below on .... Then prob- Then prob- Then prob- Then prob-

lem lem lem lem (2.1) has a unique mild solution on an has a unique mild solution on an has a unique mild solution on an has a unique mild solution on an

arbitrary segment arbitrary segment arbitrary segment arbitrary segment ....

Proof.

Let be an approximate solution to problem (2.1) on a segment ,where does not depend on (see Exercise 2.5):

, . (2.20)

Multiplying (2.20) by scalarwise in the space , we find that

Since is bounded below, we obtain that

with constants and independent of , where

.

Therefore, Gronwall’s lemma gives us that

for all in the segment of the existence of approximate solutions. Firstly,this estimate enables us to prove the global existence of approximate solutions(cf. Exercise 2.1). Secondly, by virtue of the continuity of the functional on

Theorem 2.2 enables us to pass to the limit on an arbitrary seg-ment and prove the global solvability of limit problem (2.1). Theorem 2.4

is proved.

B u� � B0 u� �� B1 u t��

B1 u t�� / 1 24�B0 u� � V �1 24�

F u� � V B0 u� � F � u� ��

1v 1 24�� F u v�� F u� �� B0 u� � v��

v 1 24 0�lim 0�

/ 1 24�F u� � V �1 24�

u t� � C s s T�� D A1 2�� 2��s s T��

um

t� � s s T�� T m

dtd��u

mt� � Au

mt� �� P

mB u

mt� � t�� u

ms� � P

mu0�

u�m t� � dtd

��um

t� �� H

u�m2 d

td��

12�� A1 2� u

m2

F um

� ��

� B1 um

t�� u�m��

12�� u�m

2B1 0 t�� 2

M2 A1 2� um

2 .� �

�

�

�

F u� �

dtd

��W um

t� �� a W um

t� �� b B1 0 t�� 2� ��

a b m

W u� � 12�� A1 2� u

2F u� ��

W um

t� �� W um

s� �� ba��) *

- . ea t s�� 2 ea t 1�� B1 0 1�� 2 1d

s

t

3��

t s s T��

W

V �1 24� m ��s s T��

Examples 93

Let be a mild solution to problem (2.1) such that for . Use Lemma 2.1 to prove that

, (2.21)

for all , where is a positive constant.

Let and be solutions to problem (2.1) with the ini-tial conditions and such that for lying in a segment . Then

,

, .

Thus, if and the hypotheses of Theorem 2.3 or 2.4 hold, then equa-tion (2.1) generates a dynamical system with the evolutionary operator which is defined by the equality , where is the solution to problem(2.1). The semigroup property of follows from the assertion on the uniquenessof solution.

Show that Theorem 2.4 holds even if we replace the assump-tion of semiboundedness of by the condition

for some and .

§ 3 Examples§ 3 Examples§ 3 Examples§ 3 Examples

Here we consider several examples of an application of theorems of Section 2. Ourpresentation is brief here and is organized in several cycles of exercises. More de-tailed considerations as well as other examples can be found in the books by Henry,Babin and Vishik, and Temam from the list of references to Chapter 2 (see also Sec-tions 6 and 7 of this chapter).

We first remind some definitions and notations. Let be a domain in . The Sobolev space of the order is defined

by the formula

,

where , , ,

, .

The space is a separable Hilbert space with the inner product

E x e r c i s e 2.8 u t� �u t� � / C

T� s t s T��

u t� � � CT

t / �� u0 /� t s s T��

/ � 1�� CT

CT� /��

E x e r c i s e 2.9 u t� � v t� �u0 v0� �/� u t� � / v t� � /� CT�

t s s T��

u t� � v t� �� CT / �� t / �� u0 v0� /�

t s s T�� / � 1��

B u t�� B u� �#�/ S

t�� S

t

Stu0 u t� �� u t� �S

t


F u� � F u� � &� A1 2� u

2� �& � 1 24� 0!

Rd

d 1&� � Hm � � m m 0 1 2 ��

Hm � � f L2 � � : D j f L2 � � j m��

j j1 j2 � jd

� � �� jk

0 1 2 �� j j1 � jd

� ��

D j f �x1

j1 �x2

j2 ��x

d

jd f x� �� x x1 � x

d� ��

Hm � �


2

C

h

a

p

t

e

r

.

Below we also use the space which is constructed as the closure in of the set of infinitely differentiable functions with compact support. Formore detailed information we refer the reader to the handbooks on the theory of So-bolev spaces.

E x a m p l e 3.1 (nonlinear heat equation)

(3.1)

Here is a bounded domain in , is the Laplace operator, and is a posi-tive constant. Assume that is a continuous function of its vari-ables which satisfies the Lipschitz condition

(3.2)

with an absolute constant . It is clear that the operator defined bythe formula

,

can be estimated as follows:

. (3.3)

Here and below is the norm in the space . It is well-known thatthe operator with the Dirichlet boundary condition on is a positiveoperator with discrete spectrum. Its domain is . More-over, . We also note that

, .

Therefore, equation (3.3) for gives us the estimate

,

where is the first eigenvalue of the operator with the Dirichlet boundarycondition on . Therefore, we can apply Theorem 2.3 with to prob-lem (3.1). This theorem guarantees the existence and uniqueness of a mild solu-tion to problem (3.1) in the space .

u v�� m

D j u x� � D j v x� � xd

3

j m��

H0m � � Hm � �

C0� � �

�tu �<u f t x u =u� � �� x t 0 ,!��

u � 0 ut 0�� u0 .� �

�8�8�

Rd < �f t x u p� � ��

f t x u p� � �� f t x u q� � �� K u1 u2� 2p

jq

j� 2

j 1�

d

��) *+ ,- .

1 2�

�

K B u t��

B u t�� x� � f t x u x� � =u x� ��

B u1 t�� B u2 t�� K u1 u2� 2 �xj

u1 u2�� 2

j��) *

- .1 2��

. H L2 � ��A <��

D A� � H2 � � H01 � �0�

D A1 2�� H01 � ��

A1 2� u2

Au u�� u�x

j��) *- . 2

xd

3

j 1�

d

�� u D A1 2�� H01 � ��

B u t��

B u1 t�� B u2 t�� K1 1�� 1�) *

- .1 2�A1 2� u�

1 <�� / 1 24�

C R� H01 � ��

Examples 95

Assume that is a continuous func-tion of its arguments satisfying the global Lipschitz condition withrespect to the variable . Prove the global theorem on the existenceof mild solutions to problem (3.1) in the space .

E x a m p l e 3.2

Let us consider problem (3.1) in the case of one spatial variable:

(3.4)

Assume that is a continuously differentiable function with respect tothe variable and , where is a function bounded onevery compact set of the real axis. We also assume that the function is continuous and possesses property (3.2). For any element

the following estimates hold:

and ,

where . Therefore, it is easy to find that the inequality

(3.5)

is valid for

,

provided Here is the norm in the space and . Equation (3.5) and Theo-

rem 2.1 guarantee the local solvability of problem (3.4) in the space .Moreover, if the function

is bounded below, then we can use Theorem 2.4 to obtain the assertion on theexistence of mild solutions to problem (3.4) on an arbitrary segment .

It should be noted that the reasoning in Example 3.2 is also valid for several spatialvariables. However, in order to ensure the fulfilment of the estimate of the form (3.5)one should impose additional conditions on the growth of the function .For example, we can require that the equation

be fulfilled, where if and is an arbitrary number if .In this case the inequality of the form (3.5) follows from the Hölder inequality and

E x e r c i s e 3.1 f t x u p� � �� f t x u� �� #

u

H L2 � ��

�t u ��x2

u g x u�� f t x u �x u� � �� , t 0 x 0 1�� !��

ux 0� u

x 1� 0 ut 0�� u0 .� � �

�8�8�

g x u�� u g

u� x u�� h u� �� h u� �

f t x u p� � �� u D A1 2��

H01 0 1��

u x� �0 1�� max u �

L2 0 1�� u

L2 0 1�� u �

L2 0 1��

u� �x u�

B u1 t�� B u2 t�� M :� � A1 2� u1 u2��

B u t�� g x u x� �� f t x u x� � u � x� ��

A1 2� uj

uj� : .�# . H �

L2 0 1�� M :� � h ;� � : ; :� �sup K 2��H0

1 � �

� x y�� g x ;�� ;d

0

y

3�

0 T��

h u� �

h u� � C1 C2 u p��

p 2 d 2�� 4� d 2! p d 2�


2

C

h

a

p

t

e

r

the continuity of the embedding of the space into , where if and is an arbitrary number if , .

The results shown in Examples 3.1 and 3.2 also hold for the systems of parabo-lic equations. For example, a system of reaction-diffusion equations

(3.6)

can be considered in a smooth bounded domain . Here and is a continuous function from into such

that

, (3.7)

where is a constant, , and is an outer normal to .

Prove the global theorem on the existence and uniqueness ofmild solutions to problem (3.6) in

.

E x a m p l e 3.3 (nonlocal Burgers equation).

(3.8)

Here is a continuous function with the values in ,

,

and is a positive parameter. Exercises 3.3–3.6 below answer the questionon the solvability of problem (3.8).

Prove the local existence of mild solutions to problem (3.8)in the space . Hint:

, ,

.

Consider the Galerkin approximate solutions to problem (3.8)

.

Write out the system of ordinary differential equations to determine. Prove that this system is locally solvable.

H1 � � Lq � � q �2 d d 2�� 4� d dim 2!� q d 2� q 1&

�t u �<u f t u =u� �� ,��

u�n��

�0 , u

t 0� u0 x� � ,� ��8�8�

Rd% u u1 u2 �� u

m� f t u ;� �� R� R m 1�� d9 Rm

f t u ;� �� f t v >� �� M u v� ; >��

M u v� Rm� ; >� Rm d� n �

E x e r c i s e 3.2

�1 24 H1 � �� mH1 � � �99#�

H1 � �9

ut

�ux x

? u�� ux

�� f t� � , 0 x l , t 0!� ��

ux 0� u

x l� 0 , ut 0� u0 .� � ��

��

f t� � L2 0 l��

? L2 0 l�� ? u�� ? x� �u x t�� xd

0

l

3��

E x e r c i s e 3.3

�1 24 H01 0 l��

A ��x x2�� D A� � H2 0 l�� H0

1 0 l�� 0�

B u� � ? u�� ux� f t� ��

E x e r c i s e 3.4

um t� � um t x�� # 2l�� gk t� � @k

l�� xsin

k 1�

m

�"�

gk

t� � �

Examples 97

Prove that the equations

(3.9)

and

(3.10)

are valid for any interval of the existence of the approximate solution. Here is the norm in .

Use equations (3.9) and (3.10) to prove the global existenceof the Galerkin approximate solutions to problem (3.8) and to obtainthe uniform estimate of the form

, (3.11)

for any .

Thus, Theorem 2.2 guarantees the global existence and uniqueness of weak solu-tions to problem (3.8) in .

E x a m p l e 3.4 (Cahn-Hilliard equation).

(3.12)

where , and are constants. The result of the cycle of Exercises3.7–3.10 is a theorem on the existence and uniqueness of solutions to problem(3.12).

Prove that the estimate

is valid for any two functions and smooth on . Usethis estimate to ascertain that problem (3.12) is locally uniquelysolvable in the space

(Hint: , ).

E x e r c i s e 3.5

12��

dtd

�� um

t� � 2 � �x

um

t� � 2� f t� � um

t� ��

dtd

�� x

um

t� � 2 � �x x2

um

t� � 2�

2�� ? u�� 2 �x um t� � 2

f t� � 2�) *- .

�

�

um t� � . L2 0 l��

E x e r c i s e 3.6

�x um t� � C �x u0 T�� t 0 T��

T 0!

�1 24 H01 0 l��

ut

��x4

u �x2

u3 a u2 b u� �� 0 , x 0 l�� , t 0 ,!�

�x ux 0� �x

3u

x 0� 0 , �x ux l� �x

3u

x l� 0 ,� � � �

ut 0� u0 x� � ,��

88�88�

� 0! a , b R�

E x e r c i s e 3.7

�x2

u v"� � 2C u

2 �x2

u2�� v

2 �x2

v2��

u x� � v x� � 0 l��

V u H2 0 l�� : ux

x 0�u

xx l�

0� ��

�

� �x4 1 A�� V D A1 2��


2

C

h

a

p

t

e

r

Let us consider the Galerkin approximate solution to problem (3.12):

.

Prove that the equality

(3.13)

holds for any interval of the existence of approximate solutions. Here and

. (3.14)

In particular, equation (3.13) implies that approximate solutions exist for any seg-ment . For they can be estimated as follows:

, , (3.15)

where the number does not depend on .

Using (3.15) show that the inequality

,

holds for any interval and for any approximate solution to problem (3.12). (Hint: first prove that

).

Using Theorem 2.2 and the result of the previous exercise,prove the global theorem on the existence and uniqueness of weaksolutions to the Cahn-Hilliard equation (3.12) in the space (seeExercise 3.7).

E x a m p l e 3.5 (abstract form of two-dimensional system

of Navier-Stokes equations)

In a separable Hilbert space we consider the evolutionary equation

, , (3.16)

um

t� �

um

t� � um

t x�� # 1l

�� g0 t� � 2l�� g

kt� � @k

l�� xcos

k 1�

m

�"��

E x e r c i s e 3.8

dtd��W u

mt� ��

x� �

x2

um

t x�� p um

t x�� ) *- .2

xd

0

l

3� 0�

um

t� � � p u� � u3 a u2 b� ��

W u� � �2��u

x2 1

4��u4 a

3��u3 b

2��u2� � �) *

- . xd

0

l

3�

0 T�� u0 V�

�x um t� � um t x�� x 0 l��

max� CT� t 0 T��

CT

m

E x e r c i s e 3.9

dtd��

x2

um

t� � 2 �x4

um

t� � 2� CT

1 �x2

um

t� � 2�� t 0 T� ��

0 T�� u

mt� � u �

L4 0 l�� 2 �

C u x� � u �L2 0 l��

"0 l��

max�


V

H

udtd

�� Au b u u�� f t� �� ut 0� u0�

Examples 99

where is a positive operator with discrete spectrum, is a positive parame-ter and is a bilinear mapping from into

possessing the property

for all (3.17)

and such that for all the estimates

, (3.18)

and

, , (3.19)

hold. We also assume that is a continuous function with the values in .

Prove that Theorem 2.1 on the local solvability with is applicable to problem (3.16), where is a number

from the interval .

Let be the basis of eigenvectors of the operator , let bethe corresponding eigenvalues and let be the orthoprojector onto the spanof . We consider the Galerkin approximations of problem (3.16):

(3.20)

Show that the estimates

(3.21)

and

(3.22)

are valid for an arbitrary interval of the existence of solutions to prob-lem (3.20). Here . Using these estimates, provethe global solvability of problem (3.20).

Show that the estimate

(3.23)

holds for a solution to problem (3.20).

A �b u v�� D A1 2�� D A1 2�� 9 � 1 24� �

D A 1 2��

b u v�� v�� 0� u v D A1 2��

u v D A� ��

b u v�� C� A1 24 ��

u A1 24 ��

v"� 0 � 12��

A b u v�� C� � A1 24� � ��

u A1 24 �

v"� 0 12�� 0 � 1

2��

f t� � H

E x e r c i s e 3.11 / �1 24 ��

0 1 24��

ek

� A 0 1 2 �� P

m

e1 � em

� � �

dtd��um t� � �Aum t� � Pm b um t� � um t� �� Pm f t� � ,�

um

0� � Pm

u0 .��8�8�


um

t� � 2 � A1 2� um1� � 2 1d

0

t

3� u02 1

1�� f 1� � 2 1d

0

t

3��

um

t� � 2u

m0� � 2

e 1� t� 1

1��

F��) *- .2

1 e 1� t��

F A 1 2�� f t� �t 0&sup�


dtd�� A1 2� u

mt� � 2 � Au

mt� � 2�

2�� b u

mt� � u

mt� �� 2

f t� � 2��

�

�


2

C

h

a

p

t

e

r

Using the interpolation inequality (see Exercise 1.13) and estimate (3.18), it is easyto find that

.

Therefore, equation (3.23) implies that

, (3.24)

where . Hence, Gronwall’s inequality givesus that

.

It follows from equation (3.21) that the value is uniformly bounded withrespect to on an arbitrary segment . Consequently, the uniform estimates

, , (3.25)

are valid for any and .

Using equations (3.23) and (3.25), prove that if then

,

for any .

Let and let . Prove that

Use the results of Exercises 3.14 and 3.15 and inequality(3.19) to prove the global existence of mild solutions to problem(3.16) in the space , provided that and is a continuous function with the values in . Here

is an arbitrary number from the interval .

b u u�� 2C u A1 2� u

2Au" C

��) *- .2

u2

A1 2� u4 �2

4�� Au

2��

dtd�� A1 2� u

mt� � 2 $

mt� � A1 2� u

mt� � 2 2

�� f t� � 2��

$m

t� � 2 C2 �34� � um

t� � 2A1 2� u

mt� � 2"�

A1 2� um

t� � 2A1 2� u0

2 $m1� � 1d

0

t

3� �8 8� �8 8� �

exp 2�� f 1� � 2 $

m;� � ;d

0

t

3� �8 8� �8 8� �

exp 1d

0

t

3��

$m 1� � 1d0t3

m 0 T��

A1 2� um t� � CT� t 0 T�� u0 D A1 2��

T 0! m 1 2 ��

E x e r c i s e 3.14 u0 �D A1 2�� ,�

dtd�� A1 2� u

mt� � 2 �

2�� Au

mt� � 2� C

T� t 0 T��

T 0!

E x e r c i s e 3.15 0 12�� A f t� � 2

0 T�� sup C

T�

dtd

�� A1 24 �

um t� � 2 � A1 � um t� � 2�

C1 A b um

t� � um

t� �� 2C

T .�

�

�


D A1 24 �� u0 D A1 24 �� f t� � D A � �

0 1 24��

E x i s t e n c e C o n d i t i o n s a n d P r o p e r t i e s o f G l o b a l A t t r a c t o r 101

§ 4 Existence Conditions and Properties§ 4 Existence Conditions and Properties§ 4 Existence Conditions and Properties§ 4 Existence Conditions and Properties

of Global Attractorof Global Attractorof Global Attractorof Global Attractor

In this section we study the asymptotic properties of the dynamical system genera-ted by the autonomous equation

, , (4.1)

where, as before, is a positive operator with discrete spectrum and isa nonlinear mapping from into such that

(4.2)

for all possessing the property , .We assume that problem (4.1) has a unique mild (in ) solution on for any

. The theorems that guarantee the fulfilment of this requirement and alsosome examples are given in Sections 2 and 3 of this chapter. It should be also notedthat in this section we use some results of Chapter 1 for the proof of main assertions.Further triple numeration is used in the references to the assertions and formulaeof Chapter 1 (first digit is the chapter number).

Let be a dynamical system with the evolutionary operator definedby the formula , where is a mild solution to problem (4.1).As shown in Chapter 1, for the system to possess a compact global attrac-tor, it should be dissipative. It turns out that the condition of dissipativity is not onlynecessary, but also sufficient in the class of systems considered.

Lemma 4.1.

Let be dissipative and let be its absorbing

ball. Then the set is absorbing for all ,

where

. (4.3)

Proof.

Using equation (2.3) and estimate (1.17), we have

,

where . Let be a bounded set in . Then for all and . Therefore, the estimate

,

holds for . Hence, for all . This implies theassertion of the lemma.

udtd

�� Au� B u� �� ut 0� u0�

A B u� ��/ H

B u1� � B u2� �� M :� � A/ u1 u2��

u1 u2 �/�� D A/� �� A/uj

:� 0 / 1��/ R�

u0 �/�

�/ St

�� St

Stu0 u t� �� u t� �

�/ St

��

�/ St�� B0 u : A/u R0� ��B� u : A�u R�� / 1��

R� � /�� /�R0

��1 �� B u� � : u �/ A/u R0�� sup��

A�u t 1�� /�� /�A/u t� � �

t 1 1��) *- .� B u t� �� td

t

t 1�

3��

u t� � Stu0� B �/ S

ty / R0�

t tB

& y B�

B u t� �� B u� � : u B0� �sup� t tB

&

u t� � Sty� A�u t 1�� R�� t t

B&


2

C

h

a

p

t

e

r

Theorem 4.1.

Assume that the dynamical system generated by problem Assume that the dynamical system generated by problem Assume that the dynamical system generated by problem Assume that the dynamical system generated by problem (4.1)isisisis dissipative. Then it is compact and possesses a connected compact globaldissipative. Then it is compact and possesses a connected compact globaldissipative. Then it is compact and possesses a connected compact globaldissipative. Then it is compact and possesses a connected compact global

attractor attractor attractor attractor .... This attractor is a bounded set in for and has This attractor is a bounded set in for and has This attractor is a bounded set in for and has This attractor is a bounded set in for and has

a finite fractal dimensiona finite fractal dimensiona finite fractal dimensiona finite fractal dimension....

Proof.

By virtue of Theorem 1.1 the space is compactly embedded into for. Therefore, Lemma 4.1 implies that the dynamical system is com-

pact. Hence, Theorem 1.5.1 gives us that possesses a connected compactglobal attractor . Evidently, for all , where is definedby equality (4.3). Thus, we should only establish the finite dimensionality of the at-tractor. Let us apply the method used in the proof of Theorem 1.9.1 and based onTheorem 1.8.1. Let and be semitrajectories of the dynamical system

such that for all , . Then equations (2.3) and(1.17) give us that

.

Using Lemma 2.1, we find that for all such that the followingestimate holds:

for ,

where the constant depends on and . Therefore,

for

for the considered and . Consequently,

,

where is an integer part of the number . Thus, the estimate of the form

(4.4)

is valid, where the constants and depend on and . Similarly, Lemma 1.1gives

(4.5)

for all such that , where and is the ortho-projector onto the first eigenvectors of the operator . If we substitute equation(4.4) in the right-hand side of inequality (4.5), then we obtain that

�/ St

��

� �� / � 1��

�� /� /! �/ S

t��

�/ St

�� A�u R�� u �� R�

Stu1 S

tu2

�/ St

�� Stu

j / R� t 0& j 1 2��

Stu1 S

tu2� / u1 u2� / M R� � // t 1�� /� S

tu1 S

tu2� / td

0

t

3��

uj

�/� Stu

j / R�

Stu1 S

tu2� / C u1 u2� /� 0 t 1� �

C / R

Stu1 S

tu2� / C S1u1 S1u2� /� 0 1 t 1 1��

u1 u2

St u1 St u2� / C S t� �u1 S t� �u2� / C t� � 1� u1 u2� /��

t� � t

Stu1 S

tu2� / C ea t u1 u2� /�

C a / R

QN

Stu1 S

tu2�� / e

N 1� t�

u1 u2� /

M R� � /t 1��) *- ./ N 1�

/� e

N 1� t 1�� S1u1 S1u2� / 1d

0

t

3

�

�

�

u1 u2� �/� St uj / R� QN I PN�� PN

N A


,

where

.

After the change of variable it is easy to find that . Therefore,

(4.6)

for all such that . Hence, there exist and suchthat

, ,

for all such that . However, the attractor lies in the ab-sorbing ball . Therefore, this estimate and inequality (4.4) mean that the hypo-theses of Theorem 1.8.1 hold for the mapping . Hence, the attractor hasa finite fractal dimension. It can be estimated with the help of the parameters in in-equalities (4.4) and (4.6). Thus, Theorem 4.1 is proved.

Equations (4.4) and (4.6) which are valid for any enable us to prove the exis-tence of a fractal exponential attractor of the dynamical system in the sameway as in Section 9 of Chapter 1.

Theorem 4.2.

Assume that the dynamical system generated by problem Assume that the dynamical system generated by problem Assume that the dynamical system generated by problem Assume that the dynamical system generated by problem (4.1)is dissipative. Then it possesses a fractal exponential attractor (is dissipative. Then it possesses a fractal exponential attractor (is dissipative. Then it possesses a fractal exponential attractor (is dissipative. Then it possesses a fractal exponential attractor (inertialinertialinertialinertial

set).set).set).set).

Proof.

It is sufficient to verify that the hypotheses of Theorem 1.9.2 (see (1.9.12)–(1.9.14)) hold for . Let us show that

,

can be taken for the compact in (1.9.12)–(1.9.14). Here is a number from theinterval and is defined by formula (4.3). We choose the parameter

such that for . Since is a bounded invariant set,equation (4.4) is valid for any with some constants and . It is alsoeasy to verify that is a compact. Indeed, let . Then , there-with we can assume that , and either or

. In the first case with the help of (4.4) we have

QN

Stu1 S

tu2�� / e

N 1� t�

C M R� � ea t JN

t� �� u1 u2� /�

JN t� � /t 1��) *- ./ N 1�

/� e

N 1� t 1�� 1d

0

t

3�; N 1� t 1�� JN t� � �

C0 N 1�1� /��

QN St u1 St u2�� / e

N 1� t� C R /��

N 1�1 /�� ea t�

) *+ ,- .

u1 u2� /�

u1 u2� �/� St uj / R� t0 0! N

QN St0u1 St0

u2�� /

� u1 u2� /� 0 � 1� �

u1 u2� �/� Stu

j / R� �

B0V St0� �

R 0!�/ S

t��

�/ St

��

�/ St

��

K St B�t t0&A� B� u : A�u R��

K �/ 1�� R�

t0 t0 B�� StK B�% t t0& K

u1 u2� K� C a

K kn

� K% kn

S� tny

n

kn

w� yn

y B�� tn

t*

��tn

��


2

C

h

a

p

t

e

r

.

Therefore, . In the second case

.

Here is the omega-limit set for the semitrajectories emanating from .Thus, is a compact invariant absorbing set. In particular this means that condition(1.9.12) is fulfilled, therewith we can take any number for . Conditions (1.9.13)and (1.9.14) follow from equations (4.4) and (4.6). Consequently, it is sufficientto apply Theorem 1.9.2 to conclude the proof of Theorem 4.2.

Thus, the dissipativity of the dynamical system generated by problem (4.1)guarantees the existence of a finite-dimensional global attractor and an inertial set.Under some additional conditions concerning the requirement of dissipativitycan be slightly weakened. We give the following definition. Let . The dynami-cal system is said to be -dissipative dissipative dissipative dissipative if there exists such thatfor any set bounded in there exists such that

for all and .

Lemma 4.2.

Assume that satisfies the global Lipschitz condition

. (4.7)

Let the dynamical system generated by mild solutions to prob-

lem (4.1) be -dissipative for some . Then is

a compact dynamical system, i.e. it possesses an absorbing set which is

compact in .

Proof.

By virtue of Lemma 4.1 it is sufficient to verify that the system isdissipative (i.e. -dissipative). If we use expression (2.3) and equation (1.17),then we obtain

for positive and . Here . Since wehave the estimate

Stn

yn

St*y�/

C ea t

n yn

y� / Stn

y St*y�/

��

kn S�tn

yn St*y K��

w Stn

yn

n ��lim ? B�� S

t0B�� K% %��

? B�� B�K

7 0!

�/ St

��

B u� �� /�

�/ St

�� R�* 0!

B �� t0 t0 B� ��

A�Sty S

ty � R�

*�# y B �/0� t t0&

B u� �

B u1� � B u2� �� M A/ u1 u2��

�/ St

�� / 1 /�� / S

t��

�/

�/ St

�� /

A/u t s�� / ��s

��) *- ./ ��

A�u t� � /t s 1��) *- ./ B u 1� �� 1d

t

t s�

3��

t s u t� � Stu0� B u� � B 0� � M A/u ,��

A/u t s�� 1s��) *- ./ ��

/ �� / �� A�u t� � B 0� � //1 /��

� ��

��


for . Hence, we can apply Lemma 2.1 to obtain

, .

Therefore, if for and , then the latter in-equality gives us that

for ,

i.e. is a dissipative system. Lemma 4.2 is proved.

Show that the assertion of Lemma 4.2 holds if instead of (4.7)we suppose that , where possessesproperty (4.7) and is such that

.

The following assertion contains a sufficient condition of dissipativity of the dynami-cal system generated by problem (4.1).

Theorem 4.3.

Assume that condition Assume that condition Assume that condition Assume that condition (4.2) is fulfilled with and is fulfilled with and is fulfilled with and is fulfilled with and

is a potential operator from into (the prime stands for is a potential operator from into (the prime stands for is a potential operator from into (the prime stands for is a potential operator from into (the prime stands for

the Frechét derivative)the Frechét derivative)the Frechét derivative)the Frechét derivative).... Let Let Let Let

,,,, (4.8)

for all for all for all for all ,,,, where where where where ,,,, ,,,, ,,,, and and and and are real parameters, therewithare real parameters, therewithare real parameters, therewithare real parameters, therewith

and and and and .... Then the dynamical system is dissipative in Then the dynamical system is dissipative in Then the dynamical system is dissipative in Then the dynamical system is dissipative in ....

Proof.

In view of Theorem 2.4 conditions (4.8) guarantee the existence of the evolu-tionary operator . Let us verify the dissipativity. As in the proof of Theorem 2.4 weconsider the Galerkin approximations . It is evident that

and

.

If we add these equations and use (4.8), then we obtain that

.

M/

s 1��) *- ./ A/u t 1�� 1d

0

s

3�

0 s 1� �

A/u t s�� C / � M� �� 1 A�u t� �� s� /�� 0 s 1� �

St u0 � R�*� t t0 B� �& u0 B �/0�

Stu0 / C / � M� �� 1 R�

*�� t 1 t0 B� ��&

�/ St

��

E x e r c i s e 4.1

B u� � B1 u� � B2 u� �� B1 u� �B2 u� �

B2 u� � : A�u R�*� � ��sup

/ 1 24� B u� � �F � u� �� 1 24 H

F u� � ��& F � u� � u�� F u� �� 7 A1 2� u2� ��&

u �1 24� � 7 � 0! 7 1� �1 24

St

um

t� � �

12��

dtd�� u

mt� � 2

A1 2� um

t� � 2F � u

m� � u

m�� 0�

dtd

��12�� A1 2� u

mt� � 2

F um

t� �� ) *- . u�m t� � 2� 0�

dtd

��12�� um

2 12�� A1 2� um

2F um� ��

� ��

1 7�� A1 2� um2 F um� � ��


2

C

h

a

p

t

e

r

Let

.

Therefore, it is clear that

with some positive constants and that do not depend on . This (cf. Exer-cise 1.4.1) easily implies the dissipativity of the system , moreover,

. (4.9)

Theorem 4.3 is proved.

Show that the assertion of Theorem 4.3 holds if (4.2) is ful-filled with and

,

where possesses properties (4.8) and satisfies the esti-mate for small enough.

Let us look at the examples of Section 3 again. We assume that the function

(4.10)

in Example 3.1 possesses property (3.2) and

(4.11)

for all , where is the first eigenvalue of the operator with theDirichlet boundary condition on . Here and are positive constants.

Using the Galerkin approximations of problem (3.1) and Lem-ma 4.2, prove that the dynamical system generated by equation (3.1)is dissipative in under conditions (3.2), (4.10), and (4.11).

Therefore, if conditions (3.2), (4.10) and (4.11) are fulfilled, then the dynamical sys-tem generated by a mild (in ) solution to problem (3.1) posses-ses both a finite-dimensional global attractor and an inertial set.

Prove that equation (4.11) holds if

,

where are real constants and the function possessesthe property

V u� � 12�� u

2 12�� A1 2� u

2F u� � ��

dtd�� V u

mt� �� ?V u

mt� �� C�

? C m

�1 24 St

��

12�� A1 2� St u

2V u0� � e ? t� C

?�� 1 e ? t��

E x e r c i s e 4.2

/ 1 24�

B u� � F � u� �� B0 u� ��

F u� � B0 u� �B0 u� � C' ' A1 2� u�� ' 0!

f t x u p� � �� f x u p� �� #

f x u x� � u x� �=� �� u x� � xd

3 1 �� u2 x� � xd

3 C��

u H01 � �� 1 <�

� � C

E x e r c i s e 4.3

H01 � �

H01 � � S

t�� H0

1 � �

E x e r c i s e 4.4

f x u =u� �� f0 x u�� aj

�u

�xj

��

j 1�

d

��

aj

f0 x u��


,

for any .

Consider the dynamical system generated by problem (3.4).In addition to the hypotheses of Example 3.2 we assume that

and the function possesses the pro-perties

,

for some positive , , and . Then the dynamical system genera-ted by (3.4) is dissipative in .

Find the analogue of the result of Exercise 4.5 for the systemof reaction-diffusion equations (3.6).

Using equations (3.9) and (3.10) prove that the dynamicalsystem generated by the nonlocal Burgers equation (3.8) with

is dissipative in .

Let us consider a dynamical system generated by the Cahn-Hilliard equation(3.12). We remind that

.

Let the function be defined by equality (3.14). Showthat for any positive and the set

(4.12)

is a closed invariant subset in for the dynamical system generated by problem (3.12).

Prove that the dynamical system generated bythe Cahn-Hilliard equation on the set defined by (4.12) is dis-sipative (Hint: cf. Exercise 3.9).

In conclusion of this section let us establish the dissipativity of the dynamical system generated by the abstract form of the two-dimensional Navier-

Stokes system (see Example 3.5) under the assumption that .We consider the dynamical system generated by the Galerkin approxi-mations (see (3.20)) of problem (3.16).

f x u�� u 1 �� u2 C�� u R�

x �

E x e r c i s e 4.5

f t x u �x

u� � �� 0# g x u��

g x ;�� ;d

0

y

3 ��& y g x y�� g x ;�� ;d

0

y

3� 7�&

� 7H0

1 0 1��

E x e r c i s e 4.6

E x e r c i s e 4.7

f t� � #f L2 0 l�� # H0

1 � �

V St

��

V u H2 0 l�� : uxx 0�

uxx l�

0� ��

�

E x e r c i s e 4.8 W u� �R �

X� R� u V : W u� � R , u x� � xd

0

l

3 ��

� �8 8� �8 8� �

�

V V St��

E x e r c i s e 4.9 X� R� St

�� X� R�

D A1 24 �� St�� f t� � f D A � ��#

Pm H Stm��


2

C

h

a

p

t

e

r

Using (3.24) prove that

(4.13)

for all , where and are constants independent of .

With the help of (3.21), (3.22) and (4.13) verify the propertyof dissipativity of the system in the space

. Deduce its dissipativity (Hint: see Exercise 3.14–3.16).

Thus, the dynamical system generated by the two-dimensional Navier-Stokes equa-tions possesses both a finite-dimensional compact global attractor and an inertial set.

§ 5 Systems with § 5 Systems with § 5 Systems with § 5 Systems with Lyapunov FunctionLyapunov FunctionLyapunov FunctionLyapunov Function

In this section we consider problem (4.1) on the assumption that condition (4.2)holds with and is a potential operator, i.e. there exists a functional

on such that its Frechét derivative possesses the pro-perty

, . (5.1)

Below we also assume that the conditions

, (5.2)

are fulfilled for all , where , , and . On the onehand, these conditions ensure the existence and uniqueness of mild (in )solutions to problem (4.1) (see Theorem 2.4). On the other hand, they guaranteethe existence of a finite-dimensional global attractor for the dynamical system

generated by problem (4.1) (see Theorem 4.3). Conditions (5.1) and(5.2) enable us to obtain additional information on the structure of attractor.

Theorem 5.1.

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (5.1),,,, (5.2),,,, and and and and (4.2) hold with hold with hold with hold with .... Then Then Then Then

the global attractor of the dynamical system the global attractor of the dynamical system the global attractor of the dynamical system the global attractor of the dynamical system generated bygenerated bygenerated bygenerated by


t A1 2� Stm

u02

c0 A1 2� S1m

u02 1d

0

1

3�) *+ ,+ ,- .

c1 S1m

u02

A1 2� S1m

u0) *- .2

1d

0

1

3� �8 8� �8 8� �

exp9

9�

0 t 1�� c0 c1 m


D A1 24 �� St

�� D A1 24� �

/ 1 24� B u� �F u� � �1 24 D A1 2�� F � u� �

B u� � F � u� �� u D A1 2��

F u� � ��& F � u� � u�� F u� �� 7 A1 2� u2� ��&

u D A1 2�� R�� 0! 7 1�D A1 2��

�

�1 24 St��

/ 1 24�� 1 24 S

t��

S y s t e m s w i t h L y a p u n o v F u n c t i o n 109

problem problem problem problem (4.1) is a bounded set in and it coincides with the un- is a bounded set in and it coincides with the un- is a bounded set in and it coincides with the un- is a bounded set in and it coincides with the un-

stable manifold emanating from the set of fixed points of the system, i.e.stable manifold emanating from the set of fixed points of the system, i.e.stable manifold emanating from the set of fixed points of the system, i.e.stable manifold emanating from the set of fixed points of the system, i.e.

,,,, (5.3)

where (for the definition of see Secti-where (for the definition of see Secti-where (for the definition of see Secti-where (for the definition of see Secti-

onononon 6 of Chapter of Chapter of Chapter of Chapter 1).).).).

The proof of the theorem is based on the following lemmata.

Lemma 5.1.

Assume that a semitrajectory possesses the property

for all , where . Then

(5.4)

for all .

Proof.

For the sake of definiteness we assume that . Equation (2.3) im-plies that

.

Since

,

equation (1.17) gives us that

This implies estimate (5.4).

Lemma 5.2.

There exists such that the set is absorbing

for the dynamical system .

Proof.

By virtue of Theorem 4.3 the system is dissipative. Therefore, itfollows from Lemma 4.1 that is an absorbing set,where

�1 D A� ��

� M� ��

� z D A� � : A z B z� �� M� ��

u t� � St u0�A�u t� � R�� t 0& 1 24 � 1� �

A1 2� u t� � u s� �� C R�� t s� � 1 24��

t s� 0&

t s 0&!

u t� � u s� �� et s�� A�

I�� u s� � et 1�� A�

B u 1� �� 1d

s

t

3��

et s�� A�

I� A e t 1�� A� 1d

s

t

3��

A1 2� u t� � u s� �� c0 t 1�� 3 24� 1 A�u s� �

c1 t 1�� 1 2�� 1 B u 1� �� .1 0&maxd

s

t

3

�

�

d

s

t

3�

R1 0! B1 u : Au R1� ��1 24 St��

�1 24 St

�� B� u : A�u R��


2

C

h

a

p

t

e

r

is defined by (4.3), . Thus, to prove the lemma it is sufficient toconsider semitrajectories possessing the property , .Let us present the solution in the form

Using Lemma 5.1 we find that

.

This implies that

, ,

provided that for . Therefore, the assertion of Lemma 5.2follows from Lemma 4.1.


The boundedness of the attractor in follows from Lemma 5.2. Let usprove (5.3). We consider the Galerkin approximation of solutions to problem(4.1):

, .

Here is the orthoprojector onto the span of . Let

, . (5.5)

It is clear that

.

This implies that

for and for any , where is the orthoprojector onto the span of. With the help of Theorem 2.2 and due to the continuity of the func-

R� 1 24 � 1� �u t� � A�u t� � R�� t 0&

u t� �

u t� � et s�� A�

u s� � et 1�� A�

B u 1� �� B u t� �� 1d

s

t

3

A 1� 1 et s�� A�� B u t� �� .

� �

�

�

Au t� � C0 t s�� 1 2��R1 24 C1 R�� t 1��

32��

1d

s

t

3 C2 R1 24� ��

Au t 1�� C R�� t 0&

A�u t� � R�� t 0&

� D A� �u

mt� �

um t� �d

td�� Au

mt� �� P

mB u

mt� �� u

m0� � P

mu0�

Pm

e1 e2 � em

� � � �

V u� � 12�� Au u�� F u� �� u �1 24� D A1 2��

dtd��V u

mt� �� Au

mt� � F � u

mt� �� u�m t� �� P

mAu

mt� � B u

mt� �� 2��

V um

t� �� V um

s� �� Pm

Aum1� � B u

m1� �� 2 1d

s

t

3� ��

PN

Aum1� � B u

m1� �� 2 1d

s

t

3��

t s& N m� PN

e1 � eN

� � �


tional we can pass to the limit in the latter equation. As a result,we obtain the estimate

, , (5.6)

for any solution to problem (4.1) and for any . If the semitrajectory lies in the attractor , then we can pass to the limit in (5.6)

and obtain the equation

(5.7)

for any and . Equation (5.7) implies that the functional defined by equality (5.5) is the Lyapunov function of the dynamical system

on the attractor . Therefore, Theorem 1.6.1 implies equation (5.3).Theorem 5.1 is proved.

Using (5.6) show that any solution to problem (4.1) with possesses the property

, .

Prove the validity of inequality (5.7) for any solution to prob-lem (4.1).

Using the results of Exercises 5.1 and 1.6.5 show that if thehypotheses of Theorem 5.1 hold, then a global minimal attractor

of the system has the form

.

Prove that the assertions of Theorems 4.3 and 5.1 hold if weconsider the equation

, (5.8)

instead of problem (4.1). Here is an arbitrary element from and possesses properties (5.1), (5.2) and (4.2) with

(Hint: ).

Let be an evolutionary operator of problem (5.8). Showthat for any there exist numbers and suchthat

,

provided , (Hint: ).

V u� � m ��

V u t� �� PN

A u 1� � B u 1� �� 2 1d

s

t

3� V u s� �� t s&

u t� � N 1&u t� � S

tu0� � N ��

V u t� �� Au 1� � B u 1� �� 2 1d

s

t

3� V u s� ��

t s 0& & u t� � �� V u� �

�1 24 St

��

E x e r c i s e 5.1 u t� �u0 �1 24�

Au 1� � 2 1d

0

t

3 �� t 0!

u t� �

E x e r c i s e 5.2

�min �1 24 St

��

�min w D A� � : A w B w� �� 0��

E x e r c i s e 5.3

udtd

�� Au� B u� � h�� ut 0� u0�

h H

B u� � / 1 24�V

hu� � V u� � h u��


R 0! aR

0& bR

0!

A1 2� Stu1 S

tu2�� b

Re

aR

tA1 2� u1 u2��

A1 2� uj

R� j 1 2�� Vh

Stu

j� � V

hu

j� � C

R� �


2

C

h

a

p

t

e

r

Theorem 5.1 and the reasonings of Section 6 of Chapter 1 reduce the question on thestructure of global attractor to the problem of studying the properties of stationarypoints of the dynamical system under consideration. Under some additional condi-tions on the operator it can be proved in general that the number of fixedpoints is finite and all of them are hyperbolic. This enables us to use the results ofSection 6 of Chapter 1 to specify the attractor structure. For some reasons (they willbe clear later) it is convenient to deal with the fixed points of the dynamical systemgenerated by problem (5.8).

Thus we consider the equation

, , (5.9)

where as before is a positive operator with discrete spectrum, is a nonline-ar mapping possessing properties (5.1), (5.2) and (4.2) with , and is anelement of .

Lemma 5.3.

For any problem (5.9) is solvable. If is a bounded set in ,

then the set of solutions to equation (5.9) is bounded in

for . If is compact in , then is compact in .

Proof.

Let us consider a continuous functional

(5.10)

on . Since for all , the functional possesses the property

(5.11)

In particular, this means that is bounded below. Let us consider the func-tional on the subspace ( is the orthoprojector onto the spanof elements , as before). By virtue of (5.11) there exists a mini-mum point of this functional in which obviously satisfies the equa-tion

. (5.12)

Equation (5.11) also implies that

B u� �

L u� � Au B u� ��# h� u D A� ��A B u� �

/ 1 24� h

H

h H� M H

L 1� M� � D A� �h M� M H L 1� M� � D A� �

W u� � 12�� Au u�� F u� � h u��

�1 24 D A1 2�� F u� � ��& u �1 24� W u� �

W u� � 12�� A1 2� u

2 �� A 1 2�� h A1 2� u�

14�� A1 2� u

2 �� A 1 2�� h2 .�

& &

&

W u� �W u� � Pm �1 24 Pm

e1 e2 � em� � �um Pm �1 24

A um Pm B um� �� Pm h�

14�� A1 2� u

m2

W um

� � � A 1 2�� h2� �

W u� � : u Pm

H� �inf � A 1 2�� h2 .� �

� �

�


Hence,

with a constant independent of . Thus, equations (5.12) and (4.2) give usthat

.

Therefore, if , then . This estimate enables us to extracta weakly convergent (in ) subsequence and to pass to the limit as

in (5.12) with the help of Theorem 1.1. Thus, the solvability of equation(5.9) is proved. It is obvious that every limit (in ) point of the sequence

possesses the property

if . (5.13)

This means that the complete preimage of any bounded set in isbounded in . Now we prove that the mapping is properproperproperproper, i.e. the pre-image is compact for a compact . Let be a sequence from

. Then the sequence lies in the compact and therefore thereexist an element and a subsequence such that as . By virtue of (5.13) we can also assume that is a weakly con-vergent sequence in . If we use the equation

, ,

Theorem 1.1, and property (4.2) with , then we can easily prove thatthe sequence strongly converges in to a solution to the equation

. Lemma 5.3 is proved.

Lemma 5.4.

In addition to (5.1), (5.2), and (4.2) with we assume that the

mapping is Frechét differentiable, i.e. for any there

exists a linear bounded operator from into such that

(5.14)

for every , such that . Then the operator

is a Fredholm operator for any .

We remind that a densely defined closed linear operator in is said to be Fred-Fred-Fred-Fred-

holmholmholmholm (of index zero) if(a) its image is closed; and(b) .

Proof of Lemma 5.4.

It is clear that the operator has the structure

A1 2� um F 0� � � A 1 2�� h2� � C 1 A 1 2�� h

2��

C m

Aum

B um

� � h� B 0� � h M A1 2� um

� � A1 2� um

� ��

h R� Aum

C R� ��D A� � umk

�k ��

D A� � u

um

�

Au CR� h R�

L 1� M� � M H

D A� � L

L 1� M� � M un

�L 1� M� � L u

n� � � M

h M� nk

� L unk� � h� 0�

k �� A unk �

D A� �

AunkB u

nk

� �� h� 0� k ��

/ 1 24�unk � D A� � u

L u� � h�

/ 1 24�B .� � u �1 24�

B � u� � �1 24 H

B u v�� B u� �� B � u� � v� o A1 2� v� ��

v �1 24� A1 2� v 1� A B � u� ��u �1 24� D A1 2��

G H

Ker Gdim KerG*dim ��

G A B � u� ��#


2

C

h

a

p

t

e

r

,

where is a bounded operator in . By virtue of Theorem1.1 the operator is compact. This implies the closedness of theimage of . Moreover, it is obvious that

and

.

Therefore, the Fredholm alternative for the compact operator gives us that

.

Let us introduce the notion of a regular valueregular valueregular valueregular value of the operator seen as an ele-ment possessing the property that for every theoperator is invertible on . Lemmata 5.3 and 5.4 enable us to usethe Sard-Smale theorem (for the statement and the proof see, e.g., the book byA. V. Babin and M. I. Vishik [1]) and state that the set of regular values of themapping is an open everywhere dense set in . The followingassertion is valid for regular values of the operator .

Lemma 5.5.

Let be a regular value of the operator . Then the set of solutions to

equation (5.9) is finite.

Proof.

By virtue of Lemma 5.3 the set is compact. Since, the operator is invertible on for . It is also

evident that has a domain . Therefore, by virtue of the uniformboundedness principle is a bounded operator for . Hence,it follows from (5.14) that

for any and in . This implies that for every there exists a vicinitythat does not contain other points of the set . Therefore, the compact set has no condensation points. Hence, consists of a finite number of elements.Lemma 5.5 is proved.

In order to prove the hyperbolicity (for the definition see Section 6 of Chapter 1) offixed points we should first consider linearization of problem (5.8) at these points.Assume that the hypotheses of Lemma 5.4 hold and is a stationary solu-tion. We consider the problem

G A C A1 2�� A1 2� I A 1 2�� C�� A1 2��

C H C B � u� � A 1 2�� K A 1 2�� C�

G

Ker Gdim Ker I K�� dim�

Ker G* dim Ker I K*�� dim�

Ker G* dim Ker G dim ��

L

h H� u L 1� h v : L v� � h� �#�L� u� � A B � u� �� H

�

L u� � Au B u� �� H

L

h L

� v : L v� � h� ��h �� L� u� � A B � u� �� H u ��

L� u� � D A� �A L� u� � 1� u ��

A v w�� AL� v� � 1� L� v� � v w�� "�

AL� v� � 1� B v� � B w� �� B � v� � v w�� o A1 2� v w��

�

� �

v w � v ��

�

v0 D A� ��


, . (5.15)

Its solution can be regarded as a continuous function in which satis-fies the equation

.

If then we can apply Theorem 2.3 on the existence and uniquenessof solution. Let stand for the evolutionary operator of problem (5.15).

Prove that is a compact operator in every space ,.

Prove that for any and the set of points of thespectrum of the operator that are lying outside the disk

is finite and the corresponding eigensubspace is finite-di-mensional.

Assume that is a symmetric operator in . Prove thatthe spectrum of the operator is real.

The next assertion contains the conditions wherein the evolutionary operator be-longs to the class , on the set of stationary points.

Theorem 5.2.

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (5.1),,,, (5.2),,,, and and and and (4.2) are fulfilled with . are fulfilled with . are fulfilled with . are fulfilled with .

Let possess a Let possess a Let possess a Let possess a Frechét derivative inFrechét derivative inFrechét derivative inFrechét derivative in such that for any such that for any such that for any such that for any

,,,, ,,,, (5.16)

provided that provided that provided that provided that ,,,, where the constant depends on where the constant depends on where the constant depends on where the constant depends on

and only. Then the evolutionary operator of problem and only. Then the evolutionary operator of problem and only. Then the evolutionary operator of problem and only. Then the evolutionary operator of problem (5.8) has a has a has a has a

Frechét derivativeFrechét derivativeFrechét derivativeFrechét derivative at every stationary point at every stationary point at every stationary point at every stationary point .... Moreover, Moreover, Moreover, Moreover,

, where , where , where , where is the evolutionary operator of linear problem is the evolutionary operator of linear problem is the evolutionary operator of linear problem is the evolutionary operator of linear problem (5.15)....

Proof.

It is evident that

.

Therefore, using (1.17) and (5.16) we have

udtd

�� A B � v0� �� u� 0� ut 0� u0�

�1 24 D A1 2��

u t� � e t A� u0 et 1�� A�

B � v0� �u 1� � 1d

0

t

3��

u0 D A1 2�� ,�T

t

E x e r c i s e 5.5 Tt ��0 � 1 ,�� t 0!

E x e r c i s e 5.6 : 0! t 0!Tt :

:� �

E x e r c i s e 5.7 B � v0� � H

Tt

St

C1 �� 0 ,!

/ 1 24�B u� � �1 24 R 0!

B u v�� B u� � B � u� � v�� C A1 2� v1 �� 0!

A1 2� v R� C C u R�� u

R St

v0 St v0� �� u�� Tt u� Tt

St

v0 u�� v0� Ttu� e t 1�� A� B S1 v0 u�� B v0� �� B � v0� �T1u�

� ��

1d

0

t

3�


2

C

h

a

p

t

e

r

where . Using the result of Exercise 5.4 we obtain that

if .

Thus,

.

Consequently, Lemma 2.1 gives us the estimate

, .

This implies the assertion of Theorem 5.2.

Assume that the constant in (5.16) depends on only,provided that and . Prove that for any .

The reasoning above leads to the following result on the properties of the set of fixedpoints of problem (5.8).

Theorem 5.3.

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (5.1),,,, (5.2) and and and and (4.2) are fulfilled with are fulfilled with are fulfilled with are fulfilled with

and the operator possesses a and the operator possesses a and the operator possesses a and the operator possesses a Frechét derivativeFrechét derivativeFrechét derivativeFrechét derivative in such that in such that in such that in such that

equation equation equation equation (5.16) holds with the constant depending only on for holds with the constant depending only on for holds with the constant depending only on for holds with the constant depending only on for

and and and and .... Then there exists an open dense (in ) set Then there exists an open dense (in ) set Then there exists an open dense (in ) set Then there exists an open dense (in ) set

such that for the set of fixed points of the system gen- such that for the set of fixed points of the system gen- such that for the set of fixed points of the system gen- such that for the set of fixed points of the system gen-

erated by problem erated by problem erated by problem erated by problem (5.8) is finite. If in addition we assume that is finite. If in addition we assume that is finite. If in addition we assume that is finite. If in addition we assume that is ais ais ais a

symmetric operator for , then fixed points are symmetric operator for , then fixed points are symmetric operator for , then fixed points are symmetric operator for , then fixed points are hyperbolichyperbolichyperbolichyperbolic....

In particular, this theorem means that if then the global attractor of the dy-namical system generated by equation (5.8) possesses the properties given in Exer-cises 1.6.9–1.6.12. Moreover, it is possible to apply Theorem 1.6.3 as well as the otherresults related to finiteness and hyperbolicity of the set of fixed points (see, e.g., thebook by A. V. Babin and M. I. Vishik [1]).

A1 2� B t� � 2 e t 1�� 1 24�C A1 2� S1 v0 u�� v0�� 1 ��

0

t

3�

B � v0� � A1 2� B 1� ��

1 ,d"�

B t� � St

v0 u�� v0� Ttu��

A1 2� St

v0 u�� v0�� bR

ea

Rt

A1 2� u� A1 2� u R�

A1 2� B t� � C0 t e1 �� a

Rt

A1 2� u1 ��

C1 t 1�� 1 24�A1 2� B 1� � 1d

0

t

3��

A1 2� St

v0 u�� v0� Ttu�� C

TA1 2� u

1 �� t 0 T��

E x e r c i s e 5.8 C R

A1 2� u R� A1 2� v R� St

C1 ��0 � 1� �

/ 1 24�B u� � �1 24

C C R� �� R

A1 2� u R� A1 2� v R� H

� h �� 1 24 St�� B � z� �

z D A� ��

h � ,�


E x a m p l e 5.1

Let us consider a dynamical system generated by the nonlinear heat equation

(5.17)

in . Assume that is twice continuously differentiable with re-spect to its variables and the conditions

,

are fulfilled with some positive constants , and .

Prove that the dynamical system generated by equation(5.17) possesses a global attractor , where isthe set of stationary solutions to problem (5.17).

Prove that there exists a dense open set in suchthat for every the set of fixed points of the dynamicalsystem generated by problem (5.17) is finite and all the points arehyperbolic.

It should be noted that if a property of a dynamical system holds for the parametersfrom an open and dense set in the corresponding space, then it is frequently said thatthis property is a generic propertygeneric propertygeneric propertygeneric property.

However, it should be kept in mind that the generic property is not the one thatholds almost always. For example one can build an open and dense set in ,the Lebesgue measure of which is arbitrarily small . To do that we shouldtake

,

where is a sequence of all the rational numbers of the segment . There-fore, it should be remembered that generic properties are quite frequently encoun-tered and stay stable during small perturbations of the properties of a system.

u�t��

2 u

x2�� g x u x� �� h x� � 0 x 1� � t 0 ,!� ��

ux 0� u

x 1� 0 ut 0�� u0 x� � ,� � ��

8�8�

H01 0 1�� g x u��

g x ;�� ;d

0

y

3 ��& y g x y�� g x ;�� ;d

0

y

3� 7�&

a b� 7

E x e r c i s e 5.9

� M� ��

E x e r c i s e 5.10 � L2 0 1�� h x� � ��

� 0 1�� '��

� x 0 1�� : x rk

� ' 2 k� 2�"��

k 1�

�

A�r

k � 0 1��


2

C

h

a

p

t

e

r

§ 6 Explicitly Solvable Model§ 6 Explicitly Solvable Model§ 6 Explicitly Solvable Model§ 6 Explicitly Solvable Model

of Nonlinear Diffusionof Nonlinear Diffusionof Nonlinear Diffusionof Nonlinear Diffusion

In this section we study the asymptotic properties of solutions to the following non-linear diffusion equation

(6.1)

where , , and are parameters. The main feature of this problem isthat the asymptotic behaviour of its solutions can be completely described with thehelp of elementary functions. We do not know whether problem (6.1) is related toany real physical process.

Show that Theorem 2.4 which guarantees the global existenceand uniqueness of mild solutions is applicable to problem (6.1) in theSobolev space .

Write out the system of ordinary differential equations for thefunctions that determine the Galerkin approximations

(6.2)

of the order of a solution to problem (6.1).

Using the properties of the functions defined by equa-tion (6.2) prove that the mild solution possesses the proper-ties

(6.3)

and

. (6.4)

Here and below is a norm in .

Using equations (6.3) and (6.4) prove that the dynamical sys-tem generated by problem (6.1) in is dissipative.

ut �ux x C u x t�� 2x D�d

0

1

3) *+ ,+ ,- .

u :ux� �� 0 , 0 x 1 , t 0 ,!� ��

ux 0� u

x 1� 0 , ut 0� u0 x� � ,� � ��

88�88�

� 0! C 0! D :

E x e r c i s e 6.1

H01 0 1��

E x e r c i s e 6.2

gk t� � �

um

t� � 2 gk

t� � @k xsink 1�

m

��

m

E x e r c i s e 6.3 um

t� �u t x��

12�� u t� � 2 � �

xu 1� � 2 C u 1� � 4 D u 1� � 2�� 1d

0

t

3� 12�� u0

2�

u t� � 2u0

2e

2�@2 t� D2

4�C@2�� 1 e

2�@2 t��

. L2 0 1��

E x e r c i s e 6.4

H01 0 1��

E x p l i c i t l y S o l v a b l e M o d e l o f N o n l i n e a r D i f f u s i o n 119

Therefore, by virtue of Theorem 4.1 the dynamical system generatedby equation (6.1) possesses a finite-dimensional global attractor.

Let be a mild solution to problem (6.1) withthe initial condition . Then the function can be conside-red as a mild solution to the linear problem

(6.5)

where is a scalar continuous function defined by the formula

.

We consider the function

. (6.6)

Then it is easy to check that is a mild solution to the heat equation

(6.7)

The following assertion shows that the asymptotic properties of equation (6.7)completely determine the dynamics of the system generated by problem (6.1).

Lemma 6.1.

Every mild (in ) solution to problem (6.1) can be re-

written in the form

, (6.8)

where has the form

(6.9)

and is the solution to problem (6.7).

H01 0 1�� S

t��

u t� � u x t�� C R� H01 0 1�� 2� ��

u0 x� � H01 0 1�� u x t��

ut �ux x b t� �u :ux� �� 0 , 0 x 1 t 0 ,!��

ux 0� u

x 1� 0 ut 0�� u0 x� � ,� � ��

��

b t� �

b t� � C u x t�� 2xd

0

1

3 D��

v x t�� u x t�� b 1� � 1:2

4�� t:

2�� x��d

0

t

3� �8 8� �8 8� �

exp�

v x t��

vt � vx x , x 0 1�� , t 0 ,!��

vx 0� v

x 1� 0 vt 0�� u0 x� �

:2�� x

� ��

.exp� � ��88�88�

H01 0 1�� u x t��

u x t�� w x t��

1 2C w 1� � 2 1d

0

t

3�

� �8 8� �8 8� �1 2��

w x t��

w x t�� v x t�� D:2

4��) *- . t

:2��x�

� ��

exp�

v x t��


2

C

h

a

p

t

e

r

Proof.

Let have the form (6.9). Then we obtain from (6.6) that

. (6.10)

Therefore,

.

Hence,

.

This and equation (6.10) imply (6.8). Lemma 6.1 is proved.

Now let us find the fixed points of problem (6.1). They satisfy the equation

.

Therefore, , where is the solution to the problem

.

However, this problem has a nontrivial solution if and only if

and ,

where is a natural number. Since , we obtain the equation

which can be used to find the constant . After the integration we have

.

The constant can be found only when the parameter possesses the property. Thus, we have the following assertion.

w x t��

w x t�� u x t�� C u 1� � 2 1d

0

t

3� �8 8� �8 8� �

exp�

w x t�� 2u x t�� 2 2C u 1� � 2 1d

0

t

3� �8 8� �8 8� �

exp 12C��

dtd

�� 2C u 1� � 2 1d

0

t

3� �8 8� �8 8� �

exp� �

2C u 1� � 2 1d

0

t

3� �8 8� �8 8� �

exp 1 2C w 1� � 2 1d

0

t

3��

�� ux x C u2 D�� u :ux� � 0 , u

x 0� ux 1� 0� � �

u x� � w x� �:

2�� x� ��

exp� w x� �

�� wx x C u 2 D�:2

4��) *- . w� 0 , w

x 0� wx 1� 0� � �

w x� �

w x� � C @n xsin� C u2 D�

:2

4�� @n� �2� � 0�

n u w:

2�� x� ��

exp�

C C2 e

:�� x

@n xsin2 xd

0

1

3 � @n� �2 D�:2

4�� 0�

C

C C2 �:�� e

: �� 1�� 2 n2�2@2

:2 4 n2�2@2�� D

:2

4�� @n� �2��

C n

D :2 4�� 4� � @n� �2� 0!


Lemma 6.2.

Let . If , then problem (6.1) has a unique

fixed point . If for some , then

the fixed points of problem (6.1) are

, , , (6.11)

where

, (6.12)

.

Show that every subspace

(6.13)

is positively invariant for the dynamical system gene-rated by problem (6.1).

Theorem 6.1.

Let . Then for any mild solution to problem Let . Then for any mild solution to problem Let . Then for any mild solution to problem Let . Then for any mild solution to problem (6.1)the estimatethe estimatethe estimatethe estimate

,,,, ,,,, (6.14)

is valid. If and , then the subspace defined byis valid. If and , then the subspace defined byis valid. If and , then the subspace defined byis valid. If and , then the subspace defined by

formula formula formula formula (6.13) is exponentially attracting:is exponentially attracting:is exponentially attracting:is exponentially attracting:

.... (6.15)

Here Here Here Here is the evolutionary operator in corresponding to is the evolutionary operator in corresponding to is the evolutionary operator in corresponding to is the evolutionary operator in corresponding to (6.1)....

Proof.

Let be of the form (6.9). Then

, (6.16)

where and

.

7 7 D � :� �� D:2

4��#� 7 �@2�u0 x� � 0# @n� �2 7 @2 n 1�� 2�� n 1&

u0 x� � 0# ukE x� � 6

ke

:2�� x

@k xsinE� k 1 2 � n� � ��

6k 6k D : �� # 14�2@k��

2:�k D : ��

C :��) *- .exp 1�) *

- .��

�kD : �� 4D� :2� 4 @n�� 2�� :2 4 @n�� 2�� "�

E x e r c i s e 6.5

HN

Lin e

:2�� x

@k x : k 1 2 � N� � ��sin� ��

�

H01 0 1�� S

t��

7 D:2

4�� @2��# u t� �

�x

u t� � C : �� e�@2 7�� t� �

xu0� t 0&

7 �@2& �@2 N2 7! HN 1�

distH0

1 0 1�� St u0 HN 1�� C : �� e �@2 N2 7�� t� �x u0�

St

H01 0 1��

w x t��

w: t� � e

:2�� x

w t� �# e � @ k� �2 7�� t� Ck :� e

kx� �

k 1�

�

��

ek

x� � 2 @k xsin�

Ck :� e

:2�� x

u0 x� � ek

x� � xd

0

1

3�


2

C

h

a

p

t

e

r

Assume that . Then it is obvious that

(6.17)

However, equation (6.8) implies that

.

Therefore, estimate (6.14) follows from (6.17) and from the obvious equality. When and the function in (6.9)

can be rewritten in the form

, (6.18)

where

and can be estimated (cf. (6.17)) as follows:

. (6.19)

Thus, it is clear that

.

Consequently, estimate (6.15) is valid. Theorem 6.1 is proved.

In particular, Theorem 6.1 means that if then the global at-tractor of problem (6.1) consists of a single zero element, whereas if , theattractor lies in an exponentially attracting invariant subspace , where

and is a sign of the integer part of a number.The following assertion shows that the global minimal attractor of problem

(6.1) consists of fixed points of the system. It also provides a description of the cor-responding basins of attraction.

Theorem 6.2.

Let Let Let Let .... Assume that the number is not integer. Assume that the number is not integer. Assume that the number is not integer. Assume that the number is not integer.

Suppose that Suppose that Suppose that Suppose that is the greatest integer such that is the greatest integer such that is the greatest integer such that is the greatest integer such that .... Let Let Let Let

7 �@2�

�xw: t� � 2 @k� �2 C

k :�2 2 � @k� �2 7�� t� �

2 �@2 7�� t� � �x

e

:2�� x

u0) *+ ,- . 2

.exp

�

�

expk 1�

�

��

�x e

:2�� x

u t� �) *+ ,- .

�x w: t� ��

u 1 @4 �x

u� 7 �@2& N 1 @4� � 7 �4! w t� �

w t� � hN t� � e

:2�� x

w: N� t� ��

hN

t� � 2 e

:2�� x

e � @ k� �2 7�� t� Ck :� @k xsin H

N 1��k 1�

N 1�

��

w: N� t� �

�x

w: N� t� � 2 2 � @N� �2 7�� t� � �x

e

:2�� x

u0) *+ ,- . 2

exp�

Stu0 H

N 1�� H0

1 0 1�� dist C : ��

xw: N� t� ��

7 D :2 4�� 4� �@2 ,�#7 �@2&

HN0

N0 �1 @4� � 7 �4� �� .� �

7 D:2

4�� @2!# @ 1� 7 �4N0 � @N0� �2 7�


and let and let and let and let be a mild solution to problem be a mild solution to problem be a mild solution to problem be a mild solution to problem (6.1)....

(a) If for all , thenIf for all , thenIf for all , thenIf for all , then

,,,, .... (6.20)

(b) If for some between and , then there exist positiveIf for some between and , then there exist positiveIf for some between and , then there exist positiveIf for some between and , then there exist positive

numbers and such thatnumbers and such thatnumbers and such thatnumbers and such that

,,,, ,,,, (6.21)

where is defined by formula where is defined by formula where is defined by formula where is defined by formula (6.11), , and is, , and is, , and is, , and is

the smallest index between and such that the smallest index between and such that the smallest index between and such that the smallest index between and such that ....

Proof.

In order to prove assertion (a) it is sufficient to note that the value isidentically equal to zero in decomposition (6.18) when . Therefore,(6.20) follows from (6.19).

Now we prove assertion (b). In this case equation (6.18) can be rewritten in theform

, (6.22)

where

,

,

and if . Moreover, the estimate

(6.23)

is valid for . It is also evident that

. (6.24)

Since

using (6.23) and (6.24) we obtain

.

Cj

u0� � 2 e

:2�� x

u0 x� � @ j xsin xd

0

1

3�

u t� �Cj u0� � 0� j 1 2 � N0� � ��

�x

u t� � C � :�� @2 N0 1�� 2 7�� t� � �x

u0exp� t 0&

Cj

u0� � 0( j 1 N0C C � : 7 u02� �� 7 ��

�x

u t� � u$ k�� C e t�� t 0&

ukE x� � $ sign C

ku0� �� k

1 N0 Ck

u0� � 0(

hN

t� �N N0 1��

w t� � gk

t� � h�k

t� � w: N0 1�� t� ��

gk t� � 2 e

:2�� x

e� @ k� �2 7�� t�

Ck u0� � @k xsin�

hk

t� � 2 e

:2�� x

e� @ j� �2 7�� t�

Cj :� @ j xsin

j k 1��

N0

��

hk t� � 0# k N0�

�x

w: N0 1�� t� � C : �� e �@2 N0 1�� 2 7�� t� �x

u0�

w: N0 1�� t� �

�x

hk

t� � C : �� e7 �� @2 k 1�� 2� � t �

xu0�

w t� � 2gk t� � 2� w t� � gk t� �� w t� � gk t� ��

2 gk t� � hk t� � w: N0 1�� t� �� hk t� � w: N0 1�� t� �� ,

� �

�

w t� � 2g

kt� � 2� C 2 7 �@2 k2 k 1�� 2�� t � �

xu0

2exp�


2

C

h

a

p

t

e

r

Integration gives

where is defined by formula (6.12). Hence,

, (6.25)

where

, .

Let

.

We consider the case when . Equations (6.23)–(6.25) imply that

Therefore,

2C gk1� � 2 1d

0

t

3 4C Ck

u0� �� 2 e

:�� x

@k xsin2 x e2 7 � @ k� �2�� 1 1d

0

t

3d

0

1

3

2C

ku0� �

6k

��) *- .2

e2 7 � @ k� �2�� t 1�� ,

� �

�

6k

1 2C w 1� � 2 1d

0

t

3� 1 2C

ku0� �

6k

��) *- .2

e2 7 � @ k� �2�� t

ak t� ��

ak

t� � C 1 2 7 �@2 k2 k 1�� 2�2

��) *- . t

� ��

exp� �x

u02� k N0 1��

vk t� �g

kt� �

1 2C w 1� � 2 1d

0

t

3�) *+ ,+ ,- .1 2��

1 k N0 1��

�x

u t� � vk

t� �� x

w: N0 1�� t� ��

xh

kt� �

1 2C w 1� � 2 1d

0

t

3�) *+ ,+ ,- .1 2��

C : �� x u0 e�@2 N0 1�� 2 7�� t�

7 �� @2 k 1�� 2� � t �exp

1 2C

ku0� �

6k

��) *- .2

e2 7 � @ k� �2�� t

ak

t� ��) *- .1 2��

�

�

�

�

�

�

�

� .

� �

�

�x

u t� � vk

t� ��

C : � u0� �� e�@2 N0 1�� 2 7�� t�

e�@2 k 1�� 2 k2�� t�

G t� � 1 2��

,

�

�


where

.

It follows from (6.25) that for all and . Moreover, theabove-mentioned estimate for enables us to state that

.

This implies that there exists a constant such that for all . Consequently,

, (6.26)

where . Now we considerthe value . Evidently,

,

where

.

Simple calculations give us that

with the constant . This and equation (6.26) imply (6.21), provi-ded . We offer the reader to analyse the case when on his/herown. Theorem 6.2 is proved.

Theorem 6.2 enables us to obtain a complete description of the basins of attraction ofeach fixed point of the dynamical system generated by problem (6.1).

Corollary 6.1.

Let . Assume that the number is not in-

teger. Let be the greatest integer possessing the property .

We denote

and define the sets

,

G t� � 1 ak

t� �� e 2 7 � @2 k2�� t� 2Ck u0� �6

k

��) *- .

2��

G t� � 0! t 0& G 0� � 1�a

kt� �

G t� �t ��lim 2

Ck

u0� �6k

��) *- .

20!�

Gmin Gmin 7 k � u0� � �� 0!�G t� � Gmin& t 0&

�x

u t� � vk

t� �� C : � 7 u02� �� e � t��

� � k � 7� �� @2 N0 1�� 2 7� �@2 2 k 1�� min� �v

kt� �

vk

t� � 5k

t u0�� u$ kx� ��

5k t u0�� 2 C

ku0� � 7 �� @2 k2� � t �exp

6k 1 2C

ku0� �6

k

��) *- .

22 7 �� @2 k2� � t �exp ak t� ��) *

- .1 2�

��

5k

t u0�� 1� C : � 7 u0� � �� e t��

�@2 2 k 1�� k N0 1�� k N0�

H01 0 1�� S

t��

7 D :2 4�� 4� �@2!# @ 1� 7 �4N0 � @N0� �2 7�

Cj u� � 2 e

:2�� x�

u x� � @ j xsin xd

0

1

3�

�k

u H01 0 1�� : C

ju� � 0� , j 1 � k 1�� , C

ku� � 0! ��

�k� u H0

1 0 1�� : Cj

u� � 0� , j 1 � k 1�� , Ck

u� � 0� ��


2

C

h

a

p

t

e

r

for . We also assume that

.

Then for any we have that

,

where are the fixed points of problem (6.1) which are defined by

equalities (6.11).

The next assertion gives us a complete description of the global attractor of problem(6.1).

Theorem 6.3.

Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem 6.2 hold and is the same as hold and is the same as hold and is the same as hold and is the same as

in Theoremin Theoremin Theoremin Theorem 6.2.... Then the global attractor of the dynamical system Then the global attractor of the dynamical system Then the global attractor of the dynamical system Then the global attractor of the dynamical system

generated by equation generated by equation generated by equation generated by equation (6.1) is the closure of the set is the closure of the set is the closure of the set is the closure of the set

,,,, (6.27)

where , , andwhere , , andwhere , , andwhere , , and

,,,, ....

Every complete trajectory which lies in the attractor andEvery complete trajectory which lies in the attractor andEvery complete trajectory which lies in the attractor andEvery complete trajectory which lies in the attractor and

does not coincide with any of the fixed points does not coincide with any of the fixed points does not coincide with any of the fixed points does not coincide with any of the fixed points ,,,, , has , has , has , has

the formthe formthe formthe form

,,,, (6.28)

where where where where are real numbers, , are real numbers, , are real numbers, , are real numbers, , ....

Show that for all the function

, (6.29)

is nonnegative and it is monotonely nondecreasing with respect to (Hint: and as ).

k 1 2 � N0� � ��

�0 u H01 0 1�� : C

ju� � 0� , j 1 2 � N0� � ��

l 0 1E 2E � N0E� � � ��

St v ul�H0

1 0 1�� t ��lim 0� v � l�F

ul

�

N0�

H01 0 1�� S

t��

� v x� �2 ;k e

:2�� x

@k xsink 1�

N0�1 2C ;k;j

�k j

�k

�j

��k j� 1�

N0��) *- .

1 2�� : ;

jR��

� �8 88 8� �8 88 8� �

�

�k 7 � @k� �2�� k 1 2 � N0� � ��

�k j

2 e

:�� x

@k xsin @ j xsin" xd

0

1

3� k j� 1 2 � N0� � ��

u t� � : t R� �u

kk 0 1 � N0E� �E��

u t� �2 ;

ke�

kt

e

:2�� x

@k xsink 1�

N0�1 2C ;

k;

j

�k j

�k

�j

�� e�

k�

j�� t

k j� 1�

N0��) *- .

1 2��

;k

k 1 2 � N0� � �� t R�

E x e r c i s e 6.6 ; RN0�

a t ;�� 2C ;k;

j

�k j

�k

�j

�� e�

k�

j�� t

k j� 1�

N0�� t 0&

t

at� t ;�� 0& a t ;�� 0� t ��



Let belong to the set given by formula (6.27). Then by virtue of Lemma 6.1we have

, (6.30)

where

and the value is defined according to (6.29). Simple calculations show that

.

Therefore, it is easy to see that for , where has the form (6.28).It follows that and that a complete trajectory lying in has the form(6.28). In particular, this means that . To prove that it is sufficient toverify using Theorem 6.1 and the reduction principle (see Theorem 1.7.4) that forany element there exists a semitrajectory such that

uniformly with respect to from any bounded set in . To do this, it should bekept in mind that for

(6.31)

has the form (6.30) with

.

Therefore, it is easy to find that

,

where is given by formula (6.29). Hence, if we choose

,

we find that

h �

St hw x t��

1 2C w 1� � 2 1d

0

t

3�) *+ ,+ ,- .1 2��

w t� � 21 a 0 ;�� e

:2�� x

;k

e�

kt @k xsin

k 1�

N0

��

a t ;��

2C w 1� � 2 1d

0

t

3 a t ;�� a 0 ;�� 1 a 0 ;��

��

St h u t� �� t 0& u t� �St� �� u t� � �

� �% � ��

h HN0� v t� � �%

�x St h v t� �� t ��lim 0�

h HN0

h 2 ;k

e

:2�� x

@k xsink 1�

N0

� HN0��

St h

w t� � 2 e

:2�� x

;k e�

kt @k xsin

k 1�

N0

��

Sth

w t� �1 a t ;�� a 0 ;�� 1 2��

a t ;��

v t� �w t� �

1 a t ;�� 1 2��


2

C

h

a

p

t

e

r

, (6.32)

where

.

Using the obvious inequality

, ,

we obtain that

,

provided that has the form (6.31). It is evident that

.

Therefore,

.

Consequently, the dissipativity property of in and equations (6.32) giveus that

,

for all , where is an arbitrary bounded set. Thus, and there-fore Theorem 6.3 is proved.

Show that the set coincides with the unstable manifold emanating from zero, provided that the hypotheses of Theo-

rem 6.3 hold.

Show that the set from Theorem 6.3 can be de-scribed as follows:

St h v t� �� B t h�� St h�

B t h�� 1 1a 0 ;��

1 a t ;�� ) *

- .1 2��

1 1 x�� 1 2�� x� 0 x 1� �

B t h�� a 0 ;��

1 a t ;��

h

a t ;�� a 0 ;�� c0 e2�N0

11d

0

t

3 ;k2

k 1�

N0

�"�&

B t h�� C e2�N0

11d

0

t

3) *+ ,+ ,- . 1�

�

St

H01 0 1��

�x St h v t� �� C e2�N0

11d

0

t

3) *+ ,+ ,- . 1�

� t tB&

h B HN0%� B � ��

E x e r c i s e 6.7 �

M� 0� �

E x e r c i s e 6.8 � M� 0� ��

� v 2 >k e

:2�� x

@k x :

2C >k>

j

�k j

�k

�j

�� 1 , > RN0��

k j� 1�

N0

�) *+ ,- .

sink 1�

N0

��

�

�

8

8

�

�

8

8

�

� .

�


Therewith, the global attractor has the form

.

Prove that the boundary

of the set is a strictly invariant set.

Show that any trajectory lying in has the form , where

, .

Using the result of Exercise 6.10 find the unstable manifold emanating from the fixed point , .

Find out for which pairs of fixed points , , there exists a heteroclinic trajectory connec-

ting them, i.e. a complete trajectory such that

, .

Display graphically the global attractor on the plane gene-rated by the vectors and

for .

�

� v 2 >k e

:2�� x

@k x : 2C >k>j

�k j

�k

�j

�� 1�k j� 1�

N0

�sink 1�

N0

��

� �8 8� �8 8� �

�

E x e r c i s e 6.9

�� v 2 >k

e

:2�� x

@k x :

2C >k>j

�k j

�k

�j

�� 1 , > RN0��

k j� 1�

N0

�) *+ ,- .

sink 1�

N0

��

�

�

8

8

�

�

8

8

�

�

�

�

E x e r c i s e 6.10 7 �� u t� ��t R� �

u t� �

>k

e�

kt

e

:2�� x

@k xsink 1�

N0

�

C >k>

j

�k j

�k

�j

�� e�

k�

j�� t

k j� 1�

N0

�) *+ ,- .1 2�

�� > RN0�


M� uk

� � uk

k 1 2 � N0E� �E�E�

E x e r c i s e 6.12 uk

uj

� � k j� �1 2 � N0E� �E�E�

uk j

t� � : t R� �

uk

uk j

t� �t ��

lim� uj

uk j

t� �t ��

lim�

E x e r c i s e 6.13 �

e1 e : 2�� 4� � x @xsin� e2 e : 2�� 4� � x 9�2@xsin9 N0 2�


2

C

h

a

p

t

e

r

Study the structure of the global attractor of the dynamicalsystem generated by the equation

in , where are positive parameters.

§ 7 Simplified Model of Appearance§ 7 Simplified Model of Appearance§ 7 Simplified Model of Appearance§ 7 Simplified Model of Appearance

of of of of Turbulence in FluidTurbulence in FluidTurbulence in FluidTurbulence in Fluid

In 1948 German mathematician E. Hopf suggested (see the references in [3]) to con-sider the following system of equations in order to illustrate one of the possible sce-narios of the turbulence appearance in fluids:

where the unknown functions , , and are even and -periodic with respectto and

.

Here and are even -periodic functions and is a positive constant.We also set the initial conditions

. (7.4)

As in the previous section the asymptotic behaviour of solutions to problem(7.1)–(7.4) can be explicitly described.

Let us introduce the necessary functional spaces. Let

.


ut

�ux x

� C u x t�� 2v x t�� 2�� xd

0

1

3 D�) *+ ,+ ,- .

u �v�� = 0, 0 x 1� � , t 0 ,!

vt

�vx x

� C u x t�� 2v x t�� 2�� xd

0

1

3 D�) *+ ,+ ,- .

v �u� � = 0, 0 x 1� � , t 0 ,!

ux 0� = u

x 1� = vx 0� = v

x 1� =0�88888�88888�

H01 0 1�� H0

1 0 1�� 9 � C D ��

ut

6ux x

v *v w *w� u * 1 ,��

vt

6vx x

v *u v *a w * b ,� ��

wt 6wx x w *u v * b� w *a ,��8�8� (7.1)

(7.2)

(7.3)

u v w 2@x

f * g� � x� � 12@�� f x y�� g y� � yd

0

2@

3�

a x� � b x� � 2@ 6

ut 0� u0 x� � v

t 0� v0 x� � wt 0� w0 x� ��

H f Lloc2 R� � : f x� � f x�� f x 2@��

S i m p l i f i e d M o d e l o f A p p e a r a n c e o f T u r b u l e n c e i n F l u i d 131

Evidently is a separable Hilbert space with the inner product and the norm de-fined by the formulae:

,

There is a natural orthonormal basis

in this space. The coefficients of decomposition of the function withrespect to this basis have the form

, .

Let . Then and

. (7.5)

The Fourier coefficients of the functions , and obeythe equations

, . (7.6)

Let be the orthoprojector onto the span of elements in . Show that

. (7.7)

Let us consider the Hilbert space with the norm as the phase space of problem (7.1)–(7.4). We define

an operator by the formula

,

on the domain

,

where is the second order Sobolev space.

Prove that is a positive operator with discrete spectrum. Itseigenvalues have the form:

, ,

H

f g�� f x� � g x� � xd

0

2@

3� f 2 f f��

12@

��1@

�� xcos 1@

�� 2 xcos ��

Cn f� � f H�

C0 f� � 12@

�� f x� � xd

0

2@

3� Cn f� � 1@

�� f x� � n x xdcos

0

2@

3�

E x e r c i s e 7.1 f g� H� f * g H�

f * g12@

�� f g"�

Cn

f * g f , g

C0 f * g� � 12@

��C0 f� � C0 g� �� Cn

f * g� � 12 @�� C

nf� � C

ng� ��

E x e r c i s e 7.2 pm

k xcos k 0 1 � m� � �� H

pm

f * g� � pm

f� � * g f * pm

g� ��

H H3 H H H99#� u v w22� � H �u

2v

2w

2� �� 1 2��A

A u v w� �� u 6ux x v 6vx x� w 6wx x�22�� u v w22� � D A� ��

D A� � Hloc2 R� �� 3 H0�

Hloc2 R� �


n

�n 0��

3 k 3k 1� 3k 2� 1 6k2�� k 0 1 2 ��


2

C

h

a

p

t

e

r

while the corresponding eigenelements are defined by the formulae

(7.8)

where

Let

Equation (7.5) implies that , provided , , , , and are theelements of the space , . Therefore, the formula

gives a continuous mapping of the space into itself.

Prove that

,

where , .

Thus, if , then problem (7.1)–(7.4) can be rewritten in the form

, ,

where and satisfy the hypotheses of Theorem 2.1. Therefore, the Cauchy prob-lem (7.1)–(7.4) has a unique mild solution in the space

on a segment , provided that . In order to prove the global exis-tence theorem we consider the Galerkin approximations of problem (7.1)–(7.4).The Galerkin approximate solution of the order with respect to basis(7.8) can be presented in the form

, (7.9)

where , , and are scalar functions. By virtue of equations (7.7)it is easy to check that the functions , , and satisfy equa-tions (7.1)–(7.3) and the initial conditions

, , . (7.10)

e012@

�� 1 0 022� � , e112@

�� 0 1 022� � , e212@

�� 0 0 122� � ,��

e3 k1@

�� k x 0 022cos� � , e3 k 1�1@

�� 0 k xcos 022� � ,��

e3 k 2�1@

�� 0 0 k xcos22� � ,��888�888�

k 1 2 ��

b1 u v w� �� v *v w *w� u * 1 ,��

b2 u v w� �� v *u v *a w * b ,� ��

b3 u v w� �� w *u v * b� w *a .��

bj u v w� �� H� a b u v w

H j 1 2 3� ��

B u v w� �� b1 u v w� �� u b2 u v w� �� v� b3 u v w� �� w�22��

H

E x e r c i s e 7.4

B y1� � B y2� ��H

C 1 a b y1 Hy2 H

� � � �) *- . y1 y2�

H�

yj uj vj2 wj2� � H�� j 1 2��

a b� H�ydtd

�� A y� B y� �� yt 0� y0�

A B

y t� � u t� � v t� � w t� �� H 0 T�� a b� H�

ym

t� � 3 m

ym t� � u m� � t� � v m� �2 t� � w m� �2 t� �) *- . uk t� � vk2 t� � wk2 t� �) *

- . k xcosk 0�

m 1�

��

uk

t� � vk

t� � wk

t� �u m� � t� � v m� � t� � w m� � t� �

u m� � 0� � pm

u0� v m� � 0� � pm

v0� w m� � 0� � pm

w0�


Thus, approximate solutions exist, locally at least. However, if we use (7.1)–(7.3) wecan easily find that

Therefore, inequality (7.5) leads to the relation

.

This implies the global existence of approximate solutions (see Exercise 2.1).Therefore, Theorem 2.2 guarantees the existence of a mild solution to problem(7.1)–(7.4) in the space on the time interval of any length . Moreover, themild solution possesses the property

,

for any segment . Approximate solution has the structure (7.9).

Show that the scalar functions involvedin (7.9) are solutions to the system of equations:

Here , for , , the numbers and are the Fourier coefficients of the func-

tions and .

Thus, equations (7.1)–(7.3) generate a dynamical system with the evolu-tionary operator defined by the formula

,

where is a mild solution (in ) to the Cauchy problem (7.1)–(7.4), . An interesting property of this system is given in the fol-lowing exercise.

Let be the span of elements , where and are defined by equations (7.8). Then the

subspace of the phase space is positively invariant with re-spect to .

12��

dtd

��" u m� � t� � 2v m� � t� � 2

w m� � t� � 2� ��

6 uxm� � 2

vxm� � 2

wxm� � 2� �

� ��

�

�

�

u m� �* 1 u m� �� v m� �

*a v m� �� w m� �*a w m� �� .� ��

dtd�� ym t� �

H2

C 1 a�� ym t� �H2�

ym t� �

H H3� T

y t� � u t� � v t� � w t� �22� ��

y t� � ym

t� ��0 T�� max 0� m ��

0 T�� ym

t� �

E x e r c i s e 7.5 uk

t� � vk

2 t� � w2k

t� � �

u� k 6k2 uk

� vk2� w

k2� u0�k 0 ,��

v�k 6 k2 vk� vk uk vk ak wk bk ,� ��

w�k 6k2 w

k� w

ku

kv

k� b

kw

ka

k .��

�88�88� (7.11)

(7.12)

(7.13)

k 1 2 �� k0 0� k 0( �00 1� a

k�

Ck

a� �� bk

Ck

b� ��a x� � b x� �

H St

�� S

t

St y0 u t� � v t� � w t� �22� ��

u t� � v t� � w t� �22� � Hy0 u0 v0 w0� ��

E x e r c i s e 7.6 �k e3 k e3 k 1� e3 k 2�� k 0 1 2 �� en �

�k HSt St �k �k%� �


2

C

h

a

p

t

e

r

Therefore, the phase space of the dynamical system falls into the ortho-gonal sum

of invariant subspaces. Evidently, the dynamics of the system in the sub-space is completely determined by the system of three ordinary differentialequations (7.11)–(7.13).

Lemma 7.1.

Assume that and

. (7.14)

Then the dynamical system is dissipative.

Proof.

Let us fix and then consider the initial conditions from the subspace

,

where are defined by equations (7.8). It is clear that is positively in-variant and the trajectory

of the system is a function satisfying (7.1)–(7.4) in the classical sense. Letbe the orthoprojector in onto the span of elements . We introduce a new variable instead of the function

. Here . Equations (7.1)–(7.3) can be rewritten in theform

where . The properties of the convolution operation (see Exer-cises 7.1 and 7.2) enable us to show that

H H St

��

H �kGk 0�

�

��

H St

��

k

a b� H�

C012@

�� a x� � xd

0

2@

3 0��

H St

��

N y0 u0 v0 w0� ��

HN �k Lin e3 k e3 k 1� e3 k 2�� Gk 0�

N

��Gk 0�

N

��

en � HN

y t� � u t� � v2 t� � w2 t� �� uk

t� � vk

2 t� � wk

2 t� �� k xcosk 0�

N

��

pm H k x : ncos 0 1 �� m� u

�t� � u t� � �m��

u t� � �m pm p0�� a�

u�

t6u�

x x� v *v w *w� u

� �m�� * 1 6�x xm ,��

vt

6vx x

� v *u�

v * qm

a� � w * b v * p0 a� � ,� � ��

wt 6wx x� w *u�

v * b� w * qm a� � w * p0 a� � ,� ��88�88� (7.15)

(7.16)

(7.17)

qm

1 pm

��


(7.18)

It is clear that

, ,

where is the zeroth Fourier coefficient of the function . Moreo-ver, the estimate

, ,

holds. We choose such that . Then equation (7.18)implies that

If we use the inequality

,

then we find that

,

where . Consequently, the estimate

(7.19)

is valid for , provided and where . By passing to the limit we can extend inequality

(7.19) over all the elements . Thus, the system possesses an ab-sorbing set

, (7.20)

where is such that and

.

Show that the ball defined by equality (7.20) is positivelyinvariant.

12��

dtd�� u

� 2 v2

w2� �� 6 u

�x

2v

x2 w

x2� ��

u� �m�� * 1 u

�� 6 �xm

u�

x�� v * q

ma� � v��

w * qm

a� � w�� v * p0 a� � v�� w * p0 a� � w�� .

� � �

� � �

�

�

u� �m�� * 1 u

�� C0 u�� 2�� v * p0a� � v�� 1

2@��C0 a� � C0 v� �� 2�

C0 f� � f x� � H�

qmv vx� m 1&

m 1& 1 2@4� � qma 6 24�

dtd�� u

� 2v

2w

2� �) *- . 6 u

�

x2

vx

2w

x2� �) *

- .

2 C0 u�� 2 2

@�� C0 a� � C0 v� � 2C0 w� � 2�) *

- . 6 �xm 2 .�

� �

� �

f 2 C0 f� � 2fx

2��

dtd�� u

� 2v

2w

2� �� u2

v2

w2� �� 6 �

xm 2��

� 6 2 2 @4 C0 a� �� min�

y�

t� � H2

y� 0� � H

2e � t� 6

�� 1 e � t�� xm 2��

y�

t� � u�

t� � v t� � w t� �� qma 6 @ 24� y�

0 y0��m 0 0� �� y0 HN�

y0 H� H St��

�m u v w� �� : u pm p0�� a� 2v

2w

2� � Rm2� ��

m a p�m

a 6 @ 24�

Rm2 6 pm p0�� a� �

x

2 6 2 2@�� C0 a� �� ) *

- .min) *- .

1�1��

E x e r c i s e 7.7 �m


2

C

h

a

p

t

e

r

Consider the restriction of the dynamical system tothe subspace

.

Show that is dissipative not depending on the validity ofcondition (7.14).

Lemma 7.2.

Assume that the hypotheses of Lemma 7.1 hold and let be the ortho-

projector onto the span of elements in ,

. Then the estimate

(7.21)

holds for all such that . Here

and is the mild solution to problem

(7.1)–(7.4) with the initial condition .

Proof.

As in the proof of Lemma 7.1 we assume that forsome . If we apply the projector to equalities (7.15)–(7.17), then we getthe equations

where , and . Therefore, we obtain

(7.22)

as in the proof of the previous lemma. It is easy to check that for every . Therefore, inequality (7.22) implies that

.

Hence, equation (7.21) is valid. Lemma 7.2 is proved.

E x e r c i s e 7.8 H St

��

H�

h x� � u x� � v x� � w x� �22� � H h x� � x 0�d

0

2@

3��

� �8 8� �8 8� �

�n

n 1&��

H�

St

��

pm

k x : k 0 1 � m� � ��cos � H

qm

1 pm

��

ym

t� �H2

y0 H2 2 m 1�� 26

qm a

2@��) *

- . t�� exp�

m qm

a 6 2@ m 1�� 2� ym

t� � qm

u t� � 2��q

mv t� � q

mw t� �2 � u t� � v2 t� � w2 t� ��

y0 u0 v0 w022� ��

y0 u0 v0 w022� � HN��N q

m

utm 6u

x xm� vm

* vm� wm* wm ,��

vtm 6 v

x xm� vm

* um wm* b vm

* qm

a ,� ��

wtm 6w

x xm� wm

* um vm� * b wm* q

ma ,��

88�88�

um qm

u� vm qm

v� wm qm

w�

12��

dtd�� um 2

vm 2wm 2� �) *

- . 6 uxm 2

vxm 2

wxm 2� �) *

- .

12@

�� qm

a vm 2wm 2�) *

- .

�

�

�

qm

h� �x

2 &m 1�� 2 h

2& h pN

H�

12��

dtd�� y

mt� �

H2 6 m 1�� 2 1

2@�� q

ma�) *

- . ym

t� �H2� 0�


Lemmata 7.1 and 7.2 enable us to prove the following assertion on the existenceof the global attractor.

Theorem 7.1.

Let and let condition Let and let condition Let and let condition Let and let condition (7.14) hold. Then the dynamical system hold. Then the dynamical system hold. Then the dynamical system hold. Then the dynamical system

generated by the mild solutions to problem generated by the mild solutions to problem generated by the mild solutions to problem generated by the mild solutions to problem (7.1)–(7.4) possesses a possesses a possesses a possesses a

global attractor . This attractor is a compact connected set. It lies in theglobal attractor . This attractor is a compact connected set. It lies in theglobal attractor . This attractor is a compact connected set. It lies in theglobal attractor . This attractor is a compact connected set. It lies in the

finite-dimensional subspacefinite-dimensional subspacefinite-dimensional subspacefinite-dimensional subspace

,,,,

where the vectors are defined by equalities where the vectors are defined by equalities where the vectors are defined by equalities where the vectors are defined by equalities (7.8) and theand theand theand the parameter parameter parameter parameter

is defined as the smallest number possessing the property is defined as the smallest number possessing the property is defined as the smallest number possessing the property is defined as the smallest number possessing the property

.... Here is the orthoprojector onto the subspace generated Here is the orthoprojector onto the subspace generated Here is the orthoprojector onto the subspace generated Here is the orthoprojector onto the subspace generated

by the elements in by the elements in by the elements in by the elements in ....

To prove the theorem it is sufficient to note that the dynamical system is compact(see Lemma 4.1). Therefore, we can use Theorem 1.5.1. In particular, it should benoted that belonging of the attractor to the subspace means that

, where is an arbitrary number possessing the property. Below we describe the structure of the attractor and evalu-

ate its dimension exactly.

According to Lemma 7.2 the subspace is a uniformly exponentially attractingand positively invariant set. Therefore, by virtue of Theorem 1.7.4 it is sufficientto study the structure of the global attractor of the finite-dimensional dynamical sys-tem . To do that it is sufficient to study the qualitative behaviour of the tra-jectory in each invariant subspace , (see Exercise 7.6). Thisbehaviour is completely described by equations (7.11)–(7.13) which get trans-formed into system (1.6.4)–(1.6.6) studied before if we take ,

, and . Therefore, the results contained in Section 6 of Chapter 1lead us to the following conclusion.

Theorem 7.2.

Let the hypotheses of Theorem Let the hypotheses of Theorem Let the hypotheses of Theorem Let the hypotheses of Theorem 7.1 hold. Then the global minimal attrac- hold. Then the global minimal attrac- hold. Then the global minimal attrac- hold. Then the global minimal attrac-

tor of the dynamical system generated by mild solutions totor of the dynamical system generated by mild solutions totor of the dynamical system generated by mild solutions totor of the dynamical system generated by mild solutions to

problem problem problem problem (7.1)–(7.4) has the form , where has the form , where has the form , where has the form , where

....

Here Here Here Here ,,,, ,,,, the values are the the values are the the values are the the values are the

a b� H�H St��

�6

HN Lin e3 ke3 k 1� e3 k 1�� G

k 0�

N

��

en

� N

qN

a �6 2@ N 1�� 2� q

N

n x : ncos N 1�& � H

�6 �N

� 6dimf 3 N 1�� N

qN a 6 2@ N 1�� 2�

HN

HN St

��

k0 k N� �

6 6 k2 �k 0��

� 6k2 ak

�� bk

�

�min6� � H St��

�min6� � 0 � H6A�

H6 uk

; rk

5 ; rk

5sincos� � k xcos@

��

� ��

0 5 2@��A

k J6 a� ��A�

uk

6 k2 ak

�� rk

k ak6 62 k2�� 1 24� a

kC

ka� ��


2

C

h

a

p

t

e

r

Fourier coefficients of the function Fourier coefficients of the function Fourier coefficients of the function Fourier coefficients of the function ,,,, and the number ranges over and the number ranges over and the number ranges over and the number ranges over

the set of indices such that the set of indices such that the set of indices such that the set of indices such that .... Topologically is a torus Topologically is a torus Topologically is a torus Topologically is a torus

(i.e. a cross product of circumferences) of the dimension (i.e. a cross product of circumferences) of the dimension (i.e. a cross product of circumferences) of the dimension (i.e. a cross product of circumferences) of the dimension .... The The The The

global attractor of the system can be obtained from by at-global attractor of the system can be obtained from by at-global attractor of the system can be obtained from by at-global attractor of the system can be obtained from by at-

taching the unstable manifold emanating from the zero element oftaching the unstable manifold emanating from the zero element oftaching the unstable manifold emanating from the zero element oftaching the unstable manifold emanating from the zero element of

the space the space the space the space .... Moreover, Moreover, Moreover, Moreover, ....

It should be noted that appearance of a limit invariant torus of high dimension pos-sesssing the structure described in Theorem 7.2 is usually assosiated with the Land-au-Hopf picture of turbulence appearance in fluids. Assume that the parameter gradually decreases. Then for some fixed choice of the function the followingpicture is sequentially observed. If is large enough, then there exists only one at-tracting fixed point in the system. While decreases and passes some critical value

, this fixed point loses its stability and an attracting limit cycle arises in the sys-tem. A subsequent decrease of leads to the appearance of a two-dimensionaltorus. It exists for some interval of values of : Then toriof higher dimensions arise sequentially. Therefore, the character of asymptotic be-haviour of typical trajectories becomes more complicated as decreases. Accordingto the Landau-Hopf scenario, movement along an infinite-dimensional torus corres-ponds to the turbulence.

§ 8 On § 8 On § 8 On § 8 On Retarded SemilinearRetarded SemilinearRetarded SemilinearRetarded Semilinear

Parabolic EquationsParabolic EquationsParabolic EquationsParabolic Equations

In this section we show how the above-mentioned ideas can be used in the study ofthe asymptotic properties of dynamical systems generated by the retarded perturba-tions of problem (2.1). It should be noted that systems corresponding to ordinary re-tarded differential equations are quite well-studied (see, e. g., the book by J. Hale [4]).However, there are only occasional journal publications on the retarded partial dif-ferential equations. The exposition in this section is quite brief. We give the readeran opportunity to restore the missing details independently.

As before, let be a positive operator with discrete spectrum in a separableHilbert space and let be the space of strongly continuous functionson the segment with the values in , . Further we also usethe notation , where is a fixed number (with the meaningof the delay time). It is clear that is a Banach space with the norm

.

a x� � k

J6 a� � 0 6k2 ak

�� H6Card J6 a� �

�6 H St

�� min6� �

M� 0� �H �6dim 2 Card J6 a� ��

6a x� �

66

616

6 63 6 62 <61� � .� �

6

A

H C a b �/� �� a b�� / D A/� �� / 0&

C/ C r 0 �/2�� r 0!C/

vC/

A/v $� � : $ r 02��

max#

O n R e t a r d e d S e m i l i n e a r P a r a b o l i c E q u a t i o n s 139

Let be a (nonlinear) mapping of the space into possessing the property

, , (8.1)

for any such that , where is an arbitrary number and is a nondecreasing function. In the space we consider a differential equation

, , (8.2)

where denotes the element from determined with the help of the function by the equality

, .

We equip equation (8.2) with the initial condition

, , (8.3)

where is an element from .The simplest example of problem (8.2) and (8.3) is the Cauchy problem for the

nonlinear retarded diffusion equation:

(8.4)

Here is a positive parameter, and are the given scalar functions.As in the non-retarded case (see Section 2), we give the following definition.

A function is called a mild (in mild (in mild (in mild (in ) solution) solution) solution) solution

to problem (8.2) and (8.3) on the half-interval if (8.3) holds and satisfies the integral equation

. (8.5)

The following analogue of Theorem 2.1 on the local solvability of problem (8.2)and (8.3) holds.

Theorem 8.1.

Assume that Assume that Assume that Assume that (8.1) holds. Then for any initial condition there holds. Then for any initial condition there holds. Then for any initial condition there holds. Then for any initial condition there

exists such that problem exists such that problem exists such that problem exists such that problem (8.2) and and and and (8.3) has a unique mild solution has a unique mild solution has a unique mild solution has a unique mild solution

on the half-interval on the half-interval on the half-interval on the half-interval ....

Proof.

As in Section 2, we use the fixed point method. For the sake of simplicity weconsider the case (for arbitrary the reasoning is similar). In the space

we consider a ball

B C/ H

B v1� � B v2� �� M R� � v1 v2�C/

� 0 / 1��

v1 v2� C/� vj C/R� R 0! M R� �

H

udtd

�� Au� B ut

� �� t t0&

ut

C/u t� �

ut $� � u t $�� $ r� 0��

ut0$� � u t0 $�� v $� �� $ r� 0��

v C/

u�t�

�� u<� f1 u t x�� f2 u t r� x�� , x , t t0 ,!��

u � 0 , u $� � $ t0 r t0�� v $� � .� ��8�8�

r f1 u� � f2 u� �

u t� � C t0 r� t0 T �/2�� /t0 t0 T�� u t� �

u t� � et t0�� A�

v 0� � et t0�� A�

B u1� � 1d

t0

t

3��

v C/�T 0!

t0 t0 T��

t0 0� t0 R�C/ 0 T�� C 0 T �/2�� #


2

C

h

a

p

t

e

r

,

where and is the initial condition for problem(8.2) and (8.3). We also use the notation

.

We consider the mapping from into itself defined by the formula

.

Here we assume that for . Using the estimate (see Exer-cise 1.23)

, , (8.6)

(for we suppose ) we have that

. (8.7)

If , then

.

This easily implies that

, ,

where, as above, is defined by the formula

, .

Hence, estimate (8.1) for gives us that

.

Since for , the last estimate can be rewritten inthe form

for . Therefore, (8.7) implies that

,

if , . Similarly, we have that

�: w C/ 0 T�� : w v�� C/ 0 T�� :� ��

v� t� � At� �v 0� �exp� v $� � C/�

wC/ 0 T�� A/w t� � : t 0 T�� max#

K C/ 0 T��

K w� � t� � v� t� � e t 1�� A� B w1� � 1d

0

t

3��

w $� � v $� �� $ r� 0� ��

A/ e s A� /e s��) *- ./� s 0!

/ 0� // 1�

A/ K w1 t� � K w2 t� �� /e t 1�� ) *- ./ B w1 1�� B w2 1�� 1d

0

t

3�

w �:�

A/w t� �0 T�� max : A/v 0� ��

w1 C/: v C/�� 1 0 T��

w1 C/�

w1 $� � w 1 $�� $ r� 0��

wj �:�

B w1 1�� B w2 1�� M : vC/

�� w1 1� w2 1��C/

�

w1 $� � w2 $� �� $ r� 0� ��

B w1 1�� B w2 1�� M : vC/

�� w1 w2�C/ 0 T��

1 0 T��

Kw1 K w2�C/ 0 T��

11 /��

/e��) *- ./T 1 /� M : v C/

�� w1 w2�C/ 0 T��

wj �:� j 1 2��

K w v�� C/ 0 T�� 1

1 /��

/e��) *- .

/T 1 /� B 0� � M : v C/

�� : v C/��

� ��

�


for . These two inequalities enable us to choose such that is a contractive mapping of into itself. Consequently, there existsa unique fixed point of the mapping . The structure of the ope-rator implies that . Therefore, the function

lies in and is a mild solution to problem (8.2), (8.3) on the segment. Thus, Theorem 8.1 is proved.

In many aspects the theory of retarded equations of the type (8.2) is similar to thecorresponding reasonings related to the problem without delay (see (2.1)). The exer-cises below partially confirm that.

Prove the assertions similar to the ones in Exercises 2.1–2.5and in Theorem 2.2.

Assume that the constant in (8.1) does not dependon . Prove that problem (8.2) and (8.3) has a unique mild solutionon , provided . Moreover, for any pair of solu-tions and the estimate

(8.8)

is valid, where is the initial condition for (see (8.3)).

For the sake of simplicity from now on we restrict ourselves to the case when themapping has the form

, , (8.9)

where is a continuous mapping from into , continu-ously maps into and possesses the property . We also assumethat is a potential operator, i.e. there exists a continuously Frechét differenti-able function on such that . We require that

, (8.10)

for all , where , , , and are real parameters, and are positive(cf. Section 2). As to the retarded term , we consider the uniform estimate

(8.11)

to be valid. Here is an absolute constant and .

w �:� T T / : vC/

� �� 0!�K �:

w t� � C/ 0 T�� K

K w � 0� � K w� � � 0� � v 0� ��

u t� �w t� � , t 0 T�� ,�

v t� � , t r 0�� ,��

�

C r� T �/2�� 0 T��

E x e r c i s e 8.1

E x e r c i s e 8.2 M R� �R

t0 �� v $� � C/�u1 t� � u2 t� �

u1 t� � u2 t� �� / a1 ea2 t t0��

v1 v2�C/

�

vj$� � u

jt� �

B

B v� � B0 v 0� �� B1 v� �� v v $� � C1 24��

B0.� � �1 24 D A1 2�� # H B1

.� �C1 24 H B1 0� � 0�

B0.� �

F u� � �1 24 B0 u� � F � u� ��

F u� � ��& F � u� � u�� F u� �� 7 u 2 ��&

u �1 24� � 7 � 7B1 v� �

B1 v1� � B2 v2� �� M A1 2� v1 $� � v2 $� �� $d

r�

0

3�

M vj$� � C C1 24#�


2

C

h

a

p

t

e

r

Assume that conditions (8.9)–(8.11) hold. Then problem (8.2)and (8.3) has a unique mild solution (in ) on any segment

for every initial condition from .

Therefore, we can define an evolutionary operator acting in the space by the formula

, , (8.12)

where is a mild solution to problem (8.2) and (8.3).

Prove that the operator given by formula (8.12) satisfiesthe semigroup property: , , and thepair is a dynamical system.

Theorem 8.2.

Let conditions Let conditions Let conditions Let conditions (8.9)–(8.11) and and and and (8.1) withwithwithwith hold. Assume thathold. Assume thathold. Assume thathold. Assume that

the parameters in the parameters in the parameters in the parameters in (8.10) and and and and (8.11) satisfy the equation satisfy the equation satisfy the equation satisfy the equation

....

Then the dynamical system generated by equality Then the dynamical system generated by equality Then the dynamical system generated by equality Then the dynamical system generated by equality (8.12) is a dissi- is a dissi- is a dissi- is a dissi-

pative compact system.pative compact system.pative compact system.pative compact system.

Proof.

We reason in the same way as in the proof of Theorem 4.3. Using the Galerkinapproximations it is easy to find that a solution to problem (8.2) and (8.3) satisfiesthe equalities

and

.

If we add these two equations and use (8.10), then we get that

(8.13)

where

. (8.14)

Using (8.11) it is easy to find that there exists a constant such that

E x e r c i s e 8.3

�1 24t0 t0 T�� v C C1 24�

St

C C1 24#

Stv� � $� � u

t$� � u t $�� #� $ r� 0��

u t� �


St IS1 S

t 1�� S0 I� t 1� 0&C1 24 S

t��

/ 1 24�

r2

27�� 2 7�� M2 r 2 7 � �� min"� ��

exp 2 7 � �� min�

C St

��

12��

dtd�� u t� � 2

A1 2� u t� � 2F � u t� �� u t� �� B1 u

t� � u t� ��

12��

dtd�� A1 2� u t� � 2 2 F u t� �� u� t� � 2� B1 u

t� � u� t� ��

dtd�� V u t� �� A1 2� u t� � 2 7 u t� � 2

u� t� � 2 F u t� ��

� B1 ut� � u t� � u� t� �� ,�

�

�

V u� � 12�� u

2 12�� A1 2� u

2F u� � ��

D0 0!


,

where

, .

Consequently, the inequality

is valid for . Therefore, we have

for the function

.

If we integrate this inequality from to , then we obtain

.

Therefore, Gronwall’s lemma gives us that

,

provided that

.

Here and are positive numbers, . This impliesthe dissipativity of the dynamical system . In order to prove its compactnesswe note that the reasoning similar to the one in the proof of Lemma 4.1 enables usto prove the existence of the absorbing set which is a bounded subset of thespace for . Using the equality (cf. (8.5))

for large enough, we can show that the equation

holds for the solution in the absorbing set . Here the constant dependson and the parameters of the problem only, . This circumstance enables

dtd

�� V u t� �� ?1V u t� �� D012��?2 A1 2� u 1� � 2 1d

t r�

t

3��

?1 2 7 � �� min� ?2r

2�� 1 2

7��) *- . M2�

B� t� � ?1B t� �� D0 ?2 B 1� � 1d

t r�

t

3��

B t� � V u t� ��

5� t� � D0 e?1t ?2 e

?1r 5 1� � 1d

t r�

t

3��

5 t� � e?1 tB t� � e

?1 tV u t� ��

0 t

5 t� � 5 0� �D0?1�� e

?1t1�� ?2 e

?1rr 5 1� � 1d

r�

t

3� ��

V u t� �� V u 0� �� C0 v C1 24

2�� e?� 3 t

C1��

?2 e?1r

r ?1�

C1 C2 ?3 ?1 ?2 e?1 r

r� 0!�C S

t��

�/C/ C r� 0 D A/� �2�� 1 24 / 1� �

u t� � e t s�� A� u s� � e t 1�� A� B u1� � 1d

s

t

3��

t s&

A1 2� u t� � u s� �� C t s� �

u t� � �/ C

�/ 0!


2

C

h

a

p

t

e

r

us to prove the existence of a compact absorbing set for the dynamical system. Theorem 8.2 is proved.

Theorem 8.2 and the results of Chapter 1 enable us to prove the following assertionon the attractor of problem (8.2) and (8.3).

Theorem 8.3.

Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem 8.2 hold. Then the dynamical hold. Then the dynamical hold. Then the dynamical hold. Then the dynamical

system possesses a compact connected global attractor which issystem possesses a compact connected global attractor which issystem possesses a compact connected global attractor which issystem possesses a compact connected global attractor which is

a bounded set in the space for each .a bounded set in the space for each .a bounded set in the space for each .a bounded set in the space for each .

It should be noted that the finite dimensionality of this attractor can be provedin some cases.

C St

��

C St

��

C/ C r� 0 �/2�� / 1�




Physics. –New York: Springer, 1988.3. HENRY D. Geometric theory of semilinear parabolic equations. Lect. Notes

in Math. 840. –Berlin: Springer, 1981.4. HALE J. K. Theory of functional differential equations. –Berlin; Heidelberg;

New York: Springer, 1977.

C h a p t e r 3

Inertial ManifoldsInertial ManifoldsInertial ManifoldsInertial Manifolds

C o n t e n t s

. . . . § 1 Basic Equation and Concept of Inertial Manifold . . . . . . . . . 149

. . . . § 2 Integral Equation for Determination of Inertial Manifold . . 155

. . . . § 3 Existence and Properties of Inertial Manifolds . . . . . . . . . . 161

. . . . § 4 Continuous Dependence of Inertial Manifold on Problem Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

. . . . § 5 Examples and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

. . . . § 6 Approximate Inertial Manifolds for Semilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . 182

. . . . § 7 Inertial Manifold for Second Order in Time Equations . . . . 189

. . . . § 8 Approximate Inertial Manifolds for Second Orderin Time Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

. . . . § 9 Idea of Nonlinear Galerkin Method . . . . . . . . . . . . . . . . . . . . 209

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

If an infinite-dimensional dynamical system possesses a global attractor of finitedimension (see the definitions in Chapter 1), then there is, at least theoretically,a possibility to reduce the study of its asymptotic regimes to the investigation of pro-perties of a finite-dimensional system. However, as the structure of attractor cannotbe described in details for the most interesting cases, the constructive investigationof this finite-dimensional system cannot be carried out. In this respect some ideasrelated to the method of integral manifolds and to the reduction principle are veryuseful. They have led to appearance and intensive use of the concept of inertial ma-nifold of an infinite-dimensional dynamical system (see [1]–[8] and the referencestherein). This manifold is a finite-dimensional invariant surface, it contains a globalattractor and attracts trajectories exponentially fast. Moreover, there is a possibilityto reduce the study of limit regimes of the original infinite-dimensional systemto solving of a similar problem for a class of ordinary differential equations.

In this chapter we present one of the approaches to the construction of inertialmanifolds (IM) for an evolutionary equation of the type:

, . (0.1)

Here is a function of the real variable with the values in a separable Hilbertspace . We pay the main attention to the case when is a positive linear operatorwith discrete spectrum and is a nonlinear mapping of subordinated to in some sense. The approach used here for the construction of inertial manifolds isbased on a variant of the Lyapunov-Perron method presented in the paper [2]. Otherapproaches can be found in [1], [3]–[7], [9], and [10]. However, it should be notedthat all the methods for the construction of IM known at present time require a quitestrong condition on the spectrum of the operator : the difference of two neighbouring eigenvalues of the operator should grow sufficiently fastas .

§ 1 Basic Equation and Concept§ 1 Basic Equation and Concept§ 1 Basic Equation and Concept§ 1 Basic Equation and Concept

of Inertial Manifoldof Inertial Manifoldof Inertial Manifoldof Inertial Manifold

In a separable Hilbert space we consider a Cauchy problem of the type

, , , , (1.1)

where is a positive operator with discrete spectrum (for the definition see Section1 of Chapter 2) and is a nonlinear continuous mapping from

udtd

�� A u� B u t�� ut 0� u0�

u t� � t

H A

B u t�� H A

A �N 1� �

N�

A

N �

H

udtd

�� A u� B u t�� t s ut s� u0� s R�

A

B . .�� D A�� R

150 I n e r t i a l M a n i f o l d s

3

C

h

a

p

t

e

r

into , possessing the properties

(1.2)

and

(1.3)

for all , , and from the domain of the operator . Here isa positive constant independent of and is the norm in the space . Further itis assumed that is the orthonormal basis in consisting of the eigenfunctionsof the operator :

, , .

Theorem 2.3 of Chapter 2 implies that for any initial condition prob-lem (1.1) has a unique mild (in ) solution on every half-interval ,i.e. there exists a unique function which satisfies the inte-gral equation

(1.4)

for all . This solution possesses the property (see (2.6) in Chapter 2)

,

for and . Moreover, for any pair of mild solutions and toproblem (1.1) the following inequalities hold (see (2.2.15)):

, (1.5)

and (cf. (2.2.18))

, (1.6)

where , and are positive numbers depending on , ,and only. Hereinafter , where is the orthoprojector onto the first

eigenvectors of the operator . Moreover, we use the notation

for and for . (1.7)

Further we will also use the following so-called dichotomy estimates provedin Lemma 1.1 of Chapter 2:

, ;

, ; (1.8)

H 0 � 1 ,��

B u t�� M 1 A�u��

B u1 t�� B u2 t�� M A� u1 u2��

u u1 u2 �� D A�� A� M

t . H

ek

� � H

A

A ek

�k

ek

� 0 �1 �2 �� k

k �lim ��

u0 �� u t� � s s T��

u t� � C s s T ��

u t� � et s�� A�

u0 et �� A�

B u �� d

s

t

��

t s s T��

A� u t �� u t� �� C�� 0 � ��

0 � 1� � t s u1 t� � u2 t� �

A�u t� � a1 ea2 t s��

A�u s� �� t s�

QN

A�u t� � e�

N 1� t s�� M 1 k�� a1 �N 1�

1� ��e

a2 t s��

A�u s� ��

u t� � u1 t� � u2 t� �� a1 a2 � �1M Q

NI P

N�� P

N

N A

k �� e

� d

0

�

�� 0 k 0� � 0�

A�e t A� PN �N�

e�

Nt� t R�

et A�

QN

e�

N 1� t�� t 0�

B a s i c E q u a t i o n a n d C o n c e p t o f I n e r t i a l M a n i f o l d 151

, , .

The inertial manifoldinertial manifoldinertial manifoldinertial manifold (IM) of problem (1.1) is a collection of surfaces in of the form

,

where is a mapping from into satisfying the Lipschitzcondition

(1.9)

with the constant independent of and . We also require the fulfillment of theinvariance condition (if , then the solution to problem (1.1) posses-ses the property , ) and the condition of the uniform exponentialattraction of bounded sets: there exists such that for any bounded set there exist numbers and such that

for all . Here is a mild solution to problem (1.1).From the point of view of applications the existence of an inertial manifold (IM)

means that a regular separation of fast (in the subspace ) and slow (in thesubspace ) motions is possible. Moreover, the subspace of slow motions turnsout to be finite-dimensional. It should be noted in advance that such separation isnot unique. However, if the global attractor exists, then every IM contains it.

When constructing IM we usually use the methods developed in the theoryof integral manifolds for central and central-unstable cases (see [11], [12]).

If the inertial manifold exists, then it continuously depends on , i.e.

for any and . Indeed, let be the solution to problem (1.1) with, . Then for and hence

.

Therefore,

Consequently, Lipschitz condition (1.9) leads to the estimate

.

Since , this estimate gives us the required continuity pro-perty of .

A� e t A� QN

� t!� �� N 1�� " e

�N 1� t�� t 0 � 0

Mt t R�� H

Mt p # p t�� : p PN

H # p t�� 1 PN

��

# p t�� PN

H R 1 PN

��

A� # p1 t�� # p2 t�� C A� p1 p2��

C pj t

u0 Ms� u t� �u t� � Mt� t s�

$ 0 B H%CB tB s

u t u0�� Mt��

: u0dist B��

sup CB

e$ t t

B��

t tB

� u t u0��

I PN

�� H

PN

H

t

A� # p s�� # p t�� t slim 0�

p PN

H� s R� u t� �u0 p # p s�� p P

NH� u t� � Mt� t s�

u t� � PN

u t� � # PN u t� � t��

# p t�� # p s�� # p t�� # PN u t� � t�� "

u t� � u0�� " p PN

u t� �� " .

�

� �

�

A� # p s�� # p t�� C A� u t� � u0��

u t� � C s �� D A�� # p t��


3

C

h

a

p

t

e

r


holds for when , , .

The notion of the inertial manifold is closely related to the notion of the inertialinertialinertialinertial

formformformform. If we rewrite the solution in the form , where, , and , then equation (1.1) can be re-

written as a system of two equations

By virtue of the invariance property of IM the condition implies that, i.e. the equality implies that .

Therefore, if we know the function that gives IM, then the solution lying in can be found in two stages: at first we solve the problem

, , (1.10)

and then we take . Thus, the qualitative behaviour of solu-tions lying in IM is completely determined by the properties of differentialequation (1.10) in the finite-dimensional space . Equation (1.10) is said to bethe inertial form (IF) of problem (1.1). In the autonomous case ( )one can use the attraction property for IM and the reduction principle (see Theorem7.4 of Chapter 1) in order to state that the finite-dimensional IF completely deter-mines the asymptotic behaviour of the dynamical system generated by problem (1.1).

Let give the inertial manifold for problem (1.1).Show that IF (1.10) is uniquely solvable on the whole real axis, i.e.there exists a unique function such thatequation (1.10) holds.

Let be a solution to IF (1.10) defined for all . Provethat is a mild solution to problem (1.1) de-fined on the whole time axis and such that .

Use the results of Exercises 1.2 and 1.3 to show that if IM exists, then it is strictly invariant, i.e. for any and there exists such that is a solution to prob-

lem (1.1).

E x e r c i s e 1.1

A� # p t�� # p t�� C� p N��

# p t�� 0 � 1� � 0 � �� t R�

u t� � u t� � p t� � q t� ��p t� � P

Nu t� �� q t� � Q

Nu t� �� Q

NI P

N��

dtd

�� p t� � A p t� �� PN

B p t� � q t� �� ,�

dtd

�� q t� � A q t� �� QN B p t� � q t� �� ,�

pt s� p0 PN u0 , q

t s� q0 QN u0 .&�&��'''�'''�

p0 q0�� Ms�p t� � q t� �� Mt� q0 # p0 s�� q t� � # p t� � t��

# p t�� u t� �Mt

dtd

�� p t� � A p t� �� PN B p t� � # p t� � t�� pt s� p0�

u t� � p t� � # p t� � t�� u t� �

PN

H

B u t�� B u� �&

E x e r c i s e 1.2 # p t��

p t� � C �� PN H��

E x e r c i s e 1.3 p t� � t R�u t� � p t� � # p t� � t��

ut s� p0 # p0 t��

E x e r c i s e 1.4

Mt� � u Mt�s t� u0 Mt� u u t� ��

B a s i c E q u a t i o n a n d C o n c e p t o f I n e r t i a l M a n i f o l d 153

In the sections to follow the construction of IM is based on a version of the Lyapunov-Perron method presented in the paper by Chow-Lu [2]. This method is based on thefollowing simple fact.

Lemma 1.1.

Let be a continuous function on with the values in such that

, .

Then for the mild solution (on the whole axis) to equation

(1.11)

to be bounded in the subspace it is necessary and sufficient that

(1.12)

for , where is an element from and is an arbitrary real

number.

We note that the solution to problem (1.11) on the whole axis is a function satisfying the equation

for any .

Proof.

It is easy to prove (do it yourself) that equation (1.12) gives a mild solutionto (1.11) with the required property of boundedness. Vice versa, let bea solution to equation (1.11) such that is bounded. Then the func-tion is a bounded solution to equation

.

Consequently, Lemma 2.1.2 implies that

.

Therefore, in order to prove (1.12) it is sufficient to use the constant variationformula for a solution to the finite-dimensional equation

, .


f t� � R H

QN

f t� � C� t R�

u t� �

dtd�� u A u� f t� ��

QN ��

u t� � e t s�� A� p e t �� A� PN

f �� d

s

t

� e t �� A� QN

f �� d

��

t

��

t R� p PN H s

u t� � �C R H��

u t� � et s�� A�

u s� � et �� A�

f �� d

s

t

��

s R�

u t� �Q

Nu t� � �

q t� � QN

u t� ��

dtd�� q t� � A q t� �� Q

Nf t� ��

q t� � et �� A�

QN

f �� d

��

t

��

pdtd

�� Ap� PN

f t� �� p t� � PN

u t� ��


3

C

h

a

p

t

e

r

Lemma 1.1 enables us to obtain an equation to determine the function .Indeed, let us assume that is bounded and there exists with the func-tion possessing the property for all and .Then the solution to problem (1.1) lying in has the form

.

It is bounded in the subspace and therefore it satisfies the equation of theform

(1.13)

Moreover,

. (1.14)

Actually it is this fact that forms the core of the Lyapunov-Perron method. It isproved below that under some conditions (i) integral equation (1.13) is uniquelysolvable for any and (ii) the function defined by equality (1.14)gives IM.

In the construction of IM with the help of the Lyapunov-Perron method an im-portant role is also played by the results given in the following exercises.

Assume that , where is anynumber from the interval and . Let be a mildsolution (on the whole axis) to equation (1.11). Show that pos-sesses the property

if and only if equation (1.12) holds for . Hint: consider the new unknown function

instead of .

Assume that is a continuous function on the semiaxis with the values in such that for some from the interval

the equation

holds. Prove that for a mild solution to equation (1.11) on thesemiaxis to possess the property

# p t�� B u t�� Mt

# p t�� A�# p t�� C� p PN

H� t R�Mt

u t� � p t� � # p t� � t��

QN

H

u t� � e t s�� A� p

et �� A�

PN

B u �� d

s

t

� et �� A�

QN

B u �� , t R�� .d

��

t

�

�

� �

�

# p s�� QN

u s� � es �� A�

QN

B u �� d

��

s

��

p PN

H� # p s��

E x e r c i s e 1.5 e$ s t��

f t� � : t s�� sup �� $�

N�

N 1�� s R� u t� �u t� �

e $ s t�� A�u t� �� t s�sup ��

t s�

w t� � e$ t s��

u t� ��u t� �

E x e r c i s e 1.6 f t� �s �� H $�

N�

N 1��

e$ s t��

f t� � : t s �� sup ��u t� �

s ��

I n t e g r a l E q u a t i o n f o r D e t e r m i n a t i o n o f I n e r t i a l M a n i f o l d 155

it is necessary and sufficient that

(1.15)

where and is an element of . Hint: see the hint toExercise 1.5.

§ 2 Integral Equation for Determination§ 2 Integral Equation for Determination§ 2 Integral Equation for Determination§ 2 Integral Equation for Determination

of Inertial Manifoldof Inertial Manifoldof Inertial Manifoldof Inertial Manifold

In this section we study the solvability and the properties of solutions to a class of in-tegral equations which contains equation (1.13) as a limit case. Broader treatment ofthe equation of the type (1.13) is useful in connection with some problems of the ap-proximation theory for IM.

For and we define the space as the setof continuous functions on the segment with the values in and such that

.

Here is a positive number. In this space we consider the integral equation

, , (2.1)

where

Hereinafter the index of the projectors and is omitted, i.e. is the ortho-projector onto and . It should be noted that the most sig-nificant case for the construction of IM is when .

e$ s t��

A�u t� � : t s �� sup ��

u t� � et s�� A�

q et �� A�

QN

f �� d

s

t

�e

t �� A�PN

f �� ,d

t

��

��

��

t s� q QN

D A��

s R� 0 L �� Cs C$ �� s L� s�� &v t� � s L� s�� " D A��

vs

e$ s t��

A�u t� �� t s L� s�� "�

sup ��&

$

v t� � Bps L�

v� " t� �� s L� t s� �

Bps L�

v� " t� � e t s�� A� p e t �� A� PB v �� d

t

s

��

e t �� A� QB v �� .d

s L�

t

�

�

�

�

N PN QN P

Lin e1 � eN� �� Q 1 P��L ��


3

C

h

a

p

t

e

r

Lemma 2.1.

Let at least one of two conditions be fulfilled:

and (2.2)

or and

, , (2.3)

where is defined by equation (1.7). Then for any fixed there

exists a unique function satisfying equation (2.1) for all

, where is an arbitrary number from the segment

in the case of (2.2) and in the case of (2.3).

Moreover,

(2.4)

and

, (2.5)

where

. (2.6)

Proof.

Let us apply the fixed point method to equation (2.1). Using (1.8) it is easyto check (similar estimates are given in Chapter 2) that

where

(2.7)

0 L �� M��

1 �� L1 �� L�

N 1��( )

* + q 1��

0 L ��

�N 1� �N� 2 Mq�� 1 k�� N 1�

� �N

�� 0 q 1� �

k s R�vs t p�� Cs�

t s L� s�� "� $ �N��N 1� " $ �N 2 M q!� � �N

��

v . p1�� v . p1�� s

1 q�� 1� A� p1 p2��

vs s1 q�� 1� D1 A�p��

D1 M 1 k�� N 1�

1� ��M�

N1� ��

A� Bp1

s L�v1� � t� � Bp2

s L�v2� � t� ��

e�

Ns t��

A� p1 p2�� N�

e�

N� t��

M v1 �� v2 �� d

t

s

� � �

�

�

�t ��( )* +� �

N 1�� e

��N 1� t ��

M v1 �� v2 �� d

s L�

t

� e

�N

s t�� A� p1 p2�� q1 s t�� q2 s t�� e

$ s t�� v1 v2�

s ,�

�

�

�

q1 s t�� M�

t ��( )* +� �N 1�

�� e�

N 1� $�� t �� d

s L�

t

��


and

. (2.8)

Therefore, if the estimate

, (2.9)

holds, then

. (2.10)

Let us estimate the values and . Assume that (2.2) is fulfilled.Then it is evident that

and

for . Therefore,

.

Consequently, equation (2.2) implies (2.9). Now let the spectral condition (2.3)be fulfilled. Then

for all . We change the variable in integration and find that

,

where the constant is defined by (1.7). It is also evident that

provided that . Equation (2.3) implies that lies inthe interval . If we choose the parameter in such way, then we get

q2 s t�� M �N�

e�

N$�� t��

�d

t

s

��

q1 s t�� q2 s t�� q� s L� t s� �

Bp1

s L�v1� " Bp2

s L�v2� "�

sA� p1 p2�� q v1 v2�

s��

q1 s t�� q2 s t��

q1 s t�� M�� t �� d

s L�

t

� M�N 1��

t s� L��

M��

1 �� t s� L�� 1 �� M�N 1�

�t s� L��

�

�

q2 s t�� M�N�

s t�� M�N 1��

s t��

�N $ �N 1��

q1 s t�� q2 s t�� M��

1 �� t s� L�� 1 ��

N 1��

L�( )* +�

q1 s t�� M��

t �� e

�N 1� $�� t �� d

��

t

� M�N 1��

�N 1� $�

��

$ �N 1��

N 1� $�� t ��

q1 s t�� M k

�N 1� $�� 1 ��

��M�

N 1��

�N 1� $��

k

q2 s t�� M�N

�

$ �N

��

$ �N $ �N 2 M q!� � �N��

�N �N 1�� $


3

C

h

a

p

t

e

r

.

Hence, equation (2.3) implies (2.9). Therefore, estimate (2.10) is valid, provi-ded that the hypotheses of the lemma hold. Moreover, similar reasoning enablesus to show that

, (2.11)

where is defined by formula (2.6). In particular, estimates (2.10) and (2.11)mean that when , , and are fixed, the operator maps into itselfand is contractive. Therefore, there exists a unique fixed point . Evi-dently it possesses properties (2.4) and (2.5). Lemma 2.1 is proved.

Lemma 2.1 enables us to define a collection of manifolds by the formula

,

where

. (2.12)

Here is the solution to integral equation (2.1). Some properties ofthe manifolds and the function are given in the following assertion.

Theorem 2.1.

Assume that at leaAssume that at leaAssume that at leaAssume that at leasssst one of two conditions t one of two conditions t one of two conditions t one of two conditions (2.2) and and and and (2.3) is satisfied.is satisfied.is satisfied.is satisfied.

Then the mapping from into possesses the propertiesThen the mapping from into possesses the propertiesThen the mapping from into possesses the propertiesThen the mapping from into possesses the properties

a) (2.13)

for any , hereinafter is defined by formula for any , hereinafter is defined by formula for any , hereinafter is defined by formula for any , hereinafter is defined by formula (2.6) and and and and

;;;; (2.14)

b) the manifold is a Lipschitzian surface andthe manifold is a Lipschitzian surface andthe manifold is a Lipschitzian surface andthe manifold is a Lipschitzian surface and

(2.15)

for all and ;for all and ;for all and ;for all and ;

c) if if if if is the solution to problem is the solution to problem is the solution to problem is the solution to problem (1.1) with the with the with the with the

initial data initial data initial data initial data ,,,, ,,,, then then then then

for for for for .... In case of the inequality In case of the inequality In case of the inequality In case of the inequality

(2.16)

q1 s t�� q2 s t�� M 1 k��

N 1��

�N 1� �

N� 2 M

q��

N��

��q

2��

Bps L�

v� "s

D1 A�p q vs

� ��

D1s L p Bp

s L�C

s

vs

t p��

MsL� �

MsL p #L p s�� : p P H��

#L p s�� es �� A�

Q B v �� d

s L�

s

� Q v s p�� &�

v t� � v t p�� Ms

L� � #L p s��

#L . s�� P H Q H

A�#L p s�� D2 q 1 q�� 1� D1 A�p��

p PH� D1

D2 M 1 k�� N 1�

1� ��

MsL

A�#L p1 s�� #L p2 s�� q

1 q�� A� p1 p2��

p1 p2� P H� s R�u t� � u t s p #

sL

p� �� &u0 p #L p s�� p PH� Qu t� � #L Pu t� � t��

L �� L ��

A� Qu t� � #L Pu t� � t��

D2 1 q�� 1�e

$L�q 1 q�� 2�

e$ t s��

D1 A�p��

�

�


holds for all , where is an arbitrary number from theholds for all , where is an arbitrary number from theholds for all , where is an arbitrary number from theholds for all , where is an arbitrary number from the

segment if segment if segment if segment if (2.2) is fulfilled and when is fulfilled and when is fulfilled and when is fulfilled and when

(2.3) is fulfilled; is fulfilled; is fulfilled; is fulfilled;

d) if does not depend on if does not depend on if does not depend on if does not depend on ,,,, then , i.e. then , i.e. then , i.e. then , i.e.

is independent of is independent of is independent of is independent of ....

Proof.

Equations (2.12) and (1.8) imply that

By virtue of (2.9) we have that . Therefore, when we change the vari-able in integration with the help of equation (2.5) we obtain (2.13).Similarly, using (2.4) and (1.8) one can prove property (2.15).

Let us prove assertion (c). We fix and assume that is afunction on the segment such that for and

for . Here is the solution to integral equation (2.1). Usingequations (1.4) and (2.1) we obtain that

(2.17)

for . Evidently, equation (2.17) also remains true for . Equa-tion (1.4) gives us that

.

Therefore, the substitution in (2.17) gives us that

(2.18)

for all , where and

. (2.19)

s t s L�� $�

N�

N 1�� " $ �N

2 M q!� � �N��

B u t�� B u� �& t #L p s�� #L p� �&#L p t�� t

A�#L p s�� M�

s ��( )* +� �

N 1�� e

�N 1� s ��

1 A�v �� d

s L�

s

�M

�s ��( )* +� �

N 1�� e

�N 1� s �� d

s L�

s

� q1 s s�� vs .�

� �

�

q1 s s�� q� �

N 1� s ��

t0 s s L�� "� w t� �s s L�� " w t� � u t� �� t s t0�� "� w t� � �

vs

t� �� t s L� s�� "� vs

t� �

w t� � e t s�� A� p #L p s�� et �� A�

B w �� d

s

t

��

et s�� A�

p et �� A�

PB w �� d

s

t

� et �� A�

QB w �� d

s L�

t

��

� �

�

s t t0� � t s L� s�� "�

p es t0�� A�

p t0� � es �� A�

PB w �� d

t0

s

��

w t� � Bp t0� �t0 L�

w� " t� � bL

t0 s t��

t t0 L� t0�� "� p t� � P u t� ��

bL

t0 s t�� et �� A�

QB vs�� d

s L�

t0 L�

��


3

C

h

a

p

t

e

r

In particular, if equation (2.18) turns into equation (2.1) with and. Therefore, equation (2.12) implies the invariance property

. Let us estimate the value (2.19). If we reason in the same way asin the proof of Lemma 2.1, then we obtain that

,

where is defined by formula (2.7) and

. (2.20)

Therefore, simple calculations give us that

, (2.21)

where is defined by formula (2.14). Let be the solution to integral equa-tion (2.1) for and . Then using (2.12), (2.18), and (2.1) we findthat

. (2.22)

However, for all we have that

.

Therefore, the contractibility property of the operator gives us that

.

Hence, it follows from (2.21) and (2.22) that

This and equation (2.5) imply (2.16). Therefore, assertion (c) is proved.In order to prove assertion (d) it should be kept in mind that if

, then the structure of the operator enables us to state that

for , where . Therefore, if is a solution to integral equation (2.1), then the function

L �� s t0�p p t0� �� Q u t0� � �#� P u t0� � t0��

A�bL t0 s t�� et t0� L��

N 1��q1* s t0 L�� q1 s t0 L�� e

$ s t0� L�� vs s

��

�

q1 s t��

q1* s t�� M

�t ��( )

* +� �N 1��( )

* + e�

N 1� t �� d

s L�

t

��

A�bL

t0 s t�� et t0� L��

N 1��D2 e

$ s t0� L�� q v

s s�

� ��

�

D2 vt0t� �

s t0� p Pu t0� ��

Q u t0� � #L Pu t0� � t0�� Q w t0� � vt0

t0� ��

t t0 L� t0�� "�

w t� � vt0t� �� Bp t0� �

t0 L�w� " t� � Bp t0� �

t0 L�vt0� " t� �� bL t0 s t��

Bp

t0 L�

1 q�� w vt0�

t0e

$ t0 t�� A�bL t0 s t��

� ��

t t0 L t0�� "�sup�

A� Q u t0� �� #L P u t0� � t0�� A� w t0� � vt0t0� ��

w vt0

�t0

1 q�� 1�e $L� D2 q e

$ t0 s�� v

s s�

� ��

.

� �

� �

� u t�� &� u� �& Bp

s L�

Bps L�

v� " t h�� Bps h L��

vh� " t� ��

s h L t s h�� vh

t� � v t h�� v t� � C$ �� s L s��

vh t� � v t h�� C$ �� s h L�� s h�� &

E x i s t e n c e a n d P r o p e r t i e s o f I n e r t i a l M a n i f o l d s 161

is its solution when is written instead of . Consequently, equation (2.12)gives us that

.


Show that if , then inequalities (2.13) and(2.16) can be replaced by the relations

, (2.23)

, (2.24)

where is defined by formula (2.14).

§ 3 Existence and Properties§ 3 Existence and Properties§ 3 Existence and Properties§ 3 Existence and Properties

of Inertial Manifoldsof Inertial Manifoldsof Inertial Manifoldsof Inertial Manifolds

In particular, assertion (c) of Theorem 2.1 shows that if the spectral gap condition

, , (3.1)

is fulfilled, then the collection of surfaces

, , (3.2)

is invariant, i.e.

, . (3.3)

Here is defined by formula (2.12) for and isthe evolutionary operator corresponding to problem (1.1). It is defined by the for-mula , where is a mild solution to problem (1.1).

In this section we show that collection (3.2) possesses the property of exponen-tial uniform attraction. Hence, is an inertial manifold for problem (1.1). More-over, Theorem 3.1 below states that is an exponentially asymptoticallyexponentially asymptoticallyexponentially asymptoticallyexponentially asymptotically

completecompletecompletecomplete IM, i.e. for any solution there exists a solution lying in the manifold such that

, , .

In this case the solution is said to be an induced trajectoryinduced trajectoryinduced trajectoryinduced trajectory for on themanifold . In particular, the existence of induced trajectories means that the so-lution to original infinite-dimensional problem (1.1) can be naturally associated withthe solution to the system of ordinary differential equations (1.10).

s h� s

#L p s h�� Q vh

s h�� Q v s� � #L p s��

E x e r c i s e 2.1 B u t�� M�

A�#L p s�� D2�

A� Qu t� � #L Pu t� � t�� D2 1 q�� 1�e

$L��

D2

�N 1� �

N2 Mq�� 1 k��

N 1��

N�� 0 q 1� �

Ms p # p s�� : p P H�� s R�

U t s�� Ms Mt% �� s t ��

# p s�� #� p s�� L �� U t s��

U t s�� u0 u t� �� u t� �

Mt� �Mt� �

u t� � U t s�� u0� u�

t� � �U t s�� u

�0� u

�t� � Mt t s��

A� u t� � u�

t� �� C e, t s�� , 0 t s

u�

t� � u t� �Mt


3

C

h

a

p

t

e

r

Theorem 3.1.

Assume that spectral gap condition Assume that spectral gap condition Assume that spectral gap condition Assume that spectral gap condition (3.1) is valid for some is valid for some is valid for some is valid for some ....

Then the manifold given by formula Then the manifold given by formula Then the manifold given by formula Then the manifold given by formula (3.2) is inertial for prob- is inertial for prob- is inertial for prob- is inertial for prob-

lem lem lem lem (1.1).... Moreover, for any solution there exists an in- Moreover, for any solution there exists an in- Moreover, for any solution there exists an in- Moreover, for any solution there exists an in-

duced trajectory such that for andduced trajectory such that for andduced trajectory such that for andduced trajectory such that for and

,,,, (3.4)

where and where and where and where and ....

Proof.

Obviously it is sufficient just to prove the existence of an induced trajectory possessing property (3.4). Let be a mild solution to problem (1.1),

. We construct the induced trajectory for in the form , where lies in the space

of continuous functions on the semiaxis such that

, (3.5)

where . We introduce the notation

(3.6)

and consider the integral equation (cf. (1.15))

(3.7)

in the space . Here the value is chosen from the condition

,

i.e. such that

.

Therefore, by virtue of (3.7) we have

. (3.8)

Thus, in order to prove inequality (3.4) it is sufficient to prove the solvability of inte-gral equation (3.7) in the space and to obtain the estimate of the solution. Thepreparatory steps for doing this are hidden in the following exercises.

q 2 2��Ms s R��

u t� � U t s�� u0�u* t� � U t s�� u0

*� u* t� � Mt� t s�

A� u t� � u* t� �� 2 1 q��

2 q�� 2 2�� e

$ t s�� A� Qu0 # P0u s��

$ �N

2 Mq��

N�� t s�

u* t� � Mt� u t� �u t� � U t s�� u0� u* t� � U t s�� u0

*� u t� �u* t� � u t� � w t� �� w t� � Cs

� Cs $� s �� &D A�� s ��

ws �� e

$ t s�� A�w t� ��

t s�sup ��&

$ �N

2 M q!� � �N��

F w t�� B u t� � w� t�� B u t� ��

w t� � Bs�

w� " t� � et s�� A�

q w� � et �� A�

Q F w ��

et �� A�

P F w �� , t s �� ,��d

t

��

��

�d

s

t

��&�

Cs� q w� � Q D A��

u* s� � u s� � w s� �� Ms��

Q u0 Q w s� �� # Pu0 P w s� �� s��

q w� � Q u0� # P u0 es �� A�

P F w �� d

s

��

�� s�( )- .- .* +

��

Cs�


Assume that has the form (3.6). Show that for any

and for the following inequalities hold:

, (3.9)

. (3.10)

Using (1.8) prove that the equations

, (3.11)

(3.12)

hold for and . Here is defined by formula(1.7).

Lemma 3.1.

Assume that spectral gap condition (3.1) holds with . Then

is a continuous contractive mapping of the space into itself.

The unique fixed point of this mapping satisfies the estimate

. (3.13)

Proof.

If we use (3.7), then we find that

for . Therefore, (3.9), (3.11), and (3.12) give us that

E x e r c i s e 3.1 F w t��

w t� � w t� � Cs�� C

s $� s �� D A��

t s�

F w t� � t�� e$ t s��

M w s ��

F w t� � t�� F w t� � t�� e$ t s��

M w w� s ��

E x e r c i s e 3.2

A�et �� A�

P e$ � s�� d

t

��

�( )- .- .* + �N

�

$ �N

�� e

$ t s�� /�

A�et �� A�

Q e$ � s�� d

s

t

�( )- .- .* +

k �N 1� $��

N 1��

�N 1� $�

�� e$ t s�� /

�

�

�N $ �N 1�� t s� k

q 2 2��Bs�

Cs�

w

w s ��2 1 q��

2 q�� 2 2�� A� Q u0 # P u0 s��

A�Bs�

w� " t� � et s��

N 1��A�q w� �

A�et �� A�

Q F w �� A�et �� A�

P F w �� d

t

��

��d

s

t

�

�

�

�

t s

A�Bs�

w� " t� � et s��

N 1��A�q w� �

�N�

$ �N

��

1 k�� N 1��

�N 1� $�

��

M e$ t s��

ws �� .

�

�

�


3

C

h

a

p

t

e

r

Since , spectral gap condition (3.1) implies that

. (3.14)

Similarly with the help of (3.10)–(3.12) we have that

(3.15)

for any . From equations (3.8), (3.9), and (2.15) we obtain that

.

Therefore, (3.11) implies that

.

Similarly we have that

. (3.17)

It follows from (3.14)–(3.17) that

, (3.18)

.

Therefore, if , then the operator is continuous and contractive in. Estimate (3.13) of its fixed point follows from (3.18). Lemma 3.1 is proved.

In order to complete the proof of Theorem 3.1 we must prove that the function

is a mild solution to problem (1.1) lying in (here is a solutionto integral equation (3.7) ). We can do that by using the result of Exercise 1.2, the in-variance of the collection , and the fact that equality (3.8) is equivalent to theequation . Theorem 3.1 is completely proved.

Show that if the hypotheses of Theorem 3.1 hold, then the in-duced trajectory is uniquely defined in the following sense: ifthere exists a trajectory such that for and

with , then for .

$ �N

2 M q!� � �N��

A�Bs�

w� " t� � et s��

N 1��A�q w� � q e

$ t s�� w

s ��

A� Bs�

w� " t� � Bs�

w� " t� ��

et s��

N 1��A� q w� � q w� �� q e

$ t s�� w w�

s ��

�

�

w w� Cs��

A� q w� � Q u0 # Pu0 s�� q M

1 q�� A�es �� A�

P e$ � s�� d

s

��

� ws ��/�

A�q w� � A� Q u0 # Pu0 s�� q2

2 1 q�� w

s ��

A� q w� � q w� �� q2

2 1 q�� w w�

s ��

Bs�

w� "s �� A� Q u0 # Pu0 s��

q

2��

2 q�1 q�� w

s ��/��

Bs�

w� " Bs�

w� "�s ��

q

2��

2 q�1 q�� w w�

s ��/�

q 2 2�� Bs�

Cs�

u* t� � u t� � w t� ��

Mt t� s�� w t� �

Mt� �u* s� � Ms�

E x e r c i s e 3.3

u* t� �u** t� � u** t� � Mt� t s�

A� u t� � u** t� �� C e$ t s��

$ �N2 Mq��N

�� u** t� � u* t� �& t s�


The construction presented in the proof of Theorem 3.1 shows that in order to buildthe induced trajectory for a solution with the exponential order of decrease given, it is necessary to have the information on the behaviour of the solution for allallallall values . In this connection the following simple fact on the exponentialcloseness of the solution to its projection onto the mani-fold appears to be useful sometimes.

Show that if the hypotheses of Theorem 3.1 hold, then the es-timate

is valid for any solution to problem (1.1). Here and (Hint: add the value

to the expression under the norm sign in the left-handside. Here is the induced trajectory for ).

It is evident that the inertial manifold consists of the solutions to problem(1.1) which are defined for all real (see Exercises 1.3 and 1.4). These solutions canbe characterized as follows.

Theorem 3.2.

Assume that spectral gap condition Assume that spectral gap condition Assume that spectral gap condition Assume that spectral gap condition (3.1) holds with and holds with and holds with and holds with and

is the inertial manifold for problem is the inertial manifold for problem is the inertial manifold for problem is the inertial manifold for problem (1.1) constructed in Theorem constructed in Theorem constructed in Theorem constructed in Theorem 3.1....

Then for a solution to problem Then for a solution to problem Then for a solution to problem Then for a solution to problem (1.1) defined for all to lie in the defined for all to lie in the defined for all to lie in the defined for all to lie in the

inertial manifold , it is necessary and sufficient thatinertial manifold , it is necessary and sufficient thatinertial manifold , it is necessary and sufficient thatinertial manifold , it is necessary and sufficient that

(3.19)

for each , where for each , where for each , where for each , where ....

Proof.

If , then . Therefore, equation (2.13)implies that

. (3.20)

The function satisfies the equation

for all real and . Therefore, we have that

u t� � $u t� �

t s�u t� � Pu t� � # P u t� � t��

E x e r c i s e 3.4

A� Q u t� � # Pu t� � t��

22 q�� 2 2�� e

$ t s�� A� Pu0 # P u0 t��

�

�

u t� � $ �N ��2 M q!� � �N

�� t s� # Pu* t� � t�� Qu* t� �� 0�

u* t� � u t� �

Mt� � u t� �t

q 2 2��Mt� �

u t� � t R�u t� � Mt��

us

e$ s t��

A�u t� � : �� t s�� sup ��&

s R� $ �N2 Mq��N

��

u t� � Mt� u t� � Pu t� � # Pu t� � t��

A�u t� � D2

q D11 q��

11 q�� A�Pu t� ��

p t� � Pu t� ��

p t� � e t s�� A� p s� � et �� A�

PB u �� d

s

t

��

t s


3

C

h

a

p

t

e

r for . With the help of (3.20) we find that

for , where

.

Hence, the inequality

holds for the function and . If we introduce the func-tion , then the last inequlity can be rewritten in the form

, ,

or

, .

After the integration over the segment and a simple transformation it is easyto obtain the estimate

. (3.21)

Obviously for we have that

.

Therefore, equations (3.21) and (3.20) imply (3.19).Vice versa, we assume that equation (3.19) holds for the solution . Then

, . (3.22)

It is evident that is a bounded (on ) solution to theequation

,

A�p t� � es t��

N A�p s� � M�N

�e� t��

N 1 A�u �� d

t

s

��

t s�

A�p t� � C s N q� �� es t��

NM�N

�

1 q�� e� t��

N A�p �� d

t

s

��

t s�

C s N q� �� A�p s� � M 1 D2

q D11 q�� ( )

* + 1

�N1 ��

��

0 t� � C s N q� �� M�

N�

1 q�� 0 �� d

t

s

��

0 t� � A�p t� � et s��

N� t s�1 t� � 0 �� d

t

s

��

1� t� �M�

N�

1 q��1 t� �� C s N q� �� t s�

dtd

�� 1 t� �M�

N

�

1 q�� t� ��

exp C s N q� �� M �

N

�

1 q�� t� ��

exp�� t s�

t s�� "

A�p t� � C s N q� �� N

M�N�

1 q��( )* + s t�� exp�

q 2 2��

�N

M�N�

1 q�� $�� N

2 Mq��

N��

u t� �

B u t� �� e$ s t��

M 1 u s�� t s�

q t� � e$ s t��

Q u t� �� s� "�

qdtd

�� A $�� q� F t� ��


where . By virtue of (3.22) the function isbounded in . It is also clear that is a positive operator with discretespectrum in . Therefore, Lemma 1.1 is applicable. It gives

.

Using the equation for it is now easy to find that

, ,

where and is the integral operator similar to the one in (2.1).Hence, we have that accoring to definition (2.12) of thefunction . Thus, Theorem 3.2 is proved.

The following assertion shows that IM can be approximated by the mani-folds , , with the exponential accuracy (see (2.12)).

Theorem 3.3.

Assume that spectral gap condition Assume that spectral gap condition Assume that spectral gap condition Assume that spectral gap condition (3.1) is fulfilled with is fulfilled with is fulfilled with is fulfilled with .... We also We also We also We also

assume that the function is defined by equality assume that the function is defined by equality assume that the function is defined by equality assume that the function is defined by equality (2.12) for for for for

. Then the estimate. Then the estimate. Then the estimate. Then the estimate

(3.23)

is valid with , , ; the constants and is valid with , , ; the constants and is valid with , , ; the constants and is valid with , , ; the constants and

are defined by equations are defined by equations are defined by equations are defined by equations (2.6) and and and and (2.14);;;;

,,,, ....

Proof.

Let . Definition (2.12) implies that

, (3.24)

where is a solution to integral equation (2.1) with , . The ope-rator acting in (see (2.1)) can be represented in the form

, ,

where

F t� � $ s t�� exp Q B u t� �� F t� �Q H A$ A $��Q H

Q u t� � e t �� A� Q B u �� d

��

t

��

P u t� �

u t� � Bps ��

u� " t� �� t s�

p P u s� �� Bps ��

u� "Q u s� � # Pu s� � s��

# p s�� #� p s��

Ms Ms�&

MsL� � L ��

q 1�#L p s��

0 L ��

A� #L1 p s�� #L2 p s��

D2 1 q�� 1� e$

NLmin� 1 q�

2 1 q�� 2�� D1 A�p�� e

2N

Lmin� ,�

�

�

Lmin L1 L2�� min� 0 L1� L2 �� D1 D2

$N

�N

2 Mq��

N�� 2

N

2 M 1 q�� q 1 q��

N��

0 L1 L2 ��

#L1 p s�� #L2 p s�� Q v1 s� � v2 s� ��

vj t� � L Lj� j 1 2��Bp

s L2�C$ �� s L2� s��

Bp

s L2�v� " t� � B

p

s L1�v� " t� � b v t s�� t s L1� s�� "�

b v t� s�� e t �� A� Q B v �� d

s L2�

s L1�

��


3

C

h

a

p

t

e

r

and is an arbitrary element in . Therefore, if is a solutionto problem (2.1) with , then

(3.25)

for all . Let us estimate the value . As before it is easy toverify that

for all , where

,

,

and the norm is defined using the constants and by the formula

.

Evidently, spectral gap condition (3.1) implies the same equation with the parameter instead of . Therefore, simple calculations based on (1.8) give us that

and ,

where is defined by formula (2.14). Using Lemma 2.1 under condition (2.3) with instead of we obtain that

,

where is given by formula (2.6). Therefore, finaly we have that

for all . Consequently,

v t� � C$ �� s L2� s�� vj

t� �L L

j�

v1 t� � v2 t� �� Bp

s L1�v1� " Bp

s L1�v2� " b v2 t� s��

s L1� t s� � b v2 t� s��

A�b v2 t� s�� et s� L1��

N 1��r1 s L1�� r2 s L1�� v2 s *�

��

t s L1� s�� "�

r1 t� � M A�e t �� A� Q �d

��

t

��

r2 t� � M A�e t �� A� Q e$*

s �� d

��

t

��

v2 s *�q*

1 q�2

�� $* �N

2Mq*

��N��

v2 s *�e

$*

s t�� A�v2 t� � : t s L2� s�� "�

� ��

sup�

q* q

r1 t� � D2� r2 e$*

s t�� q*

2��

D2q* q

v2 s *�1 q*�� 1�

D1 A�p��

D1

A�b v2 t� s�� et s� L1��

N 1��D2

q*

2 1 q*�� e$*L1 D1 A�p��

� ��

t s L1� s�� "�

e$ t s��

A�b v2 t� s�� : t s L1� s�� "��

sup

e$L1�

D2q*

2 1 q*�� e

$*L1 D1 A�p��

� �� .

�

�


Therefore, since is a contractive operator in , equation (3.25)gives us that

Here we also use the equality . Hence, estimate (3.23) follows from(3.24). Theorem 3.3 is proved.

Show that in the case when equation (3.23)can be replaced by the inequality

.

Assume that the hypotheses of Theorem 3.1 hold. Then theestimate

holds for and for any solution to problem (1.1) possess-ing the dissipativity property: for and forsome and . Here and the constant does not depend on .

Therefore, if the hypotheses of Theorem 3.1 hold, then a bounded solution to prob-lem (1.1) gets into the exponentially small (with respect to and ) vicinity ofthe manifold at an exponential velocity.

According to (2.12) in order to build an approximation of the inertialmanifold we should solve integral equation (2.1) for large enough. Thisequation has the same structure both for and for . Therefore, it is im-possible to use the surfaces directly for the effective approximation of .However, by virtue of contractiveness of the operator in the space

, its fixed point which determines can be found with thehelp of iterations. This fact enables us to construct the collection of appro-ximations for as follows. Let be an element of . We take

, ,

and define the surfaces by the formula

,

where ,

Bp

s L1�C$ �� s L1� s��

1 q�� v1 v2�C$ �� s L1� s��

e$L1�

D212��

1 q�1 q�� e

$ $*

�� L1� D1 A�p�( )* + ./�

�

�

q*12�� 1 q��


A� #L1 p s�� #L2 p s�� D2 1 q�� 1�e

$N

Lmin��

E x e r c i s e 3.6

A� Q u t� � #L P u t� � t��

C 1 A�u0�( )* + e

$N

t s�� C

Re

3�N

�L��

�

�

t t*

� u t� �A�u t� � R� t t

*s� �

R t*

$N

�N

2 Mq��

N�� 3 0

N

�N�

L

MsL : �� s ��

MsL� �

Ms� � L

L �� L ��Ms

L� � Ms� �Bp

s ��C

s– �

C$ �� s�� vs

t� � Ms

Mn s�� Ms� � v0 v0 s� t p�� C

s–

vn

vn s� t p�� & Bp

s ��v

n 1�� " t� �� n 1 2 ��

Mn s��

Mn s� p #n

p s�� : p P H��

#n

p s�� Q vn s� p s�� n 1 2 ��


3

C

h

a

p

t

e

r

Let and let . Show that

and .

Assume that spectral gap condition (3.1) is fulfilled. Showthat

,

where is defined by formula (2.6) and is the functionthat determines the inertial manifold.

Prove the assertion for similar to the one in Exer-cise 3.5.

Theorems represented above can also be used in the case when the original system isdissipative and estimates (1.2) and (1.3) are not assumed to be uniform with respectto . The dissipativity property enables us to restrict ourselves to the con-sideration of the trajectories lying in a vicinity of the absorbing set when we studythe asymptotic behaviour of solutions to problem (0.1). In this case it is convenientto modify the original problem. Assume that the mapping is continuous withrespect to its arguments and possesses the properties

, (3.26)

for any and for all , , and lying in the ball .Let be an infinitely differentiable function on such that

, ; , ;

, , .

We define the mapping by assuming that

, . (3.27)

Show that the mapping possesses the properties

,

, (3.28)

where and is a constant from (3.26).

Let us now assume that satisfies condition (3.26) and the problem

, , (3.29)

has a unique mild solution on any segment and possesses the followingdissipativity property: there exists such that for any the relation

E x e r c i s e 3.7 v0 p& B u t�� B u� �&

#0 p s�� 0& #1 p s�� A 1� Q B p� ��

E x e r c i s e 3.8

A� #n

p s�� # p s�� qn v0 s1 q�� 1�

D1 A�p��( )* +�

D1 # p s��

E x e r c i s e 3.9 #n

p s��

u D A��

B u t��

B u t�� C4� B u1 t�� B u2 t�� C4 A� u1 u2��

4 0 u u1 u2 B4 v : A�v 4�� 5 s� � R� 0 ��

5 s� � 1� 0 s 1� � 5 s� � 0� s 2�

0 5 s� � 1� � 5 6 s� � 2� s R��

BR

u t��

BR u t�� 5 R 1� A�u� � B u t�� u D A��

E x e r c i s e 3.10 BR

u t��

A�BR

u t�� M�

BR

u1 t�� BR

u2 t�� M A� u1 u2��

M C2 R 1 2 R!�� C4

B u t��

udtd

�� Au� B u t�� ut 0� u0�

s s T�� "R0 0 R 0

C o n t i n u o u s D e p e n d e n c e o f I n e r t i a l M a n i f o l d o n P r o b l e m P a r a m e t e r s 171

for all (3.30)

holds, provided that . Here is the solution to problem (3.29).

Show that the asymptotic behaviour of solutions to problem(3.29) completely coincides with the asymptotic behaviour of solu-tions to the problem

, , (3.31)

where is defined by formula (3.27) and is the constantfrom equation (3.30).

Assume that for a solution to problem (3.29) the invarianceproperty of the absorbing ball is fulfilled: if , then

for all . Let be the invariant manifoldof problem (3.31). Then the set is in-variant for problem (3.29): if , then ,

.

Thus, if the appropriate spectral gap condition for problem (3.29) is fulfilled, thenthere exists a finite-dimensional surface which is a locally invariant exponentially at-tracting set.

In conclusion of this section we note that the version of the Lyapunov-Perron me-thod represented here can also be used for the construction (see [13]) of inertialmanifolds for retarded semilinear parabolic equations similar to the ones consideredin Section 8 of Chapter 2. In this case both the smallness of retardation and the fulfil-ment of the spectral gap condition of the form (3.1) are required.

§ 4 Continuous Dependence of Inertial§ 4 Continuous Dependence of Inertial§ 4 Continuous Dependence of Inertial§ 4 Continuous Dependence of Inertial

Manifold on Problem ParametersManifold on Problem ParametersManifold on Problem ParametersManifold on Problem Parameters

Let us consider the Cauchy problem

, , (4.1)

in the space together with problem (1.1). Assume that is a nonlinearmapping from into possessing properties (1.2) and (1.3) with thesame constant as in problem (1.1). If spectral gap condition (3.1) is fulfilled, thenproblem (4.1) (as well as (1.1)) possesses an invariant manifold

A�u t s u0�� R0� t s� t0 R� ��

A�u0 R� u t s u0��


udtd

�� A u� B2 R0u t�� u

t s� u0�

B2 R0R0


A�u0 R0�A�u t s u0�� R� t s� Mt

MtR0 Mt u : A�u R0�� 7�

u0 MsR0� u t s u0�� Ms

R0�t s�

udtd

�� A u� B* u t�� ut s� u0� s R�

H B* u t�� D A�� R H

M


3

C

h

a

p

t

e

r

, . (4.2)

The aim of this section is to obtain an estimate for the distance between themanifolds and . The main result is the following assertion.

Theorem 4.1.

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (1.2),,,, (1.3),,,, and and and and (3.1) are fulfilled both forare fulfilled both forare fulfilled both forare fulfilled both for

problems problems problems problems (1.1) and and and and (4.1).... We also assume that We also assume that We also assume that We also assume that

(4.3)

for all and for all and for all and for all and ,,,, where and are positive numbers. Then where and are positive numbers. Then where and are positive numbers. Then where and are positive numbers. Then

the equationthe equationthe equationthe equation

is valid for the functions and which give the invariantis valid for the functions and which give the invariantis valid for the functions and which give the invariantis valid for the functions and which give the invariant

manifolds for problems manifolds for problems manifolds for problems manifolds for problems (1.1) and and and and (4.1) respectively. Here the numbers respectively. Here the numbers respectively. Here the numbers respectively. Here the numbers

and do not depend on and and do not depend on and and do not depend on and and do not depend on and ....

Proof.

Equation (2.12) with implies that

,

where and are solutions to the integral equations of the type (2.1) cor-responding to problems (1.1) and (4.1) respectively. Equations (1.3) and (4.3) giveus that

(4.4)

for , where

(4.5)

and as before. Hence, after simple calculations as in Section 2 wefind that

. (4.6)

Let us estimate the value . Since and are fixed points of the correspon-ding operator , we have that

Ms*

p #* p s�� : p P H�� s R�

Ms Ms*

B v t�� B* v t�� 41 42 A�v��

v D A�� t R� 41 42

A� # p s�� #* p s�� s R�sup C1 q ��

41 42�

�N1 ��

�� C2 q � M� �� 42 A�p��

# p s�� #* p s��

C1 q �� C2 q � M� �� N 4j

L ��

A� # p s�� #* p s�� A�es �� A�

Q B v �� B* v* �� d

��

s

��

v �� v* ��

B v �� B* v* �� M A� v �� v* �� 41 42 A�v �� ( )* +�

e$ s ��

M v v*� s 42 vs

�� 41�

� �

�

� s�

ws

ess e$� s t��

A8w t� ��

t s�sup�

$ �N2 Mq��N

��

A� # p s�� #* p s�� q

2�� v v*�

s

42M�� v

s�( )

* + 411 k��N 1�

1 ��

v v*�s

v v*

Bps ��


Therefore, by using spectral gap condition (3.1) and estimate (4.4) as above it iseasy to find that

.

Consequently,

.


.

Hence, estimate (2.5) gives us the inequality

.

This implies the assertion of Theorem 4.1.

Let us now consider the Galerkin approximations of problem (1.1). We re-mind (see Chapter 2) that the Galerkin approximation of the order is defined as afunction with the values in , this function being a solution to the problem

, . (4.7)

Here is the orthoprojector onto the span of elements in .

Assume that spectral gap condition (3.1) holds and .Show that problem (4.7) possesses an invariant manifold of the form

in , where the function is de-fined by equation similar to (2.12).

A� v t� � v* t� �� A� et �� A�

P B v �� B* v* �� d

t

s

�A� e

t �� A�Q B v �� B* v* �� .d

��

s

�

�

�

�

v v*�s

q v v*�s

q42M�� v

s/ 41

1

�N1 ��

��1 k��

N 1�1 ��

��( )- .* +� ��

v v*� s

q

1 q��42M�� v s/ /

411 q��

2 k��

N1 ��

��/��

A� # p s�� #* p s�� q

1 q��422 M�� v s/ / 41

2 q�2 2 q��

2 k��N

1 ��/ /��

A� # p s�� #* p s�� 41 42�

1 q�� 2��

2 k��

N1 ��

��/q

2 1 q�� 2��

42M�� A�p/��

um

t� �m

um

t� � Pm

H

um

d

td�� A u

m� P

mB um0� �� u

mt s�

u0 m�

Pm

e1 � em

� �� H

E x e r c i s e 4.1 m N 1��

Msm� �

p # m� � p s�� : p P H��

PmH # m� � p s�� : P H Pm P�� H


3

C

h

a

p

t

e

r

The following assertion holds.

Theorem 4.2.

Assume that spectral gap condition Assume that spectral gap condition Assume that spectral gap condition Assume that spectral gap condition (3.1) holds. Let andholds. Let andholds. Let andholds. Let and

be the functions defined by the formulae of the type be the functions defined by the formulae of the type be the functions defined by the formulae of the type be the functions defined by the formulae of the type (2.12) and and and and

let these functions give invariant manifolds for problems let these functions give invariant manifolds for problems let these functions give invariant manifolds for problems let these functions give invariant manifolds for problems (1.1) and and and and (4.7) for for for for

respectively. Then the estimate respectively. Then the estimate respectively. Then the estimate respectively. Then the estimate

(4.8)

is valid, where the constant is defined by formula is valid, where the constant is defined by formula is valid, where the constant is defined by formula is valid, where the constant is defined by formula (2.6)....

Proof.

It is evident that

, (4.9)

where and are solutions to the integral equations

, ,

and

, .

Here is defined as in (2.1). Since

,

we have

.

The contractiveness property of the operator leads to the equation

.

In particular, this implies that

.

Hence, with the help of (4.9) we find that

(4.10)

# p s�� # m� � p s��

m N 1��

A� # p s�� # m� � p s�� C q M ��

�m 1�1 ��

�� 1D1 A�p�

1�

N 1��

m 1��

��

� �' '� �' '� �

�

D1

# p s�� # m� � p s�� Q v s p�� v m� � s p�� "�

v t p�� v m� � t p��

v t� � Bps ��

v� " t� �� t s��

v m� � t� � Pm

Bp

s ��v m� �� " t� �� t s��

Bps ��

v t� � v m� � t� �� I Pm�� v t� � Pm Bps ��

v� " t� � Bps ��

v m� �� " t� �� "��

A� v t� � v m� � t� �� A� 1 Pm

�� v t� � A� Bps ��

v� " t� � Bps ��

v m� �� " t� �� "��

Bps ��

A� v t� � v m� � t� �� A� 1 Pm

�� v t� � q v v m� �� s e$ s t�� /��

v v m� �� s e$ s t��

t s�sup A� v t� � v m� � t� �� 1 q�� 1� 1 Pm�� v

s�&

A� # p s�� # m� � p s�� A� v s� � v m� � s� �� v v m� �� s

1 q�� 1� 1 Pm�� vs .

� � �

�


Let us estimate the value . It is clear that

.

Therefore, Lemma 2.1.1 (see also (1.8)) gives us that

where

as above. Simple calculations analogous to the ones in Lemma 2.1 imply that

,

where the constant has the form (1.7). Consequently, using (2.5) we obtain

This and (4.10) imply estimate (4.8). Theorem 4.2 is proved.

In addition assume that the hypotheses of Theorem 4.2 holdand . Show that in this case estimate (4.8) has the form

.

1 Pm

�� vs

1 Pm�� v t� � e t �� A� 1 Pm�� B v �� d

��

t

��

A� 1 Pm

�� v t� � M�

t ��( )* +� �

m 1�� e

t �� m 1��

M�

t ��( )* +� �

m 1�� e

�m 1� t ��

et �� $ � v

sd

��

t

� e$ s t�� ,/

�

�

d

��

t

��

$ �N

2 Mq��

N

� �N 1� �

m 1�� &

A� 1 Pm

�� v t� �M 1 k��

�m 1�1 ��

��M 1 k��

m 1��

�m 1� $�

�� e$ s t��

vs

��

k

1 Pm�� vs

M 1 k��

�m 1�1 ��

�� 1 1�

N 1��m 1��( )

* + 1�v s�( )

* +

M 1 k��

�m 1�1 ��

�� 1 1�

N 1��m 1��( )

* + 1�1 q�� 1�

D1 A�p�� ( )* + .

� �

�

E x e r c i s e 4.2

B u t�� M�

A� # p s�� # m� � p s�� C q M �� m 1�1� ��


3

C

h

a

p

t

e

r

§ 5 Examples and Discussion§ 5 Examples and Discussion§ 5 Examples and Discussion§ 5 Examples and Discussion

E x a m p l e 5.1

Let us consider the nonlinear heat equation

Assume that is a positive parameter and is a continuous functionof its variables which possesses the properties

, .

Problem (5.1)–(5.3) generates a dynamical system in (see Section 3of Chapter 2). Therewith

, ,

where is the Sobolev space of the order . The mapping givenby the formula satisfies conditions (1.2) and (1.3) with

. In this case spectral gap condition (2.3) has the form

.

Thus, problem (5.1)–(5.3) possesses an inertial manifold of the dimension ,provided that

(5.4)

for some .

Find the conditions under which the inertial manifold of prob-lem (5.1)–(5.3) is one-dimensional. What is the structure of the cor-responding inertial form?

Consider problem (5.1) and (5.3) with the Neumann bounda-ry conditions:

(5.5)

Show that problem (5.1), (5.3), and (5.5) has an inertial manifoldof the dimension , provided condition (5.4) holds for some

u9t9�� : 9

2 u

x29�� f x u t� �� , 0 x l , t 0 ,� ��

ux 0� u

x l� 0 ,� �

ut 0� u0 x� � .��

''�''�

5.1� �

5.2� �

5.3� �

: f x u t� ��

f x u1 t� �� f x u2 t� �� M u1 u2�� f x 0 t� �� M

l��

L2 0 l��

A : d2

x2d�� D A� � H0

1 0 L�� H2 0 l�� 7�

Hs 0 l�� s B . t�� u x� � f x u x� � t� ��

� 0�

:;2

l2�� N 1�� 2 N2�� 4 M

q��

N

N12�� 2 M

: q��

l2

;2��

q 2 2��

E x e r c i s e 5.1

E x e r c i s e 5.2

u9x9

��x 0�

u9x9

��x l�

0� �

N 1�

E x a m p l e s a n d D i s c u s s i o n 177

. (Hint: with condition (5.5), , where is small enough).

Find the conditions on the parameters of problem (5.1), (5.3),and (5.5) under which there exists a one-dimensional inertial mani-fold. Show that if , then the corresponding iner-tial form is of the type

, .

E x a m p l e 5.2

Consider the problem

(5.6)

Here and is a continuous function of its variables such that

(5.7)

for all , and

,

where are nonnegative numbers. As in Example 5.1 we assume that

, , .

It is evident that

.

Here is the norm in . By using the obvious inequality

, ,

we find that

.

Hence, conditions (1.2) and (1.3) are fulfilled with

, .

N 0� A : d2 x2d!� �� <�� B u t�� <u� f x u t� �� < 0

E x e r c i s e 5.3

f x u t� �� f u t�� &

p� t� � f p t� � t�� pt 0� p0�

9u

t9�� : 92 u

9x2�� f x u

u9x9�� t� � �( )

* + , 0 x l , t 0 ,� ��

ux 0� u

x l� 0 , ut 0� u0 x� � .��

'�'�

: 0 f x u t� � ��

f x u1 t� � �� f x u2 t� � �� L1 u1 u2� L2 1 2��

x 0 l�� t 0�

f x 0 0 t� � �� "2 xd

0

l

� L32�

Lj

A : d2

x2d�� D A� � H0

1 0 l�� H2 0 l�� 7� B u t�� f x uu9x9�� t� � �( )

* +�

B u1 t�� B u2 t�� L1 u1 u2� L2

u19x9��

u29x9��

. L2 0 l��

u9x9��

2 ;l��( )* +2

u2� u H0

1 0 l��

B u1 t�� B u2 t�� 1:

�� L1l

;�� L2�( )* + A1 2� u1 u2��

� 12�� M L3

1:

��l

;��L1 L2�( )* +�

� ��

max�


3

C

h

a

p

t

e

r

Therewith spectral gap condition (2.3) acquires the form

,

where

.

Thus, the equation

or

must be valid for some . We can ensure the fulfilment of this con-dition only in the case when

, ,

i.e. if

. (5.9)

Thus, in order to apply the above-presented theorems to the construction of theinertial manifold for problem (5.6) one should pose some additional conditions(see (5.7) and (5.9)) on the nonlinear term or require that thediffusion coefficient be large enough.

Assume that in (5.6), wherethe function possesses properties (5.7) and (5.8) with arbitrary

. Show that problem (5.6) has an inertial manifold for any, where

.

Characterize the dependence of the dimension of inertial manifoldon .

Study the question on the existence of an inertial manifold forproblem (5.6) in which the Dirichlet boundary condition is replacedby the Neumann boundary condition (5.5).

: ;l�� 2 N 1�� 2 M

q�� 2 N 1 k N 1��

k12

�� 1 2��e

� d

0

x

� ;2��

1 ;2��

N 1�2 N 1��/�

; q :2 l M��

2 ;2��

;2��

12 N 1��/� � ; q :

l M��

0 q 2 2��

2 ;2��

; q0 :l M

�� q0 2 2��

M L31:

��l

;��L1 L2�( )* +�

� ��

max 2; :l��

2 1�2 2 ;��/�&

f x u u9 x9! t� � �� :

E x e r c i s e 5.4 f x u t� � �� < f x u t� � �� f

Lj 0�0 < <0� �

<0 2; :l��

2 1�2 2 ;�� L3

1:��

l

;��L1 L2�( )* +�

� ��

max1�

/ /�

<

E x e r c i s e 5.5


It should be noted that

, , ,

where are the eigenvalues of the linear part of the equation of the type

, , ,

in a multidimensional bounded domain . Therefore, we can not expect that Theo-rem 3.1 is directly applicable in this case. In this connection we point out the paper[3] in which the existence of IM for the nonlinear heat equation is proved in a boun-ded domain that satisfies the so-called “principle of spatial ave-raging” (the class of these domains contains two- and three-dimensional cubes).

It is evident that the most severe constraint that essentially restricts an applica-tion of Theorem 3.1 is spectral gap condition (3.1). In some cases it is possible toweaken or modify it a little. In this connection we mention papers [6] and [7]in which spectral gap condition (3.1) is given with the parameters and for . Besides it is not necessary to assume that the spectrum of the opera-tor is discrete. It is sufficient just to require that the selfadjoint operator pos-sess a gap in the positive part of the spectrum such that for its edges the spectralcondition holds. We can also assume the operator to be sectorial rather than self-adjoint (for example, see [6]).

Unfortunately, we cannot get rid of the spectral conditions in the constructionof the inertial manifold. One of the approaches to overcome this difficulty runs asfollows: let us consider the regularization of problem (0.1) of the form

, . (5.10)

Here and the number is chosen such that the operator possesses spectral gap condition (3.1). Therewith IM for problem (5.10) should benaturally called an approximate IM for system (0.1). Other approaches to the con-struction of the approximate IM are presented below.

It should also be noted that in spite of the arising difficulties the number of equationsof mathematical physics for which it is possible to prove the existence of IM is largeenough. Among these equations we can name the Cahn-Hillard equations in the do-main , , the Ginzburg-Landau equations ( ,

), the Kuramoto-Sivashinsky equation, some equations of the theory of oscilla-tions ( ), a number of reaction-diffusion equations, the Swift-Hohenberg equa-tion, and a non-local version of the Burgers equation. The corresponding referencesand an extended list of equations can be found in survey [8].

In conclusion of this section we give one more interesting application of thetheorem on the existence of an inertial manifold.

�n

Cd

n2 d� 1 o 1� �� n � d =dim�

�n

u9t9

�� : u f x u u> t�� ?� x =� t 0

=

= Rd% d 3��

q 2� k 0�0 � 1��A A

A

udtd

�� A u <Am u� � B u t�� ut 0� u0�

< 0 m 0 A�

A <Am��

= 0 L�� d� d =dim 2�� = 0 L�� d�d 2�

d 1�


3

C

h

a

p

t

e

r

E x a m p l e 5.3

Let us consider the system of reaction-diffusion equations

, , (5.11)

in a bounded domain . Here and the function satisfies the global Lipschitz condition:

, (5.12)

where , , and . We also assume that .Problem (5.11) can be rewritten in the form (0.1) in the space if we suppose

, .

It is clear that the operator is positive in its natural domain and it has a dis-crete spectrum. Equation (5.12) implies that the relation

is valid for . Thus,

,

where

.

Therefore, problem (5.11) generates an evolutionary semigroup (see Chap-ter 2) in the space . An important property of is the following: thesubspace which consists of constant vectors is invariant with respect to thissemigroup. The dimension of this subspace is equal to . The action of thesemigroup in this subspace is generated by a system of ordinary differentialequations

, . (5.13)

Assume that equation (5.12) holds for . Show thatequation (5.13) is uniquely solvable on the whole time axis for anyinitial condition and the equation

(5.14)

holds for any .

u9t9�� : u f u u>�� ?� u9

n9�� 9=0�

= Rd% u u1 � um

� �� f u ��

f u �� f v ,�� L u v� 2 ,� 2�� 1 2��

u v� Rm� ,� Rm d� L 0 f 0 0�� L�H L2 =� �� "m�

A u : u u�?�� B u� � u f u u>��

A

B u� � B v� �� L u v� 2u v�� > 2�

� �� 1 2�

u v�

1 L 1 1:

��

max�( )- .* +

u v� 2 : u v�� > 2�� 1 2�

�

�

��

B u� �

B u� � B v� �� M A1 2� u v��

M 1 L 1 1:

��

max��

St

D A1 2�� St

Lm

udtd

�� f u 0�� u t� � L�

E x e r c i s e 5.6 , 0� �

eL s t��

u s� ��

t s�sup ��

s R�


The subspace consists of the eigenvectors of the operator corresponding to theeigenvalue . The next eigenvalue has the form , where isthe first nonzero eigenvalue of the Laplace operator with the Neumann boundarycondition on . Therefore, spectral gap equation (3.1) can be rewritten in theform

(5.15)

for and , where . It is clear that there exists such that equation (5.15) holds for all . Therefore, we can apply Theorem 3.1to find that if is large enough, then there exists IM of the type

.

The invariance of the subspace and estimate (5.14) enable us to use Theorem 3.2and to state that . This easily implies that , i.e. . Thus,Theorem 3.1 gives us that for any solution to problem (5.11) there exists a so-lution to the system of ordinary differential equations (5.13) such that

, ,

where the constant does not depend on and is the Sobolev normof the first order.

Consider the problem

, ; , (5.16)

where the function has the form

.

Here

, ,

and are continuous functions such that

Show that if

,

L A

�1 1� �2 :@1 1�� @1

9=

:@12q�� 1 L 1 1

:��

� ��

max�( )- .* +

1 ;2��( )

* + :@1 1� 1�( )* +�

N m� � 1 2!� 0 q 2 2�� :0 0: :0�

:

M p # p� � : � p L� #� : L H L� ��

�

LL M% # p� � 0& M L�

u t� �u�

t� �

u t� � u�

t� �� 1 C e $ t�� t 0�

$ 0 u t� � .1

E x e r c i s e 5.7

u9t9�� : 9

2u

x29�� f x u�� 0 x ;� � u

x 0� ux ;� 0� �

f x u��

f x u�� g1 u1 u2�� x g2 u1 u2�� 2 xsin�sin�

uj

2;�� u x� � j xsin xd

0

;

�� j 1 2��

gj

u1 u2��

gj

u1 u2�� gj

v1 v2��

Lj u1 v1� 2u2 v2� 2�

( )* +1 2�

; gj 0 0�� 0 .�

�

�

: 25 2 1�� ; L1

2L2

2��


3

C

h

a

p

t

e

r

then the dynamical system generated by problem (5.16) has thetwo-dimensional (flat) inertial manifold

and the corresponding inertial form is:

, .

Study the question on the existence of an inertial manifoldfor the Hopf model of turbulence appearance (see Section 7 of Chap-ter 2).

§ 6§ 6§ 6§ 6 Approximate Inertial ManifoldsApproximate Inertial ManifoldsApproximate Inertial ManifoldsApproximate Inertial Manifolds

for for for for Semilinear Parabolic EquationsSemilinear Parabolic EquationsSemilinear Parabolic EquationsSemilinear Parabolic Equations

Even in the cases when the existence of IM can be proved, the question concerningthe effective use of the inertial form

(6.1)

is not simple. The fact is that it is not practically possible to find a more or less ex-plicit solution to the integral equation for even in the finite-dimensionalcase. In this connection we face the problem of approximate or asymptotic construc-tion of an invariant (inertial) manifold. Various aspects of this problem related to fi-nite-dimensional systems are presented in the book by Ya. Baris and O. Lykova [14].

For infinite-dimensional systems the problem of construction of an approxi-mate IM can be interpreted as a problem of reduction, i.e. as a problem of construc-tive description of finite-dimensional projectors and functions :

such that an equation of form (6.1) “inherits” (of course, this needsto be specified) all the peculiarities of the long-time behaviour of the original system(0.1). It is clear that the manifolds arising in this case have to be close in some senseto the real IM (in fact, the dynamics on IM reproduces all the essential features ofthe qualitative behaviour of the original system). Under such a formulation a prob-lem of construction of IM acquires secondary importance, so one can directly con-struct a sequence of approximate IMs. Usually (see the references in survey [8]) theproblem of the construction of an approximate IM can be formulated as follows: finda surface of the form

, (6.2)

which attracts all the trajectories of the system in its small vicinity. The character ofcloseness is determined by the parameter related to the decomposition

M p1 xsin p2 2 x : p1 p2� R�sin��

p� 1 :p1� g1 p1 p2�� p� 2 4:p2� g2 p1 p2��

E x e r c i s e 5.8

9tp A p� PB p # p t�� t��

# p t��

P # . t�� P H 1 P�� H

Mt p # p t�� : p P H��

�N 1�1�

A p p r o x i m a t e I n e r t i a l M a n i f o l d s f o r S e m i l i n e a r P a r a b o l i c E q u a t i o n s 183

(6.3)

We obtain the trivial approximate IM if we put in (6.2).In this case is a finite-dimensional subspace in whereas inertial form (6.1)turns into the standard Galerkin approximation of problem (0.1) corresponding tothis subspace. One can find the simplest non-trivial approximation using for-mula (6.2) and assuming that

. (6.4)

The consideration of system (0.1) on leads to the second equation of equa-tions (6.3) being replaced by the equality . The results of thecomputer simulation (see the references in survey [8]) show that the use of just thefirst approximation to IM has a number of advantages in comparison with the tradi-tional Galerkin method (some peculiarities of the qualitative behaviour of the systemcan be observed for a smaller number of modes).

There exist several methods of the construction of an approximate IM. We presentthe approach based on Lemma 2.1 which enables us to construct an approximate IMof the exponential order, i.e. the surfaces in the phase space such that their expo-nentially small (with respect to the parameter ) vicinities uniformly attract allthe trajectories of the system. For the first time this approach was used in paper [15]for a class of stochastic equations in the Hilbert space. Here we give its deterministicversion.

Let us consider the integral equation (see(2.1))

,

and assume that , where the parameter possesses the property

. (6.5)

In this case equations (2.2) hold. Hence, Lemma 2.1 enables us to construct a collec-tion of manifolds for with the help of the formula

, (6.6)

where

. (6.7)

Here is a solution to integral equation (2.1) and .

pdtd

�� A p� P B p q� t�� ,�

qdtd

�� A q� 1 P�� B p q� t�� .��'�'�

Mt0� � # p t�� #0 p t�� 0&�

Mt0� �

�

Mt1� �

# p t�� #1 p t�� A 1� 1 P�� B p t�� &�

Mt1� �

A q 1 P�� B p t��

H

�N 1�

v t� � Bps L�

v� " t� �� s L� t s� �

L 4�N 1�� 4

q M��

1 �� 1

1 �� 41 �� 4�( )* + 1�&

MsL� � L 4�N 1�

��

MsL p #L p s�� : p P H��

#L p s�� es �� A�

Q B v �� d

s L�

s

� Q v s p�� &�

v t� � v t p�� L 4�N 1��


3

C

h

a

p

t

e

r

Show that both the function and the surface do not depend on in the autonomous case .

The following assertion is valid.

Theorem 6.1.

There exist positive numbers and There exist positive numbers and There exist positive numbers and There exist positive numbers and

such that ifsuch that ifsuch that ifsuch that if

,,,, ,,,, ,,,, (6.8)

then the mappings defined by equation then the mappings defined by equation then the mappings defined by equation then the mappings defined by equation (6.7) possess possess possess possess

the propertythe propertythe propertythe property

(6.9)

for all for all for all for all .... Here is an absolute constant and is a mild Here is an absolute constant and is a mild Here is an absolute constant and is a mild Here is an absolute constant and is a mild

solution to problem solution to problem solution to problem solution to problem (1.1) such that such that such that such that

forforforfor .... (6.10)

If , then estimate If , then estimate If , then estimate If , then estimate (6.9) can be rewritten as follows: can be rewritten as follows: can be rewritten as follows: can be rewritten as follows:

(6.11)

where is defined by equality where is defined by equality where is defined by equality where is defined by equality (2.14)....

Proof.

Let

, ,

where is a mild solution to problem (1.1) with the initial condition at the moment . Therewith . It is evident that

(6.12)

Let us estimate each term in this decomposition. Equation (1.6) implies that

, (6.13)

E x e r c i s e 6.1 #L p s�� MsL

s B u t�� B t� �&� �

41 41 M � �1� �� A A M � �1� ��

�N 1�1 �� A4 1�� L 4�

N 1�� 0 4 41��

#L . s�� : P H Q H

A� Q u t� � #L P u t� � t��

CR

1� � �04��N 1�

�t t

*��

� ��

exp CR

2� � 42�� N 1�

1 ��

exp�

�

�

t t*

L 2!�� 0 0 u t� �

A�u t� � R� t t*

��

B u t�� M�

A� Q u t� � #L Pu t� � t��

CR

�04�� N 1�

�t t

*��

� ��

exp D2 4�N 1�1 �� , , , ,exp�

�

�

D2

u� t� � U t s Pu s� � #L Pu s� � s�� t*

s t� �

U t s v�� v D A�� s u t� � U t 0 u0��

Q u t� � #L Pu t� � t�� Q u t� � u� t� ��

Q u� t� � #L P u� t� � t�� #L P u� t� � t�� #L P u t� � t�� .

�

� �

�

A�Q u t� � u� t� �� 3N t s�� A� Qu s� � #L Pu s� � s��


where

.

Using (2.16) we find that

(6.14)

where

,

moreover, the second term in can be omitted if (see Exer-cise 2.1). At last equations (2.15) and (1.5) imply that

(6.15)

Thus, equations (6.12)–(6.15) give us the inequality

(6.16)

for , where

and

It follows from (6.16) that under the condition the equation

(6.17)

holds with

.

It is clear that if

,

and

, . (6.18)

Let be such that equation (6.18) holds for and forthe parameter of the form (6.5) with . Then equation (6.8) with

implies that . Let . Thenit follows from (6.17) that

3N �� e�

N 1� ��M 1 k�� a1 �

N 1�1� ��

ea2 ��

A� Q u� s� � #L P u� s� � s�� N L t s��

�N

L �� D2 1 q�� 1�e $L� q 1 q�� 2�

R D1�� e $ ��

�N

L �� B u t�� M�

A� #L P u� t� � t�� #L Pu t� � t��

a1q

1 q�� ea2 t s��

A� Qu s� � #L Pu s� � s�� .

�

�

d t� � 3N

t s�� d s� � �N

L t s��

t s t*

� �

d t� � A� Qu t� � #L Pu t� � t��

3N

�� 3N �� a1q

1 q�� ea2 ��

e�

N 1� ��a1 M 1 k��

N 1�1� ��

q 1 q�� 1�� ea2 � .�

� �

�

s L 2!� t s L��

d t� � 3N L� d s� � �

NL L 2!��

3N L� e

�N 1�

L

2��

a1 M 1 k�� N 1�

1� �� q

1 q�� ea2 L��

3N L� 1 2!�

�N 1� L 4 2ln �

N 1�1� �� 16 a1M 1 k��

a2 L 2ln� q 1 16 a1�� 1��

41 41 M � �1� �� L 4�N 1��

q 0 4 41��A 4 1 4 a1 M 1 k�� 41�� 3

N L� 1 2!� tn

t*

1 2!� �n L��


3

C

h

a

p

t

e

r

,

After iterations we find that

, (6.19)

Equation (6.17) also gives us that

, .


for all . This implies (6.9) and (6.11) if we take in the equa-tion for . Thus, Theorem 6.1 is proved.

In particular, it should be noted that relations (6.9) and (6.11) also mean that a solu-tion to problem (0.1) possessing the property (6.10) reaches the layer of the thick-ness adjacent to the surface given by equation (6.6)for large enough. Moreover, it is clear that if problem (0.1) is autonomous

and if it possesses a global attractor, then the attractor lies in thislayer. In the autonomous case does not depend on (see Exercise 6.1). Theseobservations give us some information about the position of the attractor in thephase space. Sometimes they enable us to establish the so-called localization theo-rems for the global attractor.

Let . Use equations (1.4) and (1.8) to showthat

,

where .

In particular, the result of this exercise means that assumption (6.10) holds for any and for large enough under the condition . In the general

case equation (6.10) is a variant of the dissipativity property.

Let be a function from .Assume that

,

and

,

Show that the assertions of Theorem 6.1 remain true for the function if we add the term

d tn 1�� 1

2��d t

n� � �

NL L 2!�� n 0 1 2 ��

d tn� � 2 n� d t0� � 2�N LL

2��( )

* +�� n 0 1 2 ��

d t� � 12��d t

n� � �

NL

L

2��( )

* +�� tn

L

2�� t t

nL��

d t� � 2 2L�� t t

*�� 2ln�

� ��

exp d t*

� � 2�N LL

2��( )

* +��

t t*

L 2!�� $ �N 1��

�N

L L 2!��

<N

c1 c2� �N 1�1 �� exp� Mt

L� �t

B u t�� B u� �&� �ML t


A�u t� � e�1 t s��

A�u0 R0��

R0 M 1 k�� 11 ��

R R0 t*

B u t�� M�

E x e r c i s e 6.3 v0 v0 s�L

t s�� C$ �� s L� s��

vnL

vn s�L

t s�� & Bps L�

vn 1�� " t� �� n 1 2 ��

#nL

p s�� Q vn s�L

s p�� n 0 1 2 ��

#nL

p s��


to the right-hand sides of equations (6.9) and (6.11). Here is de-fined by equality (6.5) and is the norm of the function in thespace .

Therefore, the function generates a collection of approximate inertialmanifolds of the exponential (with respect to ) order for large enough.

E x a m p l e 6.1

Let us consider the nonlinear heat equation in a bounded domain :

(6.20)

Assume that the function possesses the properties

, .

We use Theorem 6.1 and the asymptotic formula

, ,

for the eigenvalues of the operator in to obtain that in the Sobolevspace for any there exists a finite-dimensional Lipschitzian surface

of the dimension such that

for and for any mild (in ) solution to problem (6.20). Here is large enough, and are positive constants.

Consider the abstract form of the two-dimensional system ofthe Navier-Stokes equations

, (6.21)

(see Example 3.5 and Exercises 4.10 and 4.11 of Chapter 2). Assumethat for . Use the dissipativity property for(6.21) and the formula

for the eigenvalues of the operator to show that there exists a col-lection of functions from into possessing the properties

qn v0 sc0 D1 R��

q

v0 sv0

C$ �� s L� s��

#nL

p s�� N 1� n

= Rd%

u9t9�� u f u >u�� , x = , t s ,��?�

u 9= 0 , ut s� u0 x� � .� �

�'�'�

f w ��

f u1 1�� f u2 2�� C1 u1 u2� 1 2�� f u �� C2�

�N

c0 N2 d�B N �

?� = Rd%H0

1 =� � N

MN N

distH0

1 =� �u t� � MN�� C1 �1 N1 d� t t*�� exp C2 �2 N1 d�� exp��

t t*

� H01 =� � u t� � t

*C

j�

j

E x e r c i s e 6.4

udtd

�� :Au b u u�� f t� �� ut 0� u0�

A1 2� f t� � C� t 0�

c0 k�

k

�1�� c1 k� �

A

# p t�� : t 1�� PD A� � 1 P�� D A� �


3

C

h

a

p

t

e

r

a) ;

for any ;

b) for any solution to problem (6.21) thereexists such that

Here is the orthoprojector onto the first eigenelements of theoperator .

Use Theorem 6.1 to construct approximate inertial manifoldsfor (a) the nonlocal Burgers equation, (b) the Cahn-Hilliard equa-tion, and (c) the system of reaction-diffusion equations (see Sec-tions 3 and 4 of Chapter 2).

In conclusion of the section we note (see [8], [9]) that in the autonomous case the ap-proximate IM can also be built using the equation

. (6.22)

Here , , is the Frechét derivative and is itsvalue at the point on the element . At least formally, equation (6.22) can be ob-tained if we substitute the pair into equation (6.3). The second ofequations (6.3) implicitly contains a small parameter . Therefore, using (6.22)we can suggest an iteration process of calculation of the sequence giving theapproximate IM:

, , (6.23)

where the integers are such that

, , .

One should also choose the zeroth approximation and concretely define the form ofthe values (for example, we can take and ,

). When constructing a sequence of approximate IMs one has to solve only a linearstationary problem on each step. From the point of view of concrete calculations thisgives certain advantages in comparison with the construction used in Theorem 6.1.However, these manifolds have the power order of approximation only (for detaileddiscussion of this construction and for proofs see [9]).

Prove that the mapping has the form (6.4) under thecondition . Write down the equation for when

, .

A# p t�� c1 N 1 2��

A # p1 t�� # p2 t�� c2 A p1 p2��

p p� 1 p2� PD A� ��

u t� � D A� ��t* 1

AQ u t� � # Pu t� � t��

c3 �0 N1 2� t t*

�� exp c4 �1 N1 2�� .exp�

�

�

P N

A

E x e r c i s e 6.5

#6 p� � Ap� PB p # p� �� C D A# p� �� QB p # p� ��

p PH� Q I P�� #6 p� � #6 p� � w�C Dp w

p t� � # p t� ��

N 1�1�

#m

� �

A#k

p� � Q B p #:1 k� � p� ��

#:2 k� �6 p� � Ap PB p #:3 k� � p� �� C D

�

�

�

k 1�

:i

k� �

0 :i

k� � k 1�� :i

k� �k �lim �� i 1 2 3� ��

:i k� � #0 v� � 0& :i k� � k 1�� i 1 2� ��3

E x e r c i s e 6.6 #1 v� �#0 v� � 0& #2 v� �

:1 2� � 1� :2 2� � :3 2� � 0� �

I n e r t i a l M a n i f o l d f o r S e c o n d O r d e r i n T i m e E q u a t i o n s 189

§ 7 Inertial Manifold for Second Order§ 7 Inertial Manifold for Second Order§ 7 Inertial Manifold for Second Order§ 7 Inertial Manifold for Second Order

in Time Equationsin Time Equationsin Time Equationsin Time Equations

The approach to the construction of IM given in Sections 2–4 is essentially based onthe fact that the system has form (0.1) with a selfadjoint positive operator . How-ever, there exists a wide class of problems which cannot be reduced to this form.From the point of view of applications the important representatives of this class aresecond order in time systems arising in the theory of nonlinear oscillations:

(7.1)

In this section we study the existence of IM for problem (7.1). We assume that is a selfadjoint positive operator with discrete spectrum ( and are the cor-

responding eigenvalues and eigenelements) and the mapping possesses theproperties of the type (1.2) and (1.3) for , i.e. is a continuousmapping from into such that

,

, (7.2)

where and .

The simplest example of a system of the form (7.1) is the following nonlinearwave equation with dissipation:

(7.3)

Let . It is clear that is a separable Hilbert space with theinner product

, (7.4)

where and are elements of . In the space prob-lem (7.1) can be rewritten as a system of the first order:

, ; . (7.5)

A

d2 u

t2d�� 2 < ud

td�� A u� � B u t�� , t s , < 0 ,�

ut s� u0 , ud

td��

t s�u1 .� �

�''�''�

A @k

ek

B u t�� 0 � 1 2!� � B u t��

D A�� R H

B 0 t�� M0�

B u1 t�� B u2 t�� M1 A� u1 u2��

0 � 1 2!� � u1 u2� D �� &�

92u

t29�� 2 < u9

t9��92u

x29�� f x t u

u9x9��( )

* +�� 0 , 0 x L , t s ,� ��

ux 0� u

x L� 0 ,� �

ut s� u0 x� � , u9

t9��t s�

u1 x� � .� ��'''�'''�

� D A1 2�� H � �

U V�� Au0 v0�� u1 v1��

U u0 u1�� V v0 v1��

dtd

��U t� � AU t� �� B U t� � t�� t s Ut s� U0�


3

C

h

a

p

t

e

r

Here

, .

The linear operator and the mapping are defined by the equations:

, , (7.6)

, .

Prove that the eigenvalues and eigenvectors of the operator have the form:

, , , (7.7)

where and are the eigenvalues and eigenvectors of .

Display graphically the spectrum of the operator on thecomplex plane.

These exercises show that although problem (7.1) can be represented in the form(7.5) which is formally identical to (0.1) we cannot use Theorem 3.1 here. Neverthe-less, after a small modification the reasoning of Sections 2–4 enables us to prove theexistence of IM for problem (7.1). Such a modification based on an idea from [16] isgiven below.

First of all we prove the solvability of problem (7.1). Let us first consider the li-near problem

(7.8)

These equations can also be rewritten in the form (cf. (7.5))

, , (7.9)

where and . We define a mild solution mild solution mild solution mild solution toproblem (7.8) (or (7.9)) on the segment as a function from the class

which satisfies equations (7.8). Here as before (see Chapter 2). Onecan prove the existence and uniqueness of mild solutions to (7.8) using the Galerkinmethod, for example. The approximate Galerkin solution approximate Galerkin solution approximate Galerkin solution approximate Galerkin solution of the order isdefined as a function

U t� � u t� � u9 t� �t9��( )

* +� U0 u0 u1��

A B U t��

AU u1 A u0 2 <u1�� D A� � D A� � D A1 2��

B U t�� 0 B u0 t�� U u0 u1��

E x e r c i s e 7.1

A

�nE < <2 @

n�E� f

nE

en

�nE

en

�� n 1 2 ��

@n

en

A


d2u

t2d�� 2 < ud

td�� Au� � h t� � t s ,��

ut s� u0

udtd

��

t s�� u1 .� �

�''�''�

dtd

��U t� � AU t� �� H t� �� Ut s� U0�

U t� � u t� � u� t� �� H t� � 0 h t� �� s s T�� " u t� �

Ls T� C s s T �1 2!�� C1 s s T H�� C2 s s T � 1� 2!�� 7 7&

�� D A��

m


satisfying the equations

(7.10)

for . Moreover, we assume that and isabsolutely continuous. Hereinafter we use the notation . Evidentlyequations (7.10) can be rewritten in the form

(7.11)

where is the orthoprojector onto in .

In the exercises given below it is assumed that

, , . (7.12)

Show that problem (7.10) is uniquely solvable on any segment and .

Show that the energy equality

(7.13)

holds for any solution to problem (7.10).

Using (7.11) and (7.13) prove the a priori estimate

for the approximate Galerkin solution to problem (7.8).

Using the linearity of problem (7.11) show that for every twoapproximate solutions and the estimate

um

t� � gk

t� � ek

k 1�

m

F�

u��m t� � ej

�� 2< u�m t� � ej

�� Aum

t� � ej

�� h t� � ej

�� , t s ,�

um

s� � ej

�� u0 ej

�� , u�m s� � ej

�� u1 ej

�� '�'�

j 1 2 � m� � �� gj

t� � C1 s s T�� g� j t� �v� t� � vd td!�

u��m t� � 2 <u�m t� � A um

t� �� pm

h t� � ,�

um

t s�p

mu0� , u�m

t s�p

mu1 ,�

�'�'�

pm

Lin e1 � em

�� H

h t� � L� R H�� u0 D A1 2�� u1 H�

E x e r c i s e 7.3

s s T�� " um

t� � Ls T��

E x e r c i s e 7.4

12�� u�m t� � 2

A1 2� um t� � 2�( )* + 2 < u�m �� 2 �d

s

t

��

12�� p

mu1

2A1 2� pmu0

2�( )* +

h �� u�m �� d

s

t

��

�

�

E x e r c i s e 7.5

A 1 2�� u��m t� � 2u�m t� � 2

A1 2� um

t� �� C T u0 u1� ��

um

t� �

E x e r c i s e 7.6

um

t� � um 6 t� �

A 1� 2! u��m t� � u��m 6 t� �� 2

u�m t� � u�m 6 t� �� A1 2� um t� � um 6 t� �� 2 �

�

� �


3

C

h

a

p

t

e

r

holds for all , where is an arbitrary number.

Using the results of Exercises 7.5 and 7.6 show that we canpass to the limit in equations (7.11) and prove the existenceand uniqueness of mild solutions to problem (7.8) on every segment

under the condition (7.12).

For a mild solution to problem (7.8) prove the energyequation:

(7.14)

In particular, the exercises above show that for problem (7.8) generates alinear evolutionary semigroup in the space by the formula

, (7.15)

where is a mild solution to problem (7.8) for . Equation (7.14) impliesthat the semigroup is contractive for .

Assume that conditions (7.12) are fulfilled. Show that themild solution to problem (7.8) can be presented in the form

, (7.16)

where the semigroup is defined by equation (7.15).

Let us now consider nonlinear problem (7.1) and define its mild solution mild solution mild solution mild solution asa function satisfying the integral equation

(7.17)

on . Here and .

CT

pm

pm 6�� u1

2

A1 2� pm

pm 6�� u0

2 ess pm

pm 6�� h �� 2

� s s T�� "�sup

�

� �

�

t s s T�� "� T 0

E x e r c i s e 7.7

n �

s s T�� "

E x e r c i s e 7.8 u t� �

12�� u� t� � 2

A1 2� u t� � 2�( )* + 2 < u� �� 2 �d

s

t

��

12�� u1

2 A1 2� u02�( )

* + h �� u� �� .d

s

t

��

�

�

h t� � 0&e t A� � D A1 2�� H �

e t A� u0 u1�� u t� � u� t� ��

u t� � h t� � 0&e t A� < 0�

E x e r c i s e 7.9

u� t� � u t� �� e t s�� A� u0 u1�� e t s�� A� 0 h �� d

s

t

��

e t A�

U t� � u t� � u� t� �� C s s T �� &

U t� � e t s�� A� U0 e t s�� A� B U �� d

s

t

��

s s T�� " B U t� � t�� 0 B u t� � t�� U0 u0 u1��


Show that the estimates

,

hold in the space . Here is a positive constant.

Follow the reasoning used in the proof of Theorems 2.1 and2.3 of Chapter 2 to prove the existence and uniqueness of a mild so-lution to problem (7.1) on any segment .

Thus, in the space there exists a continuous evolutionary family of operators possessing the properties

, ,

and

,

where is a mild solution to problem (7.1) with the initial condition .

Let condition hold for some integer . We consider the decomposi-tion of the space into the orthogonal sum

,

where

and is defined as the closure of the set

.

Show that the subspaces and are invariant with re-spect to the operator . Find the spectrum of the restrictions of theoperator to each of these spaces.

Let us introduce the following inner products in the spaces and (the pur-pose of this introduction will become apparent further):

Here and are elements from the corresponding sub-space . Using (7.18) we define a new inner product and a norm in by theequalities:

, ,

where and are decompositions of the elements and into the orthogonal terms , .


B U t�� M 1 U�

��

B U1 t�� B U2 t��

M U1 U2��

�

� D A1 2�� H � M


s s T�� "

�

S t s�� S t t�� I� S t �� G S � s�� S t s��

S t s�� U0 u t� � u� t� ��

u t� � U0 �u0 u1��

<2 @N 1� N

�

� �1 �2H�

�1 Lin ek

0�� 0 ek

�� : k 1 2 �N� ��

�2

Lin ek 0�� 0 ek�� : k N 1��

E x e r c i s e 7.12 �1 �2A

A

�1 �2

U V�C D1 <2 u0 v0�� Au0 v0�� <u0 u1� <v0 v1�� ,��(7.18)

U V�C D2 Au0 v0�� <2 2@N 1�� u0 v0�� <u0 u1� <v0 v1�� .� ��

U u0 u1�� V v0 v1��

i�

U V�C D U1 V1�C D1

U2 V2�C D2

�� U U U�C D1 2��

U U1 U2�� V V1 V2�� U V

Vi

Ui

� �i

� i 1 2��


3

C

h

a

p

t

e

r

Lemma 7.1.

The estimates

, ; (7.19)

, (7.20)

hold for . Here

. (7.21)

Proof.

Let . It is evident that in this case for any . Therefore,

,

i.e. equation (7.19) holds. Let . Then using the inequality

, , (7.22)

for we find that

.

If we take and use (7.22), then we obtain estimate (7.20). Thelemma is proved.

In particular, this lemma implies the estimate

(7.23)

for any , where and has the form (7.21).

Prove the equivalence of the norm and the norm generatedby the inner product (7.4).

Show that we can take for in (7.20)and (7.23).

Prove that the eigenvectors of the operator (see(7.7)) possess the following orthogonal properties:

, ,

, . (7.24)

U 11@N�� <2 @

N� A�u0� U u0 u1�� 1��

U 21

@N 1�� 2N <� A�u0� U u0 u1�� 2��

0 � 1 2!� �

2N <� @

N 1� 1<2 @

N 1��@N 1�

��( )- .* +

min�

U u0 u1�� 1�� A�u0 @N

�u0�

� 0

U 12 <2 u0

2A1 2� u0

2� @N2�� <2 @N�� A�u0

2� �

U �2�

A�u0 @N 1��

u0� � 0 u0 Lin ek : k N 1��

0 2 1��

U 22 22 A1 2� u0

2 <2 1 22�� @N 1�� u02��

2 2N <� @

N 1�1 2��

A�u0 @N 1�� 2N <�

1�U�

U u0 u1�� 0 � 1 2!� � 2N <�

E x e r c i s e 7.13 .

E x e r c i s e 7.14 2N <� <2 @N 1�� 0�

E x e r c i s e 7.15 fkE� � A

fn� f

k��C D f

n– f

k–�C D f

n� f

k– �C D 0� � � k nI

fk� fk

–�C D 0� 1 k N� �


Note that the last of these equations is one of the reasons of introducing a new innerproduct.

Let be the orthoprojector onto the subspace in , .

Lemma 7.2.

The equality

, , (7.25)

is valid. Here is the operator norm which is induced by the corres-

ponding vector norm.

Proof.

Let . We consider the function . Since is inva-riant with respect to , the equation

holds, where is a solution to problem (7.8) for . After simple cal-culations we obtain that

.

It is evident that

.

Therefore,

.

Consequently,

, . (7.26)

If we now notice that

,

then equation (7.26) implies (7.25). Thus, Lemma 7.2 is proved.

Let us consider the subspaces

.

Equation (7.24) gives us that the subspaces are orthogonal to each other and there-fore . Using (7.24) it is easy to prove (do it yourself) that

P�

i�

i� i 1 2��

e At� P�2

e�

N 1��

t�� t 0�

.

U �2� 1 t� � e A t� U2� �2

e A t�

1 t� � Au t� � u t� �� <2 2@N 1�� u t� � u t� �� u� <u� 2� ��

u t� � h t� � 0&

1dtd

�� 2 <1� 4 <2 @N 1�� u� <u� u��

2 <2 @N 1�� u� <u� u�� <2 @N 1�� u2

u� <u� 2� 1 t� ��

1dtd

�� 2 <1� 2 <2 @N 1�� 1�

1 t� � e�

N 1��

t� 1 0� �� t 0

At�� exp fN 1��

e�

N 1��

t�fN 1��

�1E Lin fk

E: k N��

�1 �1�

�1–H�


3

C

h

a

p

t

e

r

, , (7.27)

, . (7.28)

We use the following pair of orthogonal (with respect to the inner product) projectors in the space

,

to construct the inertial manifold of problem (7.1) (or (7.5)). Lemma 7.2 and equa-tions (7.27) and (7.28) imply the dichotomy equations

, ; , . (7.29)

We remind that and .The assertion below plays an important role in the estimates to follow.

Lemma 7.3.

Let , where and pos-

sesses properties (7.2). Then

, ,

, , (7.30)

where

. (7.31)

The proof of this lemma follows from the structure of the mapping and fromestimates (7.2) and (7.23).

Show that one can take for in(7.30) (Hint: see Exercise 7.14).

Let us now consider the integral equation (cf. (2.1) for )

(7.32)

eAt P�1

� e�

N

�t� t R�

e A� t P�1

� e��

N

�t� t 0

. .�C D �

P P�1

�� Q I P� P�1

� P�2

��

eAtP e�

N

�t� t R� e A� t Q e

��N 1��

t� t 0

�N� < <2 @

k�� <2 @

N 1�

B U t�� 0 B u0 t�� U u0 u1�� B u0� �

B U t�� M0 KN

U�� U ��

B U1 t�� B U2 t�� KN

U1 U2�� U1 U2� ��

KN

M1 @N 1�� 1 2!� 1

@N 1�

<2 @N 1��

��( )- .* +

max�

B U t��

E x e r c i s e 7.16 KN

M1 <2 @N 1�� 1 2�� 0�

L ��

V t� � Bps

V� " t� �

e t s�� A� p e t �� A� PB V �� d

t

s

� e t �� A� QB V �� d

��

t

��

&

&

�


in the space of continuous vector-functions on with the valuesin such that the norm

, ,

is finite. Here and .

Show that the right-hand side of equation (7.32) is a continu-ous function of the variable with the values in .

Lemma 7.4.

The operator maps the space into itself and possesses the pro-

perties

(7.33)

and

. (7.34)

Proof.

Let us prove (7.34). Evidently, equations (7.29) and (7.30) imply that

Since

,

it is evident that

with

.

Simple calculations show that . Consequently, equa-tion (7.34) holds. Equation (7.33) can be proved similarly. Lemma 7.4 is proved.

Cs

U t� � �� s"��

I UI e$ s t��

U t� �t s�sup ��& $ 1

2��

N 1��

N��

p P�� t �� s��


t �

Bps

Cs

IBps

V� " I p M01�N��

1�N 1��

( )- .* + 4 K

N

�N 1�� N

�� I VI� ��

IBps

V1� " Bps

V2� " I�4 K

N

�N 1�� N

�� IV1 V2I��

Bps

V1 t� �� " Bps

V2 t� �� "� KN e�

N

� � t�� V1 �� V2 �� d

t

s

�

KN

e��

N 1��

t �� V1 �� V2 �� .d

��

t

�

�

�

�

V1 �� V2 �� e$ s �� IV1 V2� I�

Bps

V1� " t� � Bps

V2� " t� �� q e$ s t�� IV1 V2� I�

q KN

e�

N

� $�� t�� d

t

s

� e�

N 1�� $�� t �� d

��

t

��

� �' '� �' '� �

�

q 4 KN �N 1�� N

�� 1��


3

C

h

a

p

t

e

r

Thus, if for some the condition

(7.35)

holds, then equation (7.32) is uniquely solvable in and its solution can be esti-mated as follows:

. (7.36)

Therefore, we can define a collection of manifolds in the space by the for-mula

, (7.37)

where

. (7.38)

Here is a solution to integral equation (7.32). The main result of this section isthe following assertion.

Theorem 7.1.

Assume thatAssume thatAssume thatAssume that

andandandand (7.39)

for some , where and is defined by formulafor some , where and is defined by formulafor some , where and is defined by formulafor some , where and is defined by formula

(7.31).... Then the function given by equality Then the function given by equality Then the function given by equality Then the function given by equality (7.38) satisfies the Lip- satisfies the Lip- satisfies the Lip- satisfies the Lip-

schitz conditionschitz conditionschitz conditionschitz condition

(7.40)

and the manifold is invariant with respect to the evolutionary opera-and the manifold is invariant with respect to the evolutionary opera-and the manifold is invariant with respect to the evolutionary opera-and the manifold is invariant with respect to the evolutionary opera-

tor generated by the formulator generated by the formulator generated by the formulator generated by the formula

,,,, ,,,,

in in in in ,,,, where is a solution to problem where is a solution to problem where is a solution to problem where is a solution to problem (7.1) with the initial condition with the initial condition with the initial condition with the initial condition

.... Moreover, if Moreover, if Moreover, if Moreover, if ,,,, then there exist initial conditions then there exist initial conditions then there exist initial conditions then there exist initial conditions

such that such that such that such that

for , where for , where for , where for , where ....

The proof of the theorem is based on Lemma 7.4 and estimates (7.29) and (7.30).It almost entirely repeats the corresponding reasonings in Sections 2 and 3. We givethe reader an oppotunity to recover the details of the reasonings as an exercise.

q 1�

�N 1��

N��

4KN

q��

Cs

V

IV I 1 q�� 1�p M0

1�

N��

1�

N 1��( )

* +�( )* +�

Ms� � �

Ms p # p s�� : p P��

# p s�� e t �� A� QB V �� d

��

s

��

V ��

<2 @N 1� �

N 1��

N��

4KN

q��

0 q 1� � �k� < <2 @

k�� K

N

# p s��

# p1 s�� # p2 s�� q

2 1 q�� p1 p2��

Ms

S t ��

S t �� U0 u t� � u� t� �� t s�

� u t� �U0 u0 u1�� 0 q 2 2�� U0* u0

* u1*�� Ms��

S t s�� U0 S t s�� U0*� C

qe

$ t s�� Q U0 # PU0 s��

t s� $ 12��

N� �

N 1��


Let us analyse condition (7.39). Equation (7.31) implies that (7.39) holds if

, . (7.41)

However, if we assume that

, (7.42)

then for condition (7.41) to be fulfilled it is sufficient to require that

. (7.43)

Thus, if for some conditions (7.42) and (7.43) hold, then the assertions of Theo-rem 7.1 are valid for system (7.1). This enables us to formulate the assertion on theexistence of IM as follows.

Theorem 7.2.

Assume that the eigenvalues of the operator possess the propertiesAssume that the eigenvalues of the operator possess the propertiesAssume that the eigenvalues of the operator possess the propertiesAssume that the eigenvalues of the operator possess the properties

andandandand ,,,, ,,,, ,,,, (7.44)

for some sequence which tends to infinity and satisfies the estimatefor some sequence which tends to infinity and satisfies the estimatefor some sequence which tends to infinity and satisfies the estimatefor some sequence which tends to infinity and satisfies the estimate

,,,, ....

Then there exists such that the assertions of Theorem Then there exists such that the assertions of Theorem Then there exists such that the assertions of Theorem Then there exists such that the assertions of Theorem 7.1 hold for all hold for all hold for all hold for all

....

Proof.

Equation (7.44) implies that there exists such that the intervals

, ,

cover some semiaxis . Indeed, otherwise there would appear a subse-quence such that

But that is impossible due to (7.44). Consequently, for any there exists such that equations (7.42), (7.43) as well as (7.39) hold.

Consider problem (7.3) with the function possessing the property

.

Use Theorem 7.1 to find a domain in the plane of the parameters for which one can guarantee the existence of an inertial ma-

nifold.

<2 2@N 1��@

N 1� @N

�

2 <2 @N��

4q�� M1 @N 1�

� 1 2!��

@N 1� @

N� 8 2

q��M1 @N 1�

��

2@N 1� <2 2@

N 1� @N

��

N

@N

A

@N

@N 1�

��N

inf 0 @N k� � 1� c0 k

4 1 o 1� �� 4 0 k �

N k� ��

@N k� � 1� @N k� �� 8 2q

�� M1@N k� � 1�� 0 q 2 2��

<0 0< <0�

k0

2@N k� � 1� 2@N k� � 1� @N k� �� " k k0�

<0 �� N k

j� ��

@N k

j� � 2 @

N kj

1�� 1� @N k

j� � 1��

< <0�N N<�

E x e r c i s e 7.18 f x t uu9x9��

f x t u� ��

f x t u1� �� f x t u2� �� L u1 u2��

< L��


3

C

h

a

p

t

e

r

§ 8 Approximate § 8 Approximate § 8 Approximate § 8 Approximate Inertial ManifoldsInertial ManifoldsInertial ManifoldsInertial Manifolds

for Second Order in Time Equationsfor Second Order in Time Equationsfor Second Order in Time Equationsfor Second Order in Time Equations

As seen from the results of Section 7, in order to guarantee the existence of IM fora problem of the type

(8.1)

we have to require that the parameter be large enough and the spectralgap condition (see (7.41)) be valid for the operator . Therefore, as in the case withparabolic equations there arises a problem of construction of an approximate inertialmanifold without any assumptions on the behaviour of the spectrum of the operator

and the parameter which characterizes the resistance force.Unfortunately, the approach presented in Section 6 is not applicable to the

equation of the type (8.1) without any additional assumptions on . First of all, it isconnected with the fact that the regularizing effect which takes place in the case ofparabolic equations does not hold for second order equations of the type (8.1) (inthe parabolic case the solution at the moment is smoother than its initial con-dition).

In this section (see also [17]) we suggest an iteration scheme that enables us toconstruct an approximate IM as a solution to a class of linear problems. For the sakeof simplicity, we restrict ourselves to the case of autonomous equations

. The suggested scheme is based on the equation in functional derivativessuch that the function giving the original true IM should satisfy it. This approach wasdeveloped for the parabolic equation in [9] (see also [8]). Unfortunately, this ap-proach has two defects. First, approximate IMs have the power order (not the expo-nential one as in Section 6) and, second, we cannot prove the convergence ofapproximate IMs to the exact one when the latter exists.

Thus, in a separable Hilbert space we consider a differential equation of the type(8.1) where is a positive number, is a positive selfadjoint operator with discretespectrum and is a nonlinear mapping from the domain of the operator

into such that for some integer the function lies in as amapping from into and for every the following estimates hold:

, (8.2)

d2u

t2d�� $ ud

td�� A u� � B u� � ,�

ut 0� u0 , ud

td��

t 0�u1 ,� �

$ 2 < 0�A

A $ 0

$

t 0

B u t�� &�B u� �& �

H

$ A

B .� � D A1 2�� A1 2� H m 2� B u� � Cm

D A1 2�� H 4 0

B k� � u� � w1 � wk

� ��C D C4 A1 2� wj

j 1�

k

J�

A p p r o x i m a t e I n e r t i a l M a n i f o l d s f o r S e c o n d O r d e r i n T i m e E q u a t i o n s 201

, (8.3)

where , is a norm in the space , , ,and . Here is the Frechét derivative of the order of and is its value on the elements .

Let be a class of solutions to problem (8.1) possessing the followingproperties of regularity:

I) for and for all

and

, ,

where is the space of strongly continuous functions on with the values in , hereinafter ;

II) for any the estimate

(8.4)

holds for and for , where depends on and only.In fact, the classes are studied in [18]. This paper contains necessary and

sufficient conditions which guarantee that a solution belongs to a class .It should be noted that in [18] the nonlinear wave equation of the type

(8.5)

serves as the main example. Here , and the conditions set onthe function from are such that we can take or

, where for and for .In this example the classes are nonempty for all . Other examples will begiven in Chapter 4.

We fix an integer and assume to be the projector in onto the sub-space generated by the first eigenvectors of the operator . Let . If weapply the projectors and to equation (8.1), then we obtain the following sys-tem of two equations for and :

(8.6)

The reasoning below is formal. Its goal is to obtain an iteration scheme for the deter-mination of an approximate IM. We assume that system (8.6) has an invariant mani-fold of the form

(8.7)

B k� � u� � B k� � u*� �� w1 � wk

� ��C D C4 A1 2� u u*�� A1 2� wj

j 1�

k

J�

k 0 1 � m� � �� . H A1 2� u 4� A1 2� u* 4�wj D A1 2�� B k� � u� � k B u� �B k� � u� � w1 � wk� ��C D w1 � wk� �

Lm R�

k 0 1 � m 1�� T 0

u k� � t� � C 0 T D A� ��

u m� � t� � C 0 T D A1 2�� u m 1�� t� � C 0 T H��

C 0 T V�� 0 T�� "V u k� � t� � 9t

ku t� ��

u Lm R��

u k 1�� t� � 2A1 2� u k� � t� � 2

Au k 1�� t� � 2� � R2�

k 1 � m� �� t t*� t* u0 u1L

m R�L

m R�

9t2

u $ 9tu u?� g u� �� f x� � , x = , t 0 ,��

u 9= 0 , ut 0� u0 x� � , 9tu t 0� u1 x� � ,� � �

$ 0 f x� � C� =� ��g s� � C� R� � g u� � usin� g u� � �

u2 p 1�� p 0 1 2 �� d dim = 2�� p 0 1�� d 3�Lm R� m

N P PN� H

N A Q I P��P Q

p t� � PU t� �� q t� � Qu t� ��

9t2

p $ 9tp A p� � PB p q�� ,�

9t2

q $ 9t q A q� � QB p q�� .�

M p h p p�� p� l p p�� : p p�� PH��


3

C

h

a

p

t

e

r

in the phase space . Here and are smooth mappings from into . If we substitute and in the second equality of (8.6), then we obtain the following equation:

The compatibility condition

gives us that

.

Hereinafter and are the Frechét derivatives of the function withrespect to and ; and are values of the corresponding deri-vatives on an element .

Using these formal equations, we can suggest the following iteration process todetermine the class of functions giving the sequence of approximate IMswith the help of (8.7):

(8.8)

where and the integers should be choosen such that . Here is defined by the formula

, (8.9)

where . We also assume that

. (8.10)

Find the form of and for and for.

The following assertion contains information on the smoothness properties of thefunctions and which will be necessary further.

Theorem 8.1.

Assume that the class of functions is defined according toAssume that the class of functions is defined according toAssume that the class of functions is defined according toAssume that the class of functions is defined according to

(8.8)–(8.10).... Then for each Then for each Then for each Then for each the functions the functions the functions the functions and and and and belong to the class belong to the class belong to the class belong to the class

as mappings from as mappings from as mappings from as mappings from into into into into and for all integers such and for all integers such and for all integers such and for all integers such

that the estimatesthat the estimatesthat the estimatesthat the estimates

D A1 2�� H h l PH P H

QD A� � q t� � h p t� � 9tp t� �� 9

tq t� � l p t� � 9

tp t� ��

2p

l p��C D 2p�

l $p� A p� PB p h p p�� C D

$ l p p�� A h p p�� QB p h p p�� .�

� �

�

l p t� � 9t p t� �� 9t h p t� � 9t p t� ��

l p p�� 2p h p��C D 2p�

h $p� A p� PB p h p p�� C D��

2p

f 2p� f f p p�� p p� 2

pf w�C D 2

p�f w�C D

w

hk

lk

��

Ahk p p�� QB p hk 1� p p�� $ l: k� � p p�� 2p lk 1� p��C D

2p�

lk 1� $p� A p� PB p h

k 1� p p�� C D ,�

��

k 1 2 3 �� : k� � k 1� �: k� � k� � lk p p��

lk

p p�� 2p

hk 1� p��C D 2

p�h

k 1� $p� Ap� PB p hk 1� p p�� C D��

k 1 2 3 ��

h0 p p�� l0 p p�� 0& &

E x e r c i s e 8.1 h1 p p�� l1 p p�� : 1� � 0�: 1� � 1�

hn ln

hn

ln

�� n h

nln

Cm PH PH Q H 3 � 0��3 � m��


(8.11)

(8.12)

are valid for all and from such that and are valid for all and from such that and are valid for all and from such that and are valid for all and from such that and ....

Hereinafter is the mixed Frechét derivative of the function of theHereinafter is the mixed Frechét derivative of the function of theHereinafter is the mixed Frechét derivative of the function of theHereinafter is the mixed Frechét derivative of the function of the

order with respect to and of the order with respect to ; the valuesorder with respect to and of the order with respect to ; the valuesorder with respect to and of the order with respect to ; the valuesorder with respect to and of the order with respect to ; the values

and are from and are from and are from and are from .... Moreover, if or Moreover, if or Moreover, if or Moreover, if or ,,,, then the correspon- then the correspon- then the correspon- then the correspon-

ding products in ding products in ding products in ding products in (8.11) and and and and (8.12) should be omitted. should be omitted. should be omitted. should be omitted.

Proof.

We use induction with respect to . It follows from (8.10) and (8.2) that esti-mates (8.11) and (8.12) are valid for . Assume that (8.11) and (8.12) holdfor all . Then the following lemma holds.

Lemma 8.1.

Let and let

.

Then for and for all integers such that

the estimate

(8.13)

holds, where , , , and , .

Proof.

It is evident that is the sum of terms of the type

, .

Here is one of the values of the form:

,

.

A D3 �� hn p p�� w1 � w3 w�

1 � w�� C D

C3 � R� � Awi

i 1�

3

J A1 2� w�

i ,

i 1�

�

J/

�

�

A1 2� D3 �� ln

p p�� w1 � w3 w� 1 � w� �� C D

C3 � R� � Awi

i 1�

3

J A1 2� w�

i

i 1�

�

J/

�

�

p p� P H Ap R� A1 2� p� R�D3 �� f f

3 p � p�

wj w� j PH 3 0� � 0�

n

n 0 1��n k 1��

F: p p�� B p h: p p��

F:3 ��

w� � D3 �� F: p p�� w1 � w3 w� 1 � w� �� C D�

: k 1�� 3 � 0�� 3 � m��

F:3 ��

w� � C Awi

i 1�

3

J A1 2� w�

i

i 1�

�

J/�

wj

w� j p p� PH� Ap R� A1 2� p� R�

F:3 ��

w� �

B:s

y� � B s� � p h: p p�� y1 � ys

� ��C D� s 0�

y�

y*

w� 2p h: w��C D��

y**

D� �� h: w1 � w3 w� 1 � w� �� C D�


3

C

h

a

p

t

e

r


.

Therefore, the induction hypothesis gives us (8.13).

Let us prove (8.12). The induction hypothesis implies that it is sufficient to estimatethe derivatives of the second term in the right-hand side of (8.9). It has the form

, (8.14)

where

.

The Frechét derivatives of value (8.14)

are sums of the terms of the type

,

where

.

Here , and the sets of indices possess the following proper-ties:

,

;

,

.

The induction hypothesis implies that

.

Using the induction hypothesis again as well as Lemma 8.1 and the inequality

,

we obtain an estimate of the following form (if or , then the correspon-ding product should be considered to be equal to 1):

.

B:s

y� � CR A1 2� yj

j 1�

s

J�

2p�

hk 1� Dk p p�� C D

Dk

p p�� $p� A p� PB p hk 1� p p��

D3 �� 2p�

hk 1� Dk p p�� C D w1 � w3 w� 1 � w� �� C D

G � �� D� � 1�� hk 1� p p�� wj1� wj�

w� i1� w� i�

y� �� C D&

y� �� D3 �� Dk p p�� wK1� wK3 ��

w� 41� w� 4� ��

� �� C D�

0 � 3� � 0 � ��

j1 � j�� K1 � K3 �� 7 L�

j1 � j�� K1 � K3 �� M 1 2 � 3� � ��

i1 � i�� 41 � 4� �� 7 L�

i1 � i�� 41 � 4� �� M 1 2 � ��

AG � �� C Awj�

� 1�

�

J A1 2� w�

i�A1 2� y� ��/

� 1�

�

J/�

A1 2� Ph �N1 2�

Ph�

� 3� � ��

A1 2� y� �� C 1 �N1 2�� AwK�

� 1�

3 ��

J A1 2� w� 4�� 1�

� ��

J/�


Hereinafter is the -th eigenvalue of the operator . Thus, it is possible to statethat

. (8.15)

Using the inequality

, , (8.16)

and equation (8.15) it is easy to find that estimates (8.12) are valid for . If weuse (8.8), (8.12) and follow a similar line of reasoning, we can easily obtain (8.11).Theorem 8.1 is proved.

Theorem 8.1 and equation (8.4) imply the following lemma.

Lemma 8.2.

Assume that is a solution to problem (8.1) lying in , .

Let and let

, . (8.17)

Then the estimates

with and

are valid for large enough.

Proof.

It should be noted that is the sum of terms of the form

,

where , , , are nonnegative integers such that

, .

Similar equation also holds for . Further one should use Theorem 8.1 andthe estimates

, , ,

which follow from (8.4).

Let us define the induced trajectories of the system by the formula

,

where and

, . (8.18)

�k

k A

AG � �� C 1 �N1 2�� Aw

i

i

J A1 2� w� i

i

J/�

Q h �N 1�s�

AsQ h� s 0

n k�

u t� � Lm R� m 1�

p t� � Pu t� ��

qs

t� � hs

p t� � 9tp t� �� q

st� � l

sp t� � 9

tp t� ��

A1 2� qsj� �

t� � 2Aq

sj� �

t� � 2� CR m��

0 j m 1��

qsm� � t� � 2

A1 2� qsm� � t� � 2� C

R m��

t

qsj� �

t� �

D3 �� hs

p 9tp�� p

i1� �t� � � p

i3� �t� � p

�1 1�� t� � � p

�� 1�� t� �� C D

3 � i1 � i3� � �1 � ��

1 3 �� j� � i1 � i3 �1 � �� j�

qsi� �

t� �

p k 1�� t� � 2A1 2� p k� � t� � 2

Ap n 1�� t� � 2� � R2� t t*

� 1 k m� �

Us t� � us t� � us t� ��

s 0 1 2 ��

us t� � p t� � qs t� �� us t� � 9t p t� � qs t� ��


3

C

h

a

p

t

e

r

Here , is a solution to problem (8.1); and are definedwith the help of (8.17). Assume that lies in . Then Lemma 8.2 impliesthat the induced trajectories can be estimated as follows:

, ;

for large enough. Using (8.3), (8.4), and the last estimates, it is easy to prove thefollowing assertion (do it yourself).

Lemma 8.3.

Let

.

Then

for and for large enough.

The main result of this section is the following assertion.

Theorem 8.2.

Let be a solution to problem Let be a solution to problem Let be a solution to problem Let be a solution to problem (8.1) lying in with lying in with lying in with lying in with ....

Assume that and are defined by Assume that and are defined by Assume that and are defined by Assume that and are defined by (8.8)–(8.10).... Then the es- Then the es- Then the es- Then the es-

timatestimatestimatestimates

,,,, (8.19)

,,,, (8.20)

are valid for and for large enough. Here ,are valid for and for large enough. Here ,are valid for and for large enough. Here ,are valid for and for large enough. Here ,

and are defined by and are defined by and are defined by and are defined by (8.18),,,, and is the -th eigenva-and is the -th eigenva-and is the -th eigenva-and is the -th eigenva-

lue of the operator lue of the operator lue of the operator lue of the operator ....

Proof.

Let us consider the difference between the solution and the trajectory in-duced by this solution:

, , ,

where and are defined by formula (8.18). Since , equation(8.4) implies that

, , (8.21)

p t� � Pu t� �� u t� � qs

t� � qs

t� �u t� � L

m R�

A1 2� usj� �

t� � Ausj� �

t� �� CR s�� 0 j m 1��

usm� �

t� � 2A1 2� u

sm� �

t� � 2� CR s��

t

Es

t� � B p t� � q t� �� B p t� � qs

t� ��

Esj� �

t� � CR j� A1 2� q i� � t� � qsi� � t� ��

i 0�

j

F�

j 0 1 � m� � �� t

u t� � Lm R� m 2�hn p p�� ln p p��

A9tj

u t� � un t� �� Cn R� �N 1�n 2��

A1 2� 9tj 9tu t� � un t� �� Cn R� �N 1�

n 2��

n m 1�� t 0 j m n� 1�� un t� � un t� � �N 1� N 1��

A

u t� �

5s

t� � u t� � us

t� �� 5s t� � 9tu t� � u

st� �� s 0�

us t� � us t� � 50 t� � q t� ��

A1 2� 50j 1��

t� � A50j� �

t� �� C� j 0 1 2 � m 1��


for large enough. Equations (8.8)–(8.10) also give us that

.

We use Lemma 8.3 and equation (8.21) to find that

, ,

for large enough. Therefore, equation (8.19) holds for and for largeenough. From equations (8.6), (8.8), and (8.9) it is easy to find that

and

. (8.22)

Lemma 8.4.

The estimates

(8.23)

and

(8.24)

are valid for large enough and for each , where .

Proof.

Let or . It is clear that the value is thealgebraic sum of terms of the form:

Therefore, Theorem 8.1 and Lemma 8.3 imply (8.23) and (8.24). Lemma 8.4is proved.

We use Lemmata 8.3 and 8.4 as well as inequality (8.16) to obtain that

(8.25)

where and the numbers and do not depend on .

t

A51 t� � 5 N0 t� � $ 5 60 t� � QE0 t� ��

A51j� �

t� � C�N 1�

1 2�� j 0 1 � m 2��

t n 0 1�� t

A5k

9t5k 1�� $ 9

t5: k� � 1� $ 2

p�h: k� � 1� PE: k� � 1��C D��

2p�

lk 1� PE

k 1��C D Q Ek 1��

��

5k

9t5

k 1� 2p�

hk 1� PE

k 1��C D��

A 9tj 2

p�h: PE:�C D C�

N1 2�

A1 2� 5:s� � t� �

s 0�

j

F�

A1 2� 9tj 2

p�l: PE:�C D C�

N1 2�

A1 2� 5:s� � t� �

s 0�

j

F�

t : 0� j 0 1 � m 1��

f: h:� f: l:� 9tj 2

p�f: PE:�C D

D3 � 1�� f: p$1 � p

$3 p�1 � p

�� 9tsPE:� � �� C D .

A5kj� �

t� � ck j� �

N 1�1�

A5k 1�s� �

t� �s 0�

j 2�

F A5: k� � 1�s� �

t� �s 0�

j 1�

F��

dk j�

N 1�1 2��

A5k 1�s� �

t� � ,s 0�

j

F

�

�

�

j 0 1 � m 2�� ck j� d

k j� N


3

C

h

a

p

t

e

r

If we now assume that (8.19) holds for , then equation (8.25) implies(8.19) for and for . Using (8.22) and (8.23) we obtain equation(8.20). Theorem 8.2 is proved.

Corollary 8.1

Let the manifold have the form (8.7) with and

. We also assume that , where

is the solution to problem (0.1) from the class . Then

, .

Thus, the thickness of the layer that attracts the trajectories in the phase space hasthe power order with respect to unlike the semilinear parabolic equationsof Section 6.

E x a m p l e 8.1

Let us consider the nonlinear wave equation (8.5). Let . We as-sume the following (cf. [18]) about the function :

;

there exists such that

;

for any there exists such that

. (8.26)

Under these assumptions the solution lies in for largeenough if and only if the initial data satisfy some compatibility conditions [18].Moreover, the global attractor of system (8.5) exists and any trajectory lyingin possesses properties (8.4) for all and , [18]. It is easyto see that Theorem 8.2 is applicable here (the form of , and is evi-dent in this case). In particular, Theorem 8.2 gives us that for a trajectory

of problem (8.5) which lies in the global attractor theestimate

holds for all , all , and all . Here and are defined with the help of (8.18). Therewith

n k 1��n k� k m 1��

Mn h p p�� hn

p p�� l p p�� l

np p�� U t� � u t� � u� t� �� u t� �

Lm R�

U t� � Mn�� D A� � D A1 2!� �

dist Cn�N 1�n 2!�� n 0 1 2� � �m 1��

�N 1�

d =dim 2��g s� �

s 1�s �lim g �� d

0

s

� 0�

C1 0

s 1�s �lim sg s� � C1 g �� d

0

s

��( )- .- .* +

0�

m � m� � 0

g m� � s� � C2 1 s � m� ��

u t� � Lm R� R 0

�

� t R� k 1 2 �� A B .� � H

U t� � u t� � 9t u t��

A9tj

u t� � un t� �� 2A1 2� 9t

j 9tu t� � un t� �� 2�� 1 2�

Cn R j� � �N 1�n 2��

n 1 2 �� j 1 2 �� t R� un

t� �u

nt� �

I d e a o f N o n l i n e a r G a l e r k i n M e t h o d 209

, , (8.27)

where is a manifold of the type (8.7) with and .Here is the distance between and in the space

. Equation (8.27) gives us some information on the location ofthe global attractor in the phase space.

Other examples of usage of the construction given here can be found in papers [17]and [19] (see also Section 9 of Chapter 4).

§ 9 Idea of Nonlinear Galerkin Method§ 9 Idea of Nonlinear Galerkin Method§ 9 Idea of Nonlinear Galerkin Method§ 9 Idea of Nonlinear Galerkin Method

Approximate inertial manifolds have proved to be applicable to the computationalstudy of the asymptotic behaviour of infinite-dimensional dissipative dynamical sys-tems (for example, see the discussion and the references in [8]). Their usage leadsto the appearance of the so-called nonlinear Galerkin method [20] based on the re-placement of the original problem by its approximate inertial form. In this section wediscuss the main features of this method using the following example of a second or-der in time equation of type (8.1):

, , . (9.1)

If all conditions on and given in the previous section are fulfilled, thenTheorem 8.2 is valid. It guarantees the existence of a family of mappings from into possessing the properties:

1) there exist constants and , , suchthat

, e, (9.2)

, (9.3)

(9.4)

for all and from such that

, , ;

2) for any solution to problem (9.1) which lies in for the es-timate

U Mn�� : dist U �� sup cn�

N 1�n 2�� n 1 2 ��

Mn h hn p p�� l ln p p�� U Mn�� dist U Mn

D A� � D A1 2��

d2u

t2d�� $ ud

td�� A u� � B u� �� u

t 0� u0� udtd

��

t 0�u1�

A B .� �h

klk

�� PH PH QH

Mj

Mj

n 4�� & Lj

Lj

n 4�� & j 1 2��

Ahn

p0 p� 0�� M1� A1 2� ln

p0 p� 0�� M2�

A hn p1 p�1�� hn p2 p� 2�� L1 A p1 p2�� A1 2� p�1 p� 2�� ( )* +�

A1 2� ln p1 p� 1�� ln p2 p� 2�� L1 A p1 p2�� A1 2� p� 1 p� 2�� ( )* +�

pj p� j PH

Apj

2A1 2� p� j

2� 42� j 0 1�� 4 0

u t� � Lm R� m 2�


3

C

h

a

p

t

e

r

(9.5)

is valid (see Theorem 8.2) for all and large enough. Here

(9.6)

is the -th eigenvalue of , and is the constant from (8.4).

The family is defined with the help of a quite simple procedure (see (8.8)and (8.9)) which can be reduced to the process of solving of stationary equations ofthe type in the subspace . Moreover,

, , . (9.7)

In particular, estimates (9.5) and (9.6) mean (see Corollary 8.1) that trajectories of system (9.1) are attracted by a small (for large enough)

vicinity of the manifold

. (9.8)

The sequence of mappings generates a family of approximate inertialforms of problem (9.1):

. (9.9)

A finite-dimensional dynamical system in which approximates (in some sense)the original system corresponds to each form. For equation (9.9) transformsinto the standard Galerkin approximation of problem (9.1) (due to (9.7)). If ,then we obtain a class of numerical methods which can be naturally called the non-linear Galerkin methods. However, we cannot use equation (9.9) in the computa-tional study directly. The point is that, first, in the calculation of we haveto solve a linear equation in the infinite-dimensional space and, second, we canlose the dissipativity property. Therefore, we need additional regularization. It canbe done as follows. Assume that stands for one of the functions or . We define the value

, (9.10)

where is an infinitely differentiable function on such that a) ;b) for ; c) for ; is the radius of dissipativity(see (8.4) for ) of system (9.1); is the orthoprojector in onto the sub-space generated by the first eigenvectors of the operator , . We considerthe following -dimensional evolutionary equation in the subspace :

A u t� � un

t� �� 2A1 2� 9

tu t� � u

nt� �� 2�

� �� 1 2�

Cn R� �

N 1�n 2��

n m 1�� t

un

t� � p t� � hn

p t� � 9tp t� �� ,��

un t� � 9t p t� � ln p t� � 9t p t� �� ,��

�N 1� N 1�� A R

hk

lk

��

A v g� QH

h0 p p�� l0 p p�� 0& & h1 p p�� A 1� QB p� �� l1 p p�� 0&

U t� � u t� � 9t u t� �� N

Mn

p hn

p p�� p� ln

p p�� : p p�� PH��

hn p p��

9t2p $ 9

tp A p� � P B p h

np 9

tp��

PH

n 0�n 0

hn p p�� QH

fn p p�� hn p p�� ln p p��

fn* p p�� f

N M n� � p p�� & 5 R 1� Ap2

A1 2� p�2�( )

* +1 2�

( )* + P

Mfn

p p��

5 s� � R� 0 5 s� � 1� �5 s� � 1� 0 s 1� � 5 s� � 0� s 2� R

k 0� PM H

M A M NN PN H


(9.11)

Prove that problem (9.11) has a unique solution for andthe corresponding dynamical system is dissipative in .

We call problem (9.11) a nonlinear Galerkin -approximation of problem(9.1). The following assertion is valid.

Theorem 9.1.

Assume that the mappings and satisfy equationsAssume that the mappings and satisfy equationsAssume that the mappings and satisfy equationsAssume that the mappings and satisfy equations

(9.2)–(9.5) for and for some for and for some for and for some for and for some .... Moreover, we assume that Moreover, we assume that Moreover, we assume that Moreover, we assume that

(9.5) is valid for all is valid for all is valid for all is valid for all .... Let and be defined by Let and be defined by Let and be defined by Let and be defined by (9.10) with the help with the help with the help with the help

of and and letof and and letof and and letof and and let

,,,,

,,,,

where is a solution to problem where is a solution to problem where is a solution to problem where is a solution to problem (9.11).... Then the estimate Then the estimate Then the estimate Then the estimate

(9.12)

holds, where is a solution to problem holds, where is a solution to problem holds, where is a solution to problem holds, where is a solution to problem (9.1) which lies in for which lies in for which lies in for which lies in for

and possesses property and possesses property and possesses property and possesses property (8.4) for and for all . Here for and for all . Here for and for all . Here for and for all . Here

, , and are positive constants independent of and ,, , and are positive constants independent of and ,, , and are positive constants independent of and ,, , and are positive constants independent of and ,

is the -th eigenvalue of the operator is the -th eigenvalue of the operator is the -th eigenvalue of the operator is the -th eigenvalue of the operator ....

Proof.

Let . We consider the values

and

The equalities

9t2

p* $ 9tp* A p*� � P

NB p* h

n* p* 9

tp*�� ,�

p*t 0� P

Nu0 , 9

tp*

t 0� PN

u1 .� �

E x e r c i s e 9.1 t 0P

NH P

NH

n N M� ��

hn

p p�� ln

p p�� n m 1�� m 2�

t 0 hn* l

n*

hn

ln

un* t� � p* t� � h

n* p* t� � 9

tp* t� ��

un* t� � 9

tp* t� � l

n* p* t� � 9

tp* t� ��

p* t� �

A1 2� u t� � un* t� �� 2 9

tu t� � u

n* t� �� 2�

� �� 1 2�

31�N 1�n 1�� 2!� 32�M 1�

1 2!�� t� �exp

�

�

u t� � Lm R�

m 2� k 1� t 0 n �m 1�� 31 32 � M N

�k

k A

p t� � PN u t� ��

u t� � un* t� �� p t� � p* t� �� Q

Nu t� � h

np t� � 9

tp t� �� "

hn

p t� � 9tp t� �� h

n* p* t� � 9

tp* t� �� "

� �

�

�

9tu t� � u

n* t� �� 9

tp t� � p* t� �� Q

N9

tu t� � l

np t� � 9

tp t� �� "

ln

p t� � 9tp t� �� l

n* p* t� � 9

tp* t� �� " .

� �

�

�

PM hn p t� � 9t p t� �� hn* p t� � 9t p t� ��


3

C

h

a

p

t

e

r

and

are valid for the class of solutions under consideration. Therefore, we use (9.5) tofind that

(9.13)

and

(9.14)

Therefore, we must compare the solution to problem (9.11) with the value which satisfies the equation

(9.15)

with the same initial conditions as the function . Let . Thenit follows from (9.11) and (9.15) that

(9.16)

where

.

Due to the dissipativity of problems (9.11) and (9.15) we use (9.13) to obtain

for the class of solutions under consideration. Therefore, equation (9.16) impliesthat

.

Hence, Gronwall’s lemma gives us that

.

This and equations (9.13) and (9.14) imply estimate (9.12). Theorem 9.1 is proved.

If we take and in Theorem 9.1, then estimate (9.12) changes into theaccuracy estimate of the standard Galerkin method of the order . Therefore, if the

PM

ln

p t� � 9tp t� �� l

n* p t� � 9

tp t� ��

A1 2� u t� � un* t� ��

C1 A1 2� p t� � p* t� �� 9t p t� � 9t p* t� ��( )* + C2

�M 1�1 2�

��Cn R�

�N 1�n 1�� 2!

��

�

�

9tu t� � 9

tu

n* t� ��

C3 A1 2� p t� � p* t� �� 9t p t� � 9t p* t� ��( )* + C4

�M 1�1 2�

��C

n R�

�n 1�n 1�� 2!

�� .� �

�

�

p* t� �p t� � P

Nu t� ��

9t2 p $ 9

tp A p� � Q

NB p Q

Nu��

p* t� � r t� � p t� � p* t� ��

9t2 r t� � $ 9

tr t� � A r t� �� F t p* u� �� ,�

r 0� � 0 9tr 0� �� 0 ,� �

F t p* u� �� QN B u t� �� B un* t� �� "�

F t p* u� �� CR

A1 2� r t� � 2r� t� � 2�( )

* +1 2�C

n R� �N 1�

n 1�� 2!�C�

M 1�1 2��

12��

dtd

�� r� t� � 2 A1 2� r t� � 2�( )* + CR r� t� � 2

A1 2� r t� � 2�( )* + Cn R� �

N 1�n 1��

C�M 1�

1��

r� t� � 2A1 2� r t� � 2� Cn R� �N 1�

n 1�� C�M 1�

1�� eCR t�

n 0� N M�N


parameters , , and are compatible such that , then the error ofthe corresponding nonlinear Galerkin method has the same order of smallness as inthe standard Galerkin method which uses basis functions. However, if we use thenonlinear method, we have to solve a number of linear algebraic systems of the order

and the Cauchy problem for system (9.11) which consists of equations.In particular, in order to determine the value we must solve the equation

for and choose the numbers and such that . Moreover,if , , as , then the values and must be com-patible such that .

We note that Theorem 9.1 as well as the corresponding variant of the nonlinearGalerkin method can be used in the study of the asymptotic properties of solutionsto the nonlinear wave equation (8.5) under some conditions on the nonlinear term

. Other applications of Theorem 9.1 can also be pointed out.

N M n �M 1� �

N 1�n 1��

M

M N� N

h1 p p��

A h1 p p�� PM

PN

�� QB p� ��

n 1� N M �M 1� �N 1�2�

�k c0 k� 1 o 1� �� O � 0 k � N M

M c�N2�

g u� �


3

C

h

a

p

t

e

r


1. FOIAS C., SELL G. R., TEMAM R. Inertial manifolds for nonlinear evolutionary

equations // J. Diff. Eq. –1988. –Vol. 73. –P. 309–353.2. CHOW S.-N., LU K. Invariant manifolds for flows in Banach spaces // J. Diff. Eq.

–1988. –Vol. 74. –P. 285–317.3. MALLET-PARET J., SELL G. R. Inertial manifolds for reaction-diffusion equations

in higher space dimensions. // J. Amer. Math. Soc. –1988. –Vol. 1. –P. 805–866.4. CONSTANTIN P., FOIAS C., NICOLAENKO B., TEMAM R. Integral manifolds and

inertial manifolds for dissipative partial differential equations. –New York: Springer, 1989.


Physics. –New York: Springer, 1988.6. MIKLAV«I« M. A sharp condition for existence of an inertial manifold //

J. of Dyn. and Diff. Eqs. –1991. –Vol. 3. –P. 437–456.7. ROMANOV A. V. Exact estimates of dimention of inertial manifold for non-

linear parabolic equations // Izvestiya RAN. –1993. –Vol. 57. –# 4. –P. 36–54 (in Russian).

8. CHUESHOV I. D. Global attractors for non-linear problems of mathematical

physics // Russian Math. Surveys. –1993. –Vol. 48. –# 3. –P. 133–161.9. CHUESHOV I. D. Introduction to the theory of inertial manifolds. –Kharkov:

Kharkov Univ. Press., 1992 (in Russian).10. CHEPYZHOV V. V., GORITSKY A. YU. Global integral manifolds with exponential tra-

cking for nonautonomous equations // Russian J. Math. Phys. –1997. –Vol. 5. –#l.11. MITROPOLSKIJ YU. A., LYKOVA O. B. Integral manifolds in non-linear mechanics.

–Moskow: Nauka, 1973 (in Russian).12. HENRY D. Geometric theory of semilinear parabolic equations. Lect. Notes in

Math. 840. –Berlin: Springer, 1981.13. BOUTET DE MONVEL L., CHUESHOV I. D., REZOUNENKO A. V. Inertial manifolds for

retarded semilinear parabolic equations // Nonlinear Analysis. –1998. –Vol. 34. –P. 907–925.

14. BARIS YA. S., LYKOVA O. B. Approximate integral manifolds. –Kiev: Naukova dumka, 1993 (in Russian).

15. CHUESHOV I. D. Approximate inertial manifolds of exponential order for semi-

linear parabolic equations subjected to additive white noise // J. of Dyn. and Diff. Eqs. –1995. –Vol. 7. –# 4. –P. 549–566.

16. MORA X. Finite-dimensional attracting invariant manifolds for damped semi-

linear wave equations // Res. Notes in Math. –1987. –Vol. 155. –P. 172–183.17. CHUESHOV I. D. On a construction of approximate inertial manifolds for

second order in time evolution equations // Nonlinear Analysis, TMA. –1996. –Vol. 26. –# 5. –P. 1007–1021.

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equations and their attractors // Ann. della Scuola Norm. Sup. Pisa. 1987. –Vol. 14. –P. 485–511.

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C h a p t e r 4

The Problem on Nonlinear OscillationsThe Problem on Nonlinear OscillationsThe Problem on Nonlinear OscillationsThe Problem on Nonlinear Oscillations

of a Plate in a Supersonic Gas Flowof a Plate in a Supersonic Gas Flowof a Plate in a Supersonic Gas Flowof a Plate in a Supersonic Gas Flow

C o n t e n t s

. . . . § 1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

. . . . § 2 Auxiliary Linear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

. . . . § 3 Theorem on the Existence and Uniqueness of Solutions . . 232

. . . . § 4 Smoothness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

. . . . § 5 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . 246

. . . . § 6 Global Attractor and Inertial Sets . . . . . . . . . . . . . . . . . . . . . 254

. . . . § 7 Conditions of Regularity of Attractor . . . . . . . . . . . . . . . . . 261

. . . . § 8 On Singular Limit in the Problemof Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

. . . . § 9 On Inertial and Approximate Inertial Manifolds . . . . . . . . . 276

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

In this chapter we use the ideas and the results of Chapters 1 and 3 to study indetails the asymptotic behaviour of a class of problems arising in the nonlinear theo-ry of oscillations of distributed parameter systems. The main object is the followingsecond order in time equation in a separable Hilbert space :

, (0.1)

, (0.2)

where is a positive operator with discrete spectrum in , is a real function(its properties are described below), is a linear operator in , is a givenbounded function with the values in , and is a nonnegative parameter. Theproblem of type (0.1) and (0.2) arises in the study of nonlinear oscillations of a platein the supersonic flow of gas. For example, in Berger’s approach (see [1, 2]), the dy-namics of a plate can be described by the following quasilinear partial differentialequation:

, (0.3)

with boundary and initial conditions of the form

. (0.4)

Here is the Laplace operator in the domain ; , , and are con-stants; and , , and are given functions. Equations (0.3)–(0.4)describe nonlinear oscillations of a plate occupying the domain on a plane whichis located in a supersonic gas flow moving along the -axis. The aerodynamic pres-sure on the plate is taken into account according to Ilyushin’s “piston” theory (see,e. g., [3]) and is described by the term . The parameter is determined bythe velocity of the flow. The function measures the plate deflection at thepoint and the moment . The boundary conditions imply that the edges of theplate are hinged. The function describes the transverse load on the plate.The parameter is proportional to the value of compressive force acting in theplane of the plate. The value takes into account the environment resistance.

Our choice of problem (0.1) and (0.2) as the base example is conditioned by thefollowing circumstances. First, using this model we can avoid significant technicaldifficulties to demonstrate the main steps of reasoning required to construct a solu-tion and to prove the existence of a global attractor for a nonlinear evolutionary se-cond order in time partial differential equation. Second, a study of the limit regimesof system (0.3)–(0.4) is of practical interest. The point is that the most important(from the point of view of applications) type of instability which can be found in thesystem under consideration is the flutter, i.e. autooscillations of a plate subjected to

H

d2

t2d��u � d

td��u A2u M A1 2� u

2� �� A u L u� � � � p t�

ut 0 u0 , ud

td��

t 0u1

A H M s� L H p t� H �

�t2u � �

tu �2u u� 2

xd

��

� ��

�u ��x1

u� � � � p x t��

x x1 x2�� R2� � , t 0�

u �� u� �� 0 , ut 0 u0 x� , �tu

t 0u1 x�

� � � 0� � 0� p x t�� u0 x� u1 x�

�x1

��x1u �

u x t�� x t

p x t��

�

218 T h e P r o b l e m o n N o n l i n e a r O s c i l l a t i o n s o f a P l a t e i n a S u p e r s o n i c G a s F l o w

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aerodynamical loads. The modern look on the flutter instability of a plate is the fol-lowing: there arises the Andronov-Hopf bifurcation leading to the appearance ofa stable limit cycle in the system. However, there are experimental and numericaldata that enable us to conjecture that an increase in flow velocity may result in thecomplication of the dynamics and appearance of chaotic fluctuations [4]. Therefore,the study of the existence and properties of the attractor of the given problemenables us to better understand the mechanism of appearance of a nonlinear flutter.

§ 1 Spaces§ 1 Spaces§ 1 Spaces§ 1 Spaces

As above (see Chapter 2), we use the scale of spaces generated by a positive ope-rator with discrete spectrum acting in a separable Hilbert space . We remind(see Section 2.1) that the space is defined by the equation

,

where is the orthonormal basis of the eigenelements of the operator in , are the corresponding eigenvalues and is a real parameter

(for we have and for the space should be treated as a classof formal series). The norm in is given by the equality

for .

Further we use the notation for the set of measurable functionson the segment with the values in the space such that the norm

is finite. The notation has a similar meaning for .We remind that a function with the values in a separable Hilbert space

is said to be Bochner measurableBochner measurableBochner measurableBochner measurable on a segment if it is a limit of a se-quence of functions

for almost all , where are elements of and are the cha-racteristic functions of the pairwise disjoint Lebesgue measurable sets . One

�s

A H

�s

�s

D As� � v : ck

ek

: k 1

�

� ck2 �

k2 s

k 1

�

� �� ! "# $

ek

% & A H

�1 �2 '( ( s

s 0 �s

H s 0� �s

�s

vs2

ck�

k2 s

k 1

�

� v cke

k

k 1

�

�

L2 0 T� �s

)� 0 T�* + �

s

vL2 0 T� �

s)�

v t� s2

td

0

T

�� 1 2�

Lp 0 T� X)� 1 p �( (

u t� � H

0 T�* +

uN t� uN k� ,N k� t� k 1

N

�

t 0 T�* +� uN k� H ,N k� t� AN k�

Spaces 219

can prove (see, e.g., the book by K. Yosida [5]) that for separable Hilbert spaces un-der consideration a function is measurable if and only if the scalar function

is measurable for every . Furthermore, a function is saidto be Bochner integrableBochner integrableBochner integrableBochner integrable over if

, ,

where is a sequence of simple functions defined above. The integral of thefunction over a measurable set is defined by the equation

,

where is the characteristic function of the set and the integral of a simplefunction in the right-hand side of the equality is defined in an obvious way.

For the function with the values in Hilbert spaces most facts of the ordinary Le-besgue integration theory remain true.

Let be a function on with the values in a sepa-rable Hilbert space . If there exists a sequence of measurable func-tions such that almost everywhere, then is also measurable.

Show that a measurable function with the values in is integrable if and only if . Therewith

for any measurable set .

Let a function be integrable over and let bea measurable set from . Show that

for any .

Show that the space can be described as a setof series

,

u t� u t� h��

Hh H� u t�

0 T�* +

u t� uN t� �H

2td

0

T

� 0- N �-

uN

t� % &u t� � S 0 T�* +�

u t� td

S

� ,S.� u

N.� �d

0

T

�N �-lim

,S.� S

E x e r c i s e 1.1 u t� 0 T�* +H

un

t� un

t� u t� - u t�

E x e r c i s e 1.2 u t� H

u t� L1 0 T��

u t� �d

B

� u t� �d

B

�(

B 0 T�* +�

E x e r c i s e 1.3 u t� 0 T�* + B

0 T�* +

u .� h�� H �d

B

� u .� �d

B

� h��

H

h H�

E x e r c i s e 1.4 L2 0 T� �s)�

h t� � ck

t� ek

k 1

�

�


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where are scalar functions that are square-integrable over and such that

. (1.1)

Below we also use the space of strongly continuous functions on with the values in and the norm

.

Let be a function with the values in integrable over. Show that the function

lies in . Moreover, is an absolutely continuousfunction with the values in , i.e. for any there exists such that for any collection of disjoint segments the condition implies that

.

Show that for any absolutely continuous function on with the values in there exists a function with the

values in such that it is integrable over and

, .

(Hint: use the one-dimensional variant of this assertion).

The space

(1.2)

with the norm

plays an important role below. Hereinafter the derivative stands fora function integrable over and such that

ck

t� 0 T�* +

�k2 s

ck

t� * +2 td

0

T

�k 1

�

� ��

C 0 T� �s

)� 0 T�* +�

s

vC 0 T� �

s)� v t� s

t 0 T�* +�max

E x e r c i s e 1.5 u t� �s

0 T�* +

v t� u .� �d

0

t

�

C 0 T� �s

)� v t� �

s/ 0� 0 0�

1k

2k

�* + 0 T�* +�2

k1

k��

k� 0�

v 2k

� v 1k

� �s

k

� /�

E x e r c i s e 1.6 v t� 0 T�* + �

su t�

�s

0 T�* +

v t� v 0� u .� �d

0

t

�� t 0 T�* +�

WT v t� : v t� L2 0 T� �1)� v� t� L2 0 T� H)� �� ! "# $

v WT

vL2 0 T� �1)� 2

v�L2 0 T� H)� 2��

� �1 2�

v� t� vd td�0 T�* +

Spaces 221

almost everywhere for some (see Exercises 1.5 and 1.6). Evidently, the space is continuously embedded into , i.e. every function from

lies in and

,

where is a constant. This fact is strengthened in the series of exercises given be-low.

Let be the projector onto the span of the set and let . Show that is absolutely

continuous and possesses the property

.

The equations

(1.3a)

and

(1.3b)

are valid for any and .

Use (1.3) to prove that

(1.4a)

and

. (1.4b)

Use (1.4) to prove that is continuously embedded into and

.

v t� h v� .� �d

0

t

��

h H�W

TC 0 T� H)� u t� W

T

C 0 T� H)�

u t� t 0 T�* +�

max C uW

T(

C

E x e r c i s e 1.7 pm

ek

: k %1 ' m� � & v t� W

T� p

mv t�

dtd

�� pm

v t� � � pm

v� t� L2 0 T� pm�

1)� �

E x e r c i s e 1.8

pm

v t� 1 2�2

pm

v s� 1 2�2 2 p

mv .� p

mv� .� �� 1 2� �d

s

t

��

t s�� pm

v t� 1 2�2

pm v .� 1 2�2

2 . s�� pm v .� pm v� .� �� 1 2�� .d

s

t

�

0 s t T( ( ( v t� WT

�

E x e r c i s e 1.9

pm v t� 1 2�

t 0 T�* +�sup CT v W

T(

pm

pk

�� v t� 1 2�

t 0 T�* +�sup C

Tp

mp

k�� v

WT

(

E x e r c i s e 1.10 WT

C 0 T� �1 2�)�

v t� 1 2�t 0 T�* +�

max CT v WT

(


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The following three exercises result in a particular case of Dubinskii’s theorem(see Exercise 1.13).

Let be an orthonormal basis in con-sisting of the trigonometric functions

, , ,

. Show that if and only if

(1.5)

and

.

Show that the space can be described as a set of series ofthe form (1.5) such that

.

Use the method of the proof of Theorem 2.1.1 to show that is compactly embedded into the space for any

.

Show that is compactly embedded into .Hint: use Exercise 1.10 and the reasoning which is usually appliedto prove the Arzelà theorem on the compactness of a collectionof scalar continuous functions.

§ 2 Auxiliary Linear Problem§ 2 Auxiliary Linear Problem§ 2 Auxiliary Linear Problem§ 2 Auxiliary Linear Problem

In this section we study the properties of a solution to the following linear problem:

E x e r c i s e 1.11 hk

t� % &k 0�

L2 0 T� R)�

h0 t� 1T

�� h2 k 1� t� 2T��

23k

T�� xsin h2 k

t� 2T��

23k

T�� xcos

k 1 2 '� � f t� L2 0 T� �s

)� �

f t� ck j

hk

t� ej

j 1

�

�k 0

�

�

�j2 s

ck j

2

j 1

�

�k 0

�

� ��


k2 �j2��

� � ck j

2

k j� 1

�

� ��


WT

L2 0 T� �s

)� s 1�


C 0 T� H)�

(2.1)

(2.2)

d2u

t2d�� ud

td�� A2u b t� Au� � � h t� ,

ut 0 u0 ud

td��

t 0u1 .�

�44!44#

A u x i l i a r y L i n e a r P r o b l e m 223

Here is a positive operator with discrete spectrum. The vectors , , as well as the scalar function are given (for the corresponding hypotheses seethe assertion of Theorem 2.1).

The main results of this section are the proof of the theorem on the existenceand uniqueness of weak solutions to problem (2.1) and (2.2) and the constructionof the evolutionary operator for the system when . In fact, the approach weuse here is well-known (see, e.g., [6] and [7]).

A weak solution weak solution weak solution weak solution to problem (2.1) and (2.2) on a segment is a func-tion such that and the equation

(2.3)

holds for any function such that . As above, stands for thederivative of with respect to .

Prove that if a weak solution exists, then it satisfiesthe equation

(2.4)

for every (Hint: take in (2.3), whereis a scalar function from ).

Theorem 2.1

Let Let Let Let , , and . We al, , and . We al, , and . We al, , and . We alsssso assume that o assume that o assume that o assume that is a bounded is a bounded is a bounded is a bounded

continuous function on continuous function on continuous function on continuous function on and and and and , where , where , where , where is a posi- is a posi- is a posi- is a posi-

tive number. Then problem tive number. Then problem tive number. Then problem tive number. Then problem (2.1) andandandand (2.2) has a unique weak solution has a unique weak solution has a unique weak solution has a unique weak solution

on the segment on the segment on the segment on the segment .... This solution possesses the properties This solution possesses the properties This solution possesses the properties This solution possesses the properties

,,,, (2.5)

and satisfies the energy equationand satisfies the energy equationand satisfies the energy equationand satisfies the energy equation

(2.6)

A h t� u0 u1b t�

h t� 0�

0 T�* +u t� W

T� u 0� u0

u� t� �u t� � v� t� �� td

0

T

�� Au t� b t� u t� � Av t� �� t =d

0

T

��

u1 �u0� v 0� �� h t� v t� �� td

0

T

��

v t� WT� v T� 0 u�

u t

E x e r c i s e 2.1 u t�

u� t� �u t� � w�� u1 �u0� w��

Au .� b .� u .� � Aw�� .d

0

t

� h .� w�� .d

0

t

��

�

w �1� v t� 5 .� .dt

T� w65 t� C 0 T�* +

u0 �1� u1 �0� � 0� b t� 0 T�* + h t� L� 0 T� �0)� � T

u t� 0 T�* +

u t� C 0 T� �1)� � u� t� C 0 T� �0)� �

12�� u� t� 2

Au t� 2�� u� .� 2 .d

0

t

� b .� Au .� u� .� �� .d

0

t

��

12�� u1

2Au0

2�� h .� u� .� �� . .d

0

t

��


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Proof.

We use the compactness method to prove this theorem. At first we construct ap-proximate solutions to problem (2.1) and (2.2). The approximate Galerkin solution(to this problem) of the order with respect to the basis is considered to bethe function

(2.7)

satisfying the equations

, (2.8)

, , . (2.9)

Here and is absolutely continuous. Due to the orthogonalityof the basis equations (2.8) and (2.9) can be rewritten as a system of ordinarydifferential equations:

e,

, , .

Lemma 2.1

Assume that , , is continuous, and is a measu-

rable bounded function. Then the Cauchy problem

(2.10)

is uniquely solvable on any segment . Its solution possesses the

property

, (2.11)

where . Moreover, if , and for all

the conditions

, , (2.12)

hold, then the following estimate is valid:

. (2.13)

m ek% &

um

t� gk

t� ek

k 1

m

�

u��m �u�m A2 um

b t� Aum

h t� �� ej

�� 0

um

0� ej

�� u0 ej

�� u�m 0� ej

�� u1 ej

�� j 1 2 ' m� � �

gk

t� C1 0 T�� g�k t� e

k% &

g��k �g�k �k2

gk

b t� �k

gk

�� hk

t� h t� ek

��

gk 0� u0 ek�� g�k 0� u1 ek�� k 1 2 ' m� � �

� 0� a 0� b t� c t�

g�� g� a a b t� �� g� � c t� , t 0 T�* + ,�

g 0� g0 , g� 0� g1�4!4#

0 T�* +

g� t� 2 a2 g t� 2� g12

a2 g02 1

2 �� c .� 2 .d

0

t

��

eb0 t( t 0 T�* +�,

b0 b t� t

max b t� C1 0 T�� c t� 0�t 0 T�* +�

12�� a

�2

a�� b t� 1

2�� a( ( b

�t� � a

4�� b t� �� 0�

g� t� 2 a2 g t� 2� 3 g12

a2 g02��

�2�� t��

� �exp(


Proof.

Problem (2.10) is solvable at least locally, i.e. there exists such that a so-lution exists on the half-interval . Let us prove estimate (2.11) for the in-terval of existence of solution. To do that, we multiply equation (2.10) by .As a result, we obtain that

.

We integrate this equality and use the equations

, ,

to obtain that

.

This and Gronwall’s lemma give us (2.11).In particular, estimate (2.11) enables us to prove that the solution can

be extended on a segment of arbitrary length. Indeed, let us assume thecontrary. Then there exists a point such that the solution can not be extendedthrough it. Therewith equation (2.11) implies that

, .

Therefore, (2.10) gives us that the derivative is bounded on .Hence, the values

,

are continuous up to the point . If we now apply the local theorem on exis-tence to system (2.10) with the initial conditions at the point that are equal to

and , then we obtain that the solution can be extended through .This contradiction implies that the solution exists on an arbitrary segment

.Let us prove estimate (2.13). To do that, we consider the function

. (2.14)

Using the inequality

,

it is easy to find that the equation

(2.15)

t

0 t� *g� t�

12��

dtd

�� g�2 a2 g2�� g�2� a b t� g g� c t� g��

a b t� g g�12�� b t�

tmax g�2 a2 g2�

� �� ( c g� � g�2

14�� c2�(

g� t� 2 a2 g t� � g12

a2 g02 1

2�� c2 .� .d

0

t

� b0 g� .� 2 a2 g .� 2�� .d

0

t

�� (

g t� 0 T�* +

t

g� t� 2 a2 g t� 2� C T g0 g1�)� ( 0 t t T� � �

g�� t� 0 t�*

g� t� g0 g�� .� .d

0

t

�� g t� g0 g� .� .d

0

t

��

t

t

g t� g� t� t

g t� 0 T�* +

V t� 12�� g�2 a a b t� �� g2��

2�� g g�

�2�� g2��

� ��

12�� g�2�

�2�� g2� g g�

12�� g�2

�2�� g2�( (

14�� g�2 a2 g2�� V t� 3

4�� g�2 a2 g2�� ( (


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holds under the condition

, .

Further we use (2.10) with to obtain that

.

Consequently, with the help of (2.15) we get

under conditions (2.12). This implies that

.

We use (2.15) to obtain estimate (2.13). Thus, Lemma 2.1 is proved.

Assume that in Lemma 2.1. Show that problem (2.10)is uniquely solvable on any segment and the estimate

is valid for and for any , where .

Lemma 2.1 implies the existence of a sequence of approximate solutions to problem (2.1) and (2.2) on any segment .

Show that every approximate solution is a solution toproblem (2.1) and (2.2) with , , and

, where

,

,

and is the orthoprojector onto the span of elements in .

Let us prove that the sequence of approximate solutions is convergent.

12�� a b t� � 0� a

12�� a b t� � �2� 0�

c t� 0�

Vdtd

��2�� g�2� 1

2�� a b

�a � a b�� g2�

Vdtd

��2��V� 0(

V t� V 0� �2�� t� �

� �exp(

E x e r c i s e 2.2 � 0(0 T�* +

g� t� a2 g t� 2� g12

a2 g02�

� �� e

b0 2 � 0� �� t

10�� c .� 2 e

b0 2 � 0� �� t .�� .d

0

t

�

�

�

(

g t� 0 0� b0 b t� t

max

um

t� % &0 T�* +

E x e r c i s e 2.3 um

u0 u0 m u1 u1 m

h x t�� h

mx t��

ui m

pm

ui

ui

ek

�� ek

k 1

m

�

hm

pm

h h t� ek

�� ek

k 1

m

�

(2.16)

pm

ek

: k 1�%2 ' m� � & �0 H

um

% &


At first we note that

for every . Therefore, by virtue of Lemma 2.1 we have that

for . Moreover, in the case , the result of Exercise 2.2 gives that

.

These equations imply that the sequences and are the Cauchysequences in the space on any segment . Consequently, thereexists a function such that

, ,

. (2.17)

Equations (2.8) and (2.9) further imply that

for all functions from such that . Here , and , ,are defined by (2.16). We use equation (2.17) to pass to the limit in this equation andto prove that the function satisfies equality (2.3). Moreover, it follows from(2.17) that . Therefore, the function is a weak solution to problem(2.1) and (2.2).

In order to prove the uniqueness of weak solutions we consider the function

(2.18)

for . Here is a weak solution to problem (2.1) and (2.2) for ,, and . Evidently . Therefore,

�m l� t� u�m u�m l�� 2

A um

um l�� 2� g�k t� 2 �

k2 g

kt� 2��

k m 1�

m l�

�

t 0 T�* +�

�m l� t� e

b0 tu1 e

k�� 2 �

k2

u0 ek

�� 2 12�� h t� e

k�� 2 td

0

T

�� k m 1�

m l�

�(

� 0� � 0

�m l� t� e

b0 1�� tu1 e

k�� 2 �

k2

u0 ek

�� 2 h t� ek

�� 2 td

0

T

�� k m 1�

m l�

�(

u�m t� % & Aum t� % &C 0 T� H)� � 0 T�* +

u t�

u� t� C 0 T� H)� � u t� C 0 T� �1)� �

u�m t� u� t� � 2um t� u t� �

1��

� �0 T�* +max

m �-lim 0

u�m t� �um

t� � v� t� �� td

0

T

�� A um

t� b t� um

t� � Av t� �� td

0

T

��

u1 m�u0 m

� v 0� �� hm

t� v t� �� td

0

T

��

v t� WT

v T� 0 ui m

hm

t� i 0 1�

u t� u 0� u0 u t�

vs t� u .� . ,d t s ,�

t

s

��0 , s t T ,( (�

44!44#

s 0 T�* +� u t� h 0u0 0 u1 0 vs t� WT�


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.

Due to the structure of the function we obtain that

, (2.19)

where

.

It is evident that for . Therefore,

If we substitute this estimate into equation (2.19), then it is easy to find that

,

where and is a positive constant depending on the length of the seg-ment . This and Gronwall’s lemma imply that .

Let us prove the energy equation. If we multiply equation (2.8) by andsummarize the result with respect to , then we find that

.

After integration with respect to we use (2.17) to pass to the limit and obtain (2.6).Theorem 2.1 is completely proved.


(2.20)

is valid for a weak solution to problem (2.1) and (2.2). Here and .

u� t� �u t� � v�s t� �� td

0

T

�� Au t� b t� u t� � Avs

t� �� td

0

T

�� 0

vs

t�

12�� u s� 2

Avs 0� 2�� u t� 2

td

0

s

�� Js u v��

Js

u v�� b t� u t� Avs

t� �� td

0

s

�

Avs t� Avs 0� Avt 0� � t s(

Js u v�� b0 Avs 0� u t� td

0

s

� u t� Avt 0� 6 td

0

s

��

14�� Av

s0� 2

s b02

u t� 2td

0

s

�b02�� u t� 2

Avt

0� 2�� t .d

0

s

��

( (

(

u s� 2Av

s0� 2� C

Tu t� 2 Av

t0� 2��

� � td

0

s

�(

s 0 T�* +� CT

0 T�* + u t� 0�g� j t�

j

12��

dtd

�� u�m2

Aum

2�� u�m

2b t� Au

mu�m�� h u�m��

t

E x e r c i s e 2.4

u� t� 2Au t� 2� u1

2Au0

2 12�� h .� 2 .d

0

t

��

eb0 t(

u t� b0 b t� : t 0�% &max � 0�


Let be a weak solution to problem (2.1) and (2.2). Provethat and

.

Here we treat the equality as an equality of elements in . (Hint: use the results of Exercises 2.1 and 2.1.3).

Let be a weak solution to problem (2.1) and (2.2) con-structed in Theorem 2.1. Then the function is absolutely conti-nuous as a vector-function with the values in while the derivati-ve belongs to the space . Moreover, the function

satisfies equation (2.1) if we treat it as an equality of elementsin for almost all .

In particular, the result of Exercise 2.6 shows that a weak solution satisfies equation(2.1) in a stronger sense then (2.4).

We also note that the assertions of Theorem 2.1 and Exercises 2.4–2.6 with thecorresponding changes remain true if the initial condition is given at any other mo-ment which is not equal to zero.

Now we consider the case and construct the evolutionary operator of prob-lem (2.1) and (2.2). To do that, let us consider the family of spaces

, .

Every space is a set of pairs such that and .We define the inner product in by the formula

.

Prove that is compactly embedded into for .

In the space we define the evolutionary operator of problem (2.1) and(2.2) for by the equation

, (2.21)

where is a solution to (2.1) and (2.2) at the moment with initial conditionsthat are equal to at the moment .

The following assertion plays an important role in the study of asymptotic be-haviour of solutions to problem (0.1) and (0.2).

E x e r c i s e 2.5 u t� A2u t� C 0 T� � 1�)� �

u� t� �u t� � u1 �u0 A2u .� b .� Au .� h .� �� .d

0

t

��

� 1�

E x e r c i s e 2.6 u t� u� t� � 1�

u�� t� L� 0 T� � 1�)� u t�

� 1� t 0 T�* +�

t0

h t� 0�

�7 �1 7� �78 7 0�

�7 y u ; v� u �1 7�� v �7��7

y1 y2�� 7

u1 u2�� 1 7� v1 v2�� 7�

E x e r c i s e 2.7 �71�7 71 7�

�0 U t t0)� h t� 0�

U t t0)� y u t� u� t� )�

u t� t

y u0 u1)� t0


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Theorem 2.2

Assume that the function Assume that the function Assume that the function Assume that the function is continuously differentiable in is continuously differentiable in is continuously differentiable in is continuously differentiable in (2.1)and such thatand such thatand such thatand such that

, .

Then the evolutionary operator Then the evolutionary operator Then the evolutionary operator Then the evolutionary operator of problem of problem of problem of problem (2.1) andandandand (2.2) for for for for

is a linear bounded operator in each space is a linear bounded operator in each space is a linear bounded operator in each space is a linear bounded operator in each space for for for for and it and it and it and it

possesses the properties:possesses the properties:possesses the properties:possesses the properties:

a) ,,,, ,,,, ;;;;

b) for all the estimatefor all the estimatefor all the estimatefor all the estimate

(2.22)

is valid;is valid;is valid;is valid;

c) there exists a number there exists a number there exists a number there exists a number depending ondepending ondepending ondepending on , , , , , , , , and such that the and such that the and such that the and such that the

equationequationequationequation

,,,, ,,,, (2.23)

holds holds holds holds for all for all for all for all ,,,, where where where where is the orthoprojector onto the subspace is the orthoprojector onto the subspace is the orthoprojector onto the subspace is the orthoprojector onto the subspace

in the space in the space in the space in the space ....

Proof.

Semigroup property a) follows from the uniqueness of a weak solution.The boundedness property of the operator follows from (2.22). Let us proverelations (2.22) and (2.23). It is sufficient to consider the case . Accordingto the definition of the evolutionary operator we have that

, ,

where is the weak solution to problem (2.1) and (2.2) for . Due to(2.17) it can be represented as a convergent series of the form

.

Moreover, Lemma 2.1 implies that

. (2.24)

Since

,

equation (2.24) implies (2.22).

b t�

b0 b t� t

sup �� b1 b�

t� t

sup ��

U t .)� h t� 0� �7 7 0�

U t .)� U . s)� U t s)� t . s� � U t t)� I7 0�

U t .)� y�7

y�7

12�� b0 t .��

� �exp(

N0 � b0 b1

I PN�� U t .)� y�7

3 I PN�� y�7

e

�4�� t .��

( t .�

N N0� PN

LN

Lin ek

0)� 0 ek

)� � : k 1 2 ' N� � �% &

�0

U t .)� . 0

U t 0)� y u t� u� t� )� y u0 ; u1�

u t� h t� 0�

u t� gk t� ek

k 1

�

�

g�k t� 2 �k2

gk t� 2� g�k 0� 2 �k2

gk 0� 2�� eb0 t(

U t 0)� y�72

g�k t� 2 �k2

gk

t� 2��

k27

k 1

�

�


Further we use equation (2.13) to obtain that

, (2.25)

provided the conditions (cf. (2.12))

, ,

are fulfilled. Evidently, these conditions hold if

,

where and . Since

,

equation (2.25) gives us (2.23) for all , where is the smallest naturalnumber such that

. (2.26)


Show that a weak solution to problem (2.1) and (2.2)can be represented in the form

, (2.27)

where and is defined by (2.21).

Use the result of Exercise 2.2 to show that Theorem 2.1 andTheorem 2.2 (a, b) with another constant in (2.22) also remain truefor . Use this fact to prove that if the hypotheses of Theorems2.1 and 2.2 hold on the whole time axis, then problem (2.1) and (2.2)is solvable in the class of functions

with .

Show that the evolutionary operator has a boundedinverse operator in every space for . How is the operator

for related to the solution to equation (2.1) for? Define the operator using the formula

for and prove assertion (a) of Theorem 2.2 forall .

g�k t� 2 �k2

gk

t� 2� 3 gk

0� 2 �k2

gk

0� 2�� e

�2�� t�

(

12��k�

�2

�k

�� b t� 12��k( ( b

�t� �

�k

4�� b t� ��

� �� 0�

�k

4 b1�� 4 b0

�2

b0��

b0 b t� t

max b1 b t� t

max

I PN

�� U t 0)� y�7

2g�k t� 2 k4 g

kt� 2��

� � k27

k N 1�

�

�

N N0 1�� N0

�N0

4 b1�� 4 b0

�2

b0��


u t� u� t� )� U t 0)� y U t .)� 0 h .� )� .d

0

t

��

y u0 u1)� U t .)�

E x e r c i s e 2.9

� 0�

� C R �1)� C1 R �0)� 9

� 0�

E x e r c i s e 2.10 U t .�� 7 7 0�

U t .�� * + 1�t .�

h t� 0� U t .�� U t .�� U . t�� * + 1� t .�t .� R�


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§ 3 Theorem on Existence§ 3 Theorem on Existence§ 3 Theorem on Existence§ 3 Theorem on Existence

and Uniqueness of Solutionsand Uniqueness of Solutionsand Uniqueness of Solutionsand Uniqueness of Solutions

In this section we use the compactness method (see, e.g., [8]) to prove the theoremon the existence and uniqueness of weak solutions to problem (0.1) and (0.2) underthe assumption that

, , ; (3.1)

, , (3.2)

where , , is the first eigenvalue of the operator , and theoperator is defined on and satisfies the estimate

, . (3.3)

Similarly to the linear problem (see Section 2), the function is saidto be a weak solution weak solution weak solution weak solution to problem (0.1) and (0.2) on the segment if and the equation

(3.4)

holds for any function such that . Here the space is definedby equation (1.2).

Prove the analogue of formula (2.4) for weak solutions toproblem (0.1) and (0.2).

The following assertion holds.

Theorem 3.1

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (3.1)–(3.3) hold. Then on every segment hold. Then on every segment hold. Then on every segment hold. Then on every segment

problem problem problem problem (0.1) andandandand (0.2) has a weak solution has a weak solution has a weak solution has a weak solution .... This solution is unique. This solution is unique. This solution is unique. This solution is unique.

It possesses the propertiesIt possesses the propertiesIt possesses the propertiesIt possesses the properties

,,,, (3.5)

u0 �1� u1 �0� p t� L� 0 T� �0)� �

M z� C1 R�� z� M :� :d

0

z

� a z� b��

0 a �1�( b R� �1 A

L D A�

Lu C Au( u D A� �u t� W

T�

0 T�* +u 0� u0

u� t� �u t� � v� t� �� td

0

T

�� Au t� M A1 2� u t� 2� �� u t� � Av t� ��

� � t �d

0

T

�

Lu t� v t� �� td

0

T

�

�

� u1 �u0� v 0� �� p t� v t� �� td

0

T

��

v t� WT� v T� 0 WT

E x e r c i s e 3.1

0 T�* +u t�

u t� C 0 T� �1)� � u� t� C 0 T� �0)� �

T h e o r e m o n E x i s t e n c e a n d U n i q u e n e s s o f S o l u t i o n s 233

and satisfies the energy equalityand satisfies the energy equalityand satisfies the energy equalityand satisfies the energy equality

,,,, (3.6)

wherewherewherewhere

.... (3.7)

We use the scheme from Section 2 to prove the theorem.The Galerkin approximate solution of the order to problem (0.1) and (0.2)

with respect to the basis is defined as a function of the form

which satisfies the equations

(3.8)

for with and the initial conditions

, , . (3.9)

Simple calculations show that the problem of determining of approximate solutionscan be reduced to solving the following system of ordinary differential equations:

, (3.10)

, , . (3.11)

The nonlinear terms of this system are continuously differentiable with respect to. Therefore, it is solvable at least locally. The global solvability follows from the

a priori estimate of a solution as in the linear problem. Let us prove this estimate.We consider an approximate solution to problem (0.1) and (0.2) on the

solvability interval . It satisfies equations (3.8) and (3.9) on the interval. We multiply equation (3.8) by and summarize these equations with re-

spect to from 1 to . Since

,

we obtain

(3.12)

E u t� u� t� �� E u0 u1�� u� .� 2Lu t� � p t� � u� .� ��

� � .d

0

t

��

E u v�� 12�� v

2Au

2� A1 2� u

2� ��

� �

m

ek

um

t� gk

t� ek

k 1

m

�

u��m t� �u�m t� � ej

�� +

Aum t� M A1 2� u2

� �� um t� � A ej��

� � Lum t� p t� � ej�� 0

j 1 2 ' m� � � t 0 T� +��

um

0� ej

�� u0 ej

�� u�m 0� ej

�� u1 ej

�� j 1 2 ' m� � �

g��k � g�k �k2

gk

�k

M �jg

jt� 2

j 1

m

��

gk

L ej

ek

�� gj

j 1

m

�� pk

t�

gk 0� g0 k u0 ek�� g�k 0� g1 k u1 ek�� k 1 2 ' m� � �

gj

um

t� 0 t��

0 t�� g� j t� j m

dtd

�� A1 2� u2

� �� 2 M A1 2� u

2� �� Au t� u� t� ��

dtd

��E um

t� u�m t� �� u�m t� 2Lu

mt� p t� � u�m t� ��


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as a result, where is defined by (3.7). Equation (3.3) implies that

.

Condition (3.2) gives us the estimate

(3.13)

with the constants independent of . Therefore, due to Gronwall’s lemma equation(3.12) implies that

, (3.14)

with the constants , , and depending on the problem parameters only.

Use equation (3.14) to prove the global solvability of Cauchyproblem (3.10) and (3.11).

It is evident that

and .

Therefore,

,

where . Consequently, equation (3.14) givesus that

(3.15)

for any , where does not depend on . Thus, the set of approximate solu-tions is bounded in for any . Hence, there exist an element

and a sequence such that weakly in . Let usshow that the weak limit point possesses the property

(3.16)

for almost all . Indeed, the weak convergence of the sequence tothe function in means that and weakly (in ) con-verge to and respectively. Consequently, this convergence will also takeplace in for any and from the segment . Therefore, byvirtue of the known property of the weak convergence we get

.

E u u��

L um

u�m�� C Aum

u�m C Aum

2u�m

2�� ( (

12�� Au

m2

u�m2��

� � C1 C2 E um u�m�� (

m

u�m t� 2Au

mt� 2� C0 C1 E u

m0� u�m 0� ��

� � eC2 t(

C0 C1 C2

E x e r c i s e 3.2

u�m 0� u1( A1 2� um

0� 1�1

�� Aum

0� 1�1

�� u0 1( (

E um

0� u�m 0� �� 12�� u1

2u0 1

2CM u0 1

� � �� (

CM �� z� : 0 z �2 �1�( (% &max

u�m t� 2Au

mt� 2��

� �t 0 T�* +�

sup CT

(

T 0� CT m

um t� % & WT T 0�u t� WT� mk% & umk

t� u t� - WT

u t�

u� t� 2Au t� 2� CT(

t 0 T�* +� umk% &

u WT

u�mkAumk

L2 0 T� H)� u� Au

L2 a b� H)� a b 0 T�* +

u� t� 2Au t� 2��

� � td

a

b

� u�mk

t� 2Au

mk

t� 2�� td

a

b

�k �-lim(


With the help of (3.15) we find that

.

Therefore, due to the arbitrariness of and we obtain estimate (3.16).

Lemma 3.1

For any function

,

where

.

Proof.

Since

where , due to (3.15) and (3.16) we have

,

where the constant is the maximum of the function on the sufficient-ly large segment , determined by the constant from inequalities(3.15) and (3.16). Hence,

The compactness of the embedding of into (see Exer-cise 1.13) implies that as . It is evident that

u� t� 2Au t� 2��

� � td

a

b

� CT b a�� (

a b

v t� L2 0 T� H)� �

JT umk

v�� k �-lim JT u v��

JT

u v�� M A1 2� u t� 2� �� u t� v t� �� td

0

T

�

M A1 2� umk

2� �� M A1 2� u

2� ��

M�

: A1 2� um

k1 :�� A1 2� u��

� � :d

0

1

� A1 2� um

ku�� ,6

(

(

M�

z� 2 z M� z2� C R��

M A1 2� um

k

2� �� M A1 2� u

2� �� C

T

1� A1 2� u

mk

t� u t� �� (

CT1�

M�

z� 0 aT�* + CT

0k

M A1 2� um

t� 2� �� M A1 2� u t� 2

� �� u

mk

t� v t� �� td6

0

T

�

CT

vL2 0 T� H)�

A1 2� um

kt� u t� �� 2

td

0

T

�� 1 2�

.

(

(

�

WT

L2 0 T� �1 2�)� 0

k0- k �-


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.

Because of the weak convergence of to this gives us the assertion of thelemma.

Prove that the functional

is continuous on for any .

Let us prove that the limit function is a weak solution to problem (0.1) and(0.2).

Let be the orthoprojector onto the span of elements in the space . We also assume that

and

.

It is clear that an arbitrary element of the space has the form

,

where is an absolutely continuous real function on such that

, .

If we multiply equation (3.8) by , summarize the result with respect to from1 to , and integrate it with respect to from to , then it is easy to find that

for . The weak convergence of the sequence to in as well as Lem-ma 3.1 and Exercise 3.3 enables us to pass to the limit in this equality and to show

JT

um

kv�� J

Tu v�� 0

ku

mk

t� u t� � v t� �� M A1 2� u t� 2� �� 6 td

0

T

��(

um

ku

E x e r c i s e 3.3

� u* + Lu t� v t� �� td

0

T

�

WT v L2 0 T� H)� �

u t�

pl

ek

k 1 2 ' l� � ��H

W�

T v WT

: v T� 0�% &

W�

Tl

plW�

T� plv : v W

�T�% &

W�

Tl

vl x t�� ;k t� ek

k 1

l

�

;k

t� 0 T�* +

;k T� 0 ;�k

t� L2 0 T��

;j t� j

l t 0 T

u�m �um� v� l�� td

0

T

�� Aum M A1 2� um2

� �� um� Avl��

� � td

0

T

� L um vl�� td

0

T

��

u1 �u0� vl

0� �� p vl

�� td

0

T

��

m l� umku WT


that the function satisfies equation (3.4) for any function , where. Further we use (cf. Exercise 2.1.11) the formula

for any function in order to turn from the elements of to the func-tions from the space .

Prove that .

Thus, every weak limit point of the sequence of Galerkin approximations in the space is a weak solution to problem (0.1) and (0.2).

If we compare equations (3.4) and (2.3), then we find that every weak solution is simultaneously a weak solution to problem (2.1) and (2.2) with and

. (3.17)

It is evident that . Therefore, due to Theorem 2.1 equations(3.5) are valid for the function .

To prove energy equality (3.6) it is sufficient (due to (2.6)) to verify that for of form (3.17) the equality

(3.18)

holds. Here is a vector-function possessing property (3.5). We can do that byfirst proving (3.18) for the function of the form and then passing to the limit.

Let be a weak solution to problem (0.1) and (0.2). Useequation (3.6) to prove that

, , (3.19)

where are constants depending on the parametersof problem (0.1) and (0.2).

Let us prove the uniqueness of a weak solution to problem (0.1) and (0.2). We as-sume that and are weak solutions to problem (0.1) and (0.2) with theinitial conditions and , respectively. Then the function

u t� v W�

Tl�

l 1 2 '� �

plv� t� v� t� � 2

plAv t� Av t� � 2��

� � td

0

T

�l �-lim 0

v t� W�

T� v W�

Tl

v W�

T

E x e r c i s e 3.4 u t� t 0 u0

u t� um% &WT

u t� b t� 0

h t� M A1 2� u t� 2� �� Au t� � Lu t� p t� ��

h t� L� 0 T� �0)� �u t�

h t�

h .� u� .� �� .d

0

t

�Lu .� � p .� � u� .� �� .d

0

t

� 12�� A1 2� u .� 2

� ��

. 0

. t

�

u t� pl u


u� t� 2Au t� 2� C0 C1E u0 u1�� e

C2 t( t 0�

C0 C1 C2� � 0�

u1 t� u2 t� u01 u11�% & u02 u12�% &

u t� u1 t� u2 t� �


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is a weak solution to problem (2.1) and (2.2) with the initial conditions , , the function , and the right-hand

side

.

We use equation (3.19) to verify that

, ,

where is a positive monotonely increasing function of the parameter .Therefore, equation (2.20) implies that

where depends on and the problem parameters and is a function of thevariables , . We can assume that is the same for all initialdata such that , . Using Gronwall’s lemma we obtain that

, (3.20)

where and is a constant depending only on , the problem pa-rameters and the value such that . In particular, this esti-mate implies the uniqueness of weak solutions to problem (0.1) and (0.2). The proofof Theorem 3.1 is complete.

Show that a weak solution satisfies equation (0.1) if weconsider this equation as an equality of elements in for almost all .Moreover, (Hint: see Exercise 2.5).

Assume that the hypotheses of Theorem 3.1 hold. Let bea weak solution to problem (0.1) and (0.2) on the segment and let be the corresponding Galerkin approximation of theorder . Show that

as .

u0 u01�u02� u1 u11 u12� b t� M A1 2� u1 t� 2�

h t� M A1 2� u2 t� 2� �� M A1 2� u1 t� 2

� �� Au2 t� L u2 t� u1 t� ��

h t� GT

E u01 u11�� E u02 u12�� A u1 t� u2 t� �� ( t 0 T�* +�

GT :� :

u� 1 t� u� 2 t� � 2u1 t� u2 t� �

12�

CT

u11 u1 2� 2u01 u02�

12

u1 .� u2 .� �12 .d

0

t

��

,

(

(

CT

0� T

E u0 ju1 j

�� j 1 2� CT

E u0 ju1 j

�� R( j 1 2�

u� 1 t� u� 2 t� � 2u1 t� u2 t� �

12 � C1 u11 u12� 2

u01 u02�12��

� � ea C2(

t 0 T�* +� C 0� T

R 0� u0 j 12

u1 j2� R(

E x e r c i s e 3.6 u t� � 1� t

u�� t� C 0 T� � 1�)� �

E x e r c i s e 3.7 u t� 0 T�* +

um

t� m

um

t� u t� weakly in - L2 0 T� �1)� ,

u�m t� u� t� weakly in - L2 0 T� �0)� ,

um

t� u t� - strongly in L2 0 T� �s

)� , s 1 ,�

m �-


In conclusion of the section we note that in case of stationary load wecan construct an evolutionary operator of problem (0.1) and (0.2) in the space

supposing that

for , where is a weak solution to problem (0.1) and (0.2) with theinitial conditions . Due to the uniqueness of weak solutions we have

, , .

By virtue of (3.20) the nonlinear mapping is a continuous mapping of . Equa-tion (3.5) implies that the vector-function is strongly continuous with respectto for any . Moreover, for any and there exists a constant

such that

(3.21)

for all and for all .

Use equation (3.21) to show that is a continu-ous mapping from into .

Prove the theorem on the existence and uniqueness of solu-tions to problem (0.1) and (0.2) for . Use this fact to show thatthe collection of operators is defined for negative and formsa group (Hint: cf. Exercises 2.9 and 2.10).

Prove that the mapping is a homeomorphism in for eve-ry .

Let be a periodic function: , . Define the family of operators by the for-

mula

, ,

in the space . Here is a solution to problem (0.1)and (0.2) with the initial conditions . Prove that thepair is a discrete dynamical system. Moreover, and is a homeomorphism in .

p t� p H��S

t

� �0� �1 �08

St y u t� u� t� )�

y u0 u1�� u t� y u0 u1)�

St < S. S

t .� S0 I t .� 0�

St

�

Sty

t y �� R 0� T 0�C R T�� 0�

Sty1 S

ty2�

�C R T�� y1 y2�

�6(

t 0 T�* +� yj y �� : y�

R(% &�

E x e r c i s e 3.8 t y�� Sty-

R� �8 �

E x e r c i s e 3.9

� 0(St% & t


�

t 0�

E x e r c i s e 3.11 p t� L� R� H�� p t� p t t0�� t0 0� S

m

Sm

y u m t0� u� m t0� )� m 0 1 2 '� � �

� �1 �08 u t� y u0 u1��

� Sm

�� Sm

S1m

S1 �


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§ 4 Smoothness of Solutions§ 4 Smoothness of Solutions§ 4 Smoothness of Solutions§ 4 Smoothness of Solutions

In the study of smoothness properties of solutions constructed in Section 3 we usesome ideas presented in paper [9]. The main result of this section is the following as-sertion.

Theorem 4.1

Let the hypotheses of Theorem Let the hypotheses of Theorem Let the hypotheses of Theorem Let the hypotheses of Theorem 3.1 hold. We assume that hold. We assume that hold. We assume that hold. We assume that

and the load and the load and the load and the load lies in lies in lies in lies in for some for some for some for some .... Then Then Then Then

for a weak solution for a weak solution for a weak solution for a weak solution to problem to problem to problem to problem (0.1) andandandand (0.2) to possess the properties to possess the properties to possess the properties to possess the properties

(4.1)

it is necessary and sufficient that the following compatibility conditionsit is necessary and sufficient that the following compatibility conditionsit is necessary and sufficient that the following compatibility conditionsit is necessary and sufficient that the following compatibility conditions

are fulfilled:are fulfilled:are fulfilled:are fulfilled:

,,,, ;;;; .... (4.2)

Here Here Here Here is a strong derivative of the function is a strong derivative of the function is a strong derivative of the function is a strong derivative of the function with respect to with respect to with respect to with respect to

of the order of the order of the order of the order and the values and the values and the values and the values are recurrently defined by the initial are recurrently defined by the initial are recurrently defined by the initial are recurrently defined by the initial

conditions conditions conditions conditions and and and and with the help of equation with the help of equation with the help of equation with the help of equation (0.1)::::

,,,, ,,,,

(4.3)

where where where where ....

Proof.

It is evident that if a solution possesses properties (4.1) then compatibilityconditions (4.2) are fulfilled. Let us prove that conditions (4.2) are sufficient forequations (4.1) to be satisfied. We start with the case . The compatibility condi-tions have the form: , . As in the proof of Theorem 3.1 we considerthe Galerkin approximation

of the order for a solution to problem (0.1) and (0.2). It satisfies the equations

M z� �Cl 1� R�� p t� Cl 0 T� �0)� l 1�

u t�

u k� t� C 0 T� �2)� � ,,,, k 0 1 2 ' l 1 , , , ,��

u l� t� C 0 T� �1)� ,,,, u l 1�� t� C 0 T� �0)� , , , ,��

u k� 0� �2� k 0 1 2 ' l 1�� u l� 0� �1�

u j� t� u t� t

j u k� 0� u0 u1

u 0� 0� u0 u 1� 0� u1

u k� 0� �u k 1�� 0� A2u k 2�� 0� Lu k 2�� 0�

� dk 2�

tk 2�d�� M A1 2� u t� 2

� �� Au t� p t� ��

� �t 0

� � ��

!

"

#

$ ,

�

k 2 3 '� �

u t�

l 1u0 �2� u1 �1�

um t� gk t� ek

k 1

m

�

m

S m o o t h n e s s o f S o l u t i o n s 241

(4.4)

, ,

where is the orthoprojector onto the span of elements . The structureof equation (3.10) implies that . We differentiate equation(4.4) with respect to to obtain that satisfies the equation

(4.5)

and the initial conditions

,

. (4.6)

It is clear that

,

where is defined by (4.3). Therefore,

.

The compatibility conditions give us that and hence . Thus,the initial condition possesses the property

, . (4.7)

We multiply (4.5) by scalarwise in to find that

, (4.8)

where

(4.9)

Using a priori estimates (3.14) for we obtain

, .

u��

m t� �u�m t� A2 u

mt�

M A1 2� um

t� 2� �� A u

mt� p

mL u

mt�

� � �

� � pm

p t� ,

um

0� pm

u0 u�m 0� pm

u1

pm

e1 '� em

um

t� C2 0 T� �2)� �t v

mt� u�m t�

v��m �v�m A2 vm M A1 2� um2

� �� Avm pm Lvm� � � �

2� M = A1 2� um

2� �� Au

mu�m�� Au

mp

mp� t� �

vm 0� pmu1

v�m 0� pm

� �u1 A2u0 M A1 2� pm

u0

2� �� Au0 L p

mu0 p 0� ��

� ! "# $

v�m 0� pmu 2� 0� M A1 2� u02

� �� M A1 2� pmu

02

� �� Apmu

0L u0 p�

mu0� � �

u 2� 0�

vm

0� pm

u 2� 0� � C u0 1� u0 p

mu�

0 1(

u0 �2� u 2� 0� �0�v�m 0�

v�m 0� u 2� 0� � 0- m �-

v�m t� H

12��

dtd

�� v�m t� 2 Avm

t� 2�� v�m t� 2� F

mt� v�m t� ��

Fm t� M A1 2� um2

� �� Avm� pm L vm�

2� M = A1 2� um

2� �� Au

mu�m�� Au

mp

mp� t� .�

�

um t�

Fm

t� CT

1 Avm

t� �� ( t 0 T�* +�


4

C

h

a

p

t

e

r

Thus, equation (4.8) implies that

, .

Equation (4.7) and the property give us that the estimate

holds uniformly with respect to . Therefore, we reason as in Section 3 and useGronwall’s lemma to find that

, . (4.10)

Consequently,

, . (4.11)


.

Therefore, (3.14) and (4.11) imply that

, . (4.12)

Thus, the sequence of approximate solutions to problem (0.1) and (0.2)possesses the properties (cf. Exercise 3.7):

(4.13)

where is a weak solution to problem (0.1) and (0.2). Moreover (see Exer-cise 1.13),

(4.14)

for every . If we use these equations and arguments similar to the ones given inSection 3, then it is easy to pass to the limit and to prove that the function is a weak solution to the problem

(4.15)

dtd

�� v�m t� 2Av

mt� 2��

� � CT 1 v�m t� 2

Avm

t� 2� �� ( t 0 T�* +�

u1 �1�

v�m 0� 2Avm 0� 2� C�

m

Avm

t� 2v�m t� 2� C

T( t 0 T�* +�

Au�m t� 2u��m t� 2� C

T( t 0 T�* +�

A2um t� u��m t� � u�m t� M A1 2� um t� 2� Aum t� Lum t� p t� � � � �(

A2um t� CT

( t 0 T�* +�

um

t� % &

um t� u t� weakly in L2 0 T� D A2� )� ;-

u�m t� u� t� weakly in L2 0 T� D A� )� ;-

u��m t� u�� t� weakly in L2 0 T� H)� ;- 44"44$

u t�

u�m t� u� t� �s

2u

mt� u t� �

1 s�2� td

0

T

�m �-lim 0

s 1�w t� u� t�

w�� w� A2 w M A1 2� u t� 2� �� Aw L w� � � �

2� M = A1 2� u t� 2� �� Au u�� Au p� t� ,�

w 0� u1 , w� 0� u 2� 0� ,�444!444#


where is defined by (4.3). Therefore, Theorem 2.1 gives us that

.

This implies equation (4.1) for .

Further arguments are based on the following assertion.

Lemma 4.1

Let be a weak solution to the linear problem

(4.16)

where is a scalar continuously differentiable function,

and , . Then

(4.17)

and the function is a weak solution to the problem obtained

by the formal differentiation of (4.16) with respect to and equiped

with the initial conditions and

.

Proof.

Let be the Galerkin approximation of the order of a solution toproblem (4.16) (see (2.7)). It is clear that is thrice differentiable with re-spect to and satisfies the equation

, ,

and the initial conditions

, .

Therefore, as above, it is easy to prove the validity of equations (4.10)–(4.14)for the case under consideration and complete the proof of Lemma 4.1.

Assume that the hypotheses of Lemma 4.1 hold with and for some . Let the com-

patibility conditions (4.2) be fulfilled with , , and

u 2� 0�

u� t� w t� C 0 T� D A� )� C1 0 T� H)� 9�l 1

u t�

u�� t� �u� t� A2 u t� b t� Au� � � h t� ,

u 0� u0 ,�4!4#

b t� h t� �C1 0 T� �0)� � u0 �2� u1 �1�

u t� C 0 T� �2)� C1 0 T� �1)� C2 0 T� �0)� 9 9�

v t� u� t� t

v 0� u1 v� 0� u�� 0� �u1 A2u0� ��b 0� Au0 h 0� ��

um t� m

um t� t vm t� u�m t�

v��m �v�m A2vm b t� Avm� � � b�

t� Aum� pmh t� � t 0�

vm

0� pm

u1 v�m 0� pm

� �u1 A2 u0 b 0� Au0 h 0� ��

E x e r c i s e 4.1 b t� �Cl 0 T�� h t� Cl 0 T� �0)� � l 1�

u 0� 0� u0 u 1� 0� u1

u k� 0� �u k 1�� 0� A2 u k 2�� 0�

Ck 2�j

b j� 0� A u k 2� j�� 0� j 0

k 2�

�

� �

� �

�!#

�

�L u k 2�� 0� h k 2�� 0� � "$


4

C

h

a

p

t

e

r

for . Show that the weak solution to problem(4.16) possesses properties (4.1) and the function isa weak solution to the equation obtained by the formal differentia-tion of (4.16) times with respect to . Here .

In order to complete the proof of Theorem 4.1 we use induction with respect to. Assume that the hypotheses of the theorem as well as equations (4.2) for

hold. Assume that the assertion of the theorem is valid for .Since equations (4.1) hold for the solution with , we have

,

where , . Therefore, we differentiate equation(0.1) times with respect to to obtain that is a weak solutionto problem (4.16) with

and .

Consequently, Lemma 4.1 gives us that is a weak solution to the problemwhich is obtained by the formal differentiation of equation (0.1) times with re-spect to :

However, the hypotheses of Lemma 4.1 hold for this problem. Therefore (see (4.17)),

,

i.e. equations (4.1) hold for . Theorem 4.1 is proved.

Show that if the hypotheses of Theorem 4.1 hold, then thefunction is a weak solution to the problem which isobtained by the formal differentiation of equation (0.1) times withrespect to , .

Assume that the hypotheses of Theorem 4.1 hold and in equation (0.1). Show that if the conditions

, , (4.18)

are fulfilled, then a solution to problem (0.1) and (0.2) possess-es the properties

, .

k 2 3 ' l� � � u t� v

kt� u k� t�

k t k 0 1 ' l� � �

l

l n 1� l n 1�u t� l n

dk

tkd�� M A1 2� u t� 2

� �� Au t�

� ! "# $

M A1 2� u t� 2� �� Au k� t� G

kt� �

Gk t� C1 0 T� �0)� � k 1 ' n� �n 1� t v t� u n 1�� t�

b t� M A1 2� u t� 2� �� h t� G

n 1� t� � p n 1�� t� �

w t� v� t� n

t

w�� w� A2 w M A1 2� u t� 2� �� Aw� � � p n� t� G

nt� ,�

w 0� u n� 0� , w� 0� u n 1�� 0� .�4!4#

u n� t� w t� C 0 T� �2)� C1 0 T� �1)� C2 0 T� �0)� 9 9�

l n 1�

E x e r c i s e 4.2

v t� u k� t� k

t k 1 2 ' l� � �

E x e r c i s e 4.3 L 0�

p t� Ck 0 T�* + � l k�)� � k 0 1 ' l� � �

u t�

u t� Ck 0 T�* + �l 1 k��)� � k 0 1 ' l 1��


Assume that the hypotheses of Theorem 4.1 hold. We definethe sets

equation (4.2) holds with (4.19)

in the space . Prove that

and .

Show that every set given by equality (4.19) is invariant:

, .

Here is a weak solution to problem (0.1) and (0.2).

Assume that in equation (0.1) and the load pos-sesses property (4.18). Show that for the set ofform (4.19) contains the subspace .

Assume that the hypotheses of Theorem 3.1 hold and the ope-rator (in equation (0.1)) possesses the property

for some . (4.20)

Let and let . Show that the estimate

, , (4.21)

is valid for the approximate Galerkin solution to problem(0.1) and (0.2). Here the constant does not depend on (Hint: multiply equation (3.8) by and summarize the re-sult with respect to ; then use relation (3.14) to estimate the non-linear term).

Show that if the hypotheses of Exercise 4.7 hold, then prob-lem (0.1) and (0.2) possesses a weak solution such that

,

where is the number from Exercise 4.7.

E x e r c i s e 4.4

�k

u0 u1�� : ��!#

l k "$

� �1 �08

�1 �2 �18 �1 �2 ' �l

> > >

E x e r c i s e 4.5 �k

u0 u1)� �k� u t� u� t� )� �k�? k 1 ' l� �

u t�

E x e r c i s e 4.6 L 0� p t� k 1 2 ' l� � � �

k

�k 1� �

k8

E x e r c i s e 4.7

L

Lus

C u 1 s�( 0 s 1� �

u0 �1 s�� u1 �s

�

u�m t� s

2u

mt�

1 s�2� C

T( t 0 T�* +�

um

t� C

Tm

�j2 s

g� j t� j

E x e r c i s e 4.8

u t�

u t� C 0 T� �1 s�)� C1 0 T� �s

)� C2 0 T � 1� s�)�� 9 9�

s 0 1��


4

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§ 5 Dissipativity and Asymptotic§ 5 Dissipativity and Asymptotic§ 5 Dissipativity and Asymptotic§ 5 Dissipativity and Asymptotic

CompactnessCompactnessCompactnessCompactness

In this section we prove the dissipativity and asymptotic compactness of the dynam-ical system generated by weak solutions to problem (0.1) and (0.2) for

in the case of a stationary load . The phase space is . The evolutionary operator is defined by the formula

, (5.1)

where is a weak solution to problem (0.1) and (0.2) with the initial condition.

Theorem 5.1

Assume that in addition to Assume that in addition to Assume that in addition to Assume that in addition to (3.2) the following conditions are fulfilled the following conditions are fulfilled the following conditions are fulfilled the following conditions are fulfilled::::

a) there exist numbers there exist numbers there exist numbers there exist numbers such thatsuch thatsuch thatsuch that

,,,, (5.2)

with a constant with a constant with a constant with a constant ;;;;

b) there exist there exist there exist there exist and and and and such that such that such that such that

,,,, .... (5.3)

Then the dynamical system Then the dynamical system Then the dynamical system Then the dynamical system generated by problem generated by problem generated by problem generated by problem (0.1) andandandand (0.2)for for for for and for and for and for and for is dissipative is dissipative is dissipative is dissipative....

To prove the theorem it is sufficient to verify (see Theorem 1.4.1 and Exercise 1.4.1)that there exists a functional on which is bounded on the bounded sets ofthe space , differentiable along the trajectories of system (0.1) and (0.2), andsuch that

, (5.4)

, (5.5)

where and are constants. To construct the functional we use the method which is widely-applied for finite-dimensional systems (we usedit in the proof of estimate (2.13)).

Let

,

where . Here is the energy of system (0.1)and (0.2) defined by the formula (3.7),

� St

�� 0� p t� p H�� 0 �

�1 �08

Sty u t� u� t� ��

u t� y u0 u1)�

aj 0�

z M z� a1 M :� :d

0

z

�� a2 z1 1� a3�� z 0�

1 0�0 @ 1�( C 0�

Lu C A@u( u D A@� �

� St

�� 0� p t� p H��

V y� �

�

V y� 1 y�

2�1��

dtd

�� V Sty� � � 2V S

ty� � � 2(

1 2� 0� �1 � 2� 0� V y�

V y� E y� AB y� �

y u0 u1)� �� E y� E u0 u1)�


,

and the parameter will be chosen below. It is evident that

.

For this implies estimate (5.4) and the inequality

(5.6)

with the constants independent of . This inequality guarantees theboundedness of on the bounded sets of the space .

Energy equality (3.6) implies that the function , where , iscontinuously differentiable and

.

Therefore, due to (5.3) we have that

.

We use interpolation inequalities (see Exercises 2.1.12 and 2.1.13) to obtain that

, .

Thus, the estimate

(5.7)

holds for any .

Lemma 5.1

Let be a weak solution to problem (0.1) and (0.2) and let

. Then the function is continuously differenti-

able and

. (5.8)

We note that since (see Exercise 3.6), equation (5.8) is correct-ly defined.

Proof.

It is sufficient to verify that

. (5.9)

B y� u0 u1)� �2�� u0

2�

A 0�

12�� u1

2� B y� 12�� u1

2 � u02�( (

0 A ��

V y� b1E y� b2�(

b1 b2� 0� AV y� �

E y t� � y t� Sty

dtd

��E y t� � � u� t� 2� L� u t� p� u� t� ��

dtd

��E y t� � �2�� u�

2� C1 A@u2

C2� �(

A@u2 0 Au

2C0 A1 2� u

2�( 0 0�

dtd

�� E y t� � �2�� u�

2 /2�� Au

2�� C/ A1 2� u2

C2� �(

/ 0�

u t� y t� u t� u� t� �� B y t� �

dtd

��B y t� � u� t� 2u�� t� �u� t� � u t� ��

u�� t� C R� � 1��

u t� u� t� �� u0 u1�� u�� .� u .� �� u� .� 2�� ! "# $ .d

0

t

��


4

C

h

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p

t

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Let be the orthoprojector onto the span of elements in. Then it is evident that the vector-function is twice continu-

ously differentiable with respect to . Therefore,

.

The properties of the projector (see Exercise 2.1.11) enable us to pass tothe limit and to obtain (5.9). Lemma 5.1 is proved.

Since is a solution to equation (0.1), relation (5.8) implies that

.

Therefore, equation (5.2) and the evident inequality

give us that

.

Hence, (5.6) and (5.7) enable us to obtain the estimate

where and

.

Therefore, for any estimate (5.5) holds, provided and are chosenappropriately. Thus, Theorem 5.1 is proved.

Prove that if the hypotheses of Theorem 5.1 hold, then the as-sertion on the dissipativity of solutions to problem (0.1) and (0.2)remains true in the case of a nonstationary load .

Theorem 5.2

Let the hypotheses of Theorem Let the hypotheses of Theorem Let the hypotheses of Theorem Let the hypotheses of Theorem 5.1 hold and assume that for some hold and assume that for some hold and assume that for some hold and assume that for some

,,,, ,,,, .... (5.10)

Then there exists a positively invariant bounded set Then there exists a positively invariant bounded set Then there exists a positively invariant bounded set Then there exists a positively invariant bounded set in the space in the space in the space in the space

which is closed in which is closed in which is closed in which is closed in and such that and such that and such that and such that

pl

ek

k 1 2 ' l� � ��% &�0 u

lt� p

lu t�

t

ul

t� u� l t� �� ul

0� u� l 0� �� u�� l .� ul.� �� u� l .� 2�

� ! "# $

.d

0

t

��

pl

l �-

u t�

dtd

��B y t� � u�2

Au2

M A1 2� u2

� �� A1 2� u

26 Lu p� u�� ! "# $

�

Lu p� u�� 12�� Au

2C1 u

2p

2� �(

dtd

��B y t� � u�2 1

2�� Au

2� a1� A1 2� u2

� �� a2 A1 2� u

2 21�C1 u

2C2� ��(

dtd

��V y t� � 0V y t� � � 12�� 0 b1� 2A�� u�

2�( �

12�� A 0 b1� /�� Au

2 Aa1

0b12

�� A1 2� u

2� �� R u A / 0� �)� ,��

0 0�

R u A / 0� �)� Aa2 A1 2� u2 21�� C/ A1 2� u

2 AC1 u2

C0� � �

0 A � 2�� 0 /

E x e r c i s e 5.1

p t� L� R� H��

7 0�

p �7� LD A� D A7� � A7L u C Au(

K7 �7 �1 7� �78 �


(5.11)

for any bounded set for any bounded set for any bounded set for any bounded set in the space in the space in the space in the space , , , , ....

Due to the compactness of the embedding of in for , this theorem andLemma 1.4.1 imply that the dynamical system is asymptotically compact.

Proof.

Since the system is dissipative, there exists such that for all and

, (5.12)

where is a weak solution to problem (0.1) and (0.2) with the initial conditions. We consider as a solution to linear problem (2.1) and (2.2)

with and . It is easy to verify that is acontinuously differentiable function and

, .

Moreover, equation (2.27) implies that

, (5.13)

where

.

Here is the evolutionary operator of the homogeneous problem (2.1) and(2.2) with and . By virtue of Theorem 2.2 thereexists such that

, (5.14)

where , , and is the orthoprojector onto

, .

This implies that

.

Therefore, we use (5.10) to obtain that

. (5.15)

Sty K7��

�dist : y B�

� ! "# $

sup C e

�4�� t t

B��

(

B � t tB

�

�7 � 7 0�� S

t��

� St�� R 0�y B� t t0� t0 B�

u� t� 2Au t� 2� R

2(

u t� y u0 u1)� B� u t�

b t� M A1 2� u t� 2� h t� Lu t� � p� b t�

b t� b�

t� � CR( t t0�

Sty U t t0�� y t0� G t t0� y)� �

G t t0� y)� U t .�� 0 L u .� p�)� .d

t0

t

��

U t .�� h t� 0 b t� M A1 2� u t� 2�

N0 0�

1 PN

�� U t .�� h�7

3 h�7

�4�� t .��

� ! "# $exp(

N N0� t . t0� � PN

�

�N

Lin ek

0)� 0 ek

)� �% k 1 2 ' N� � � &

1 PN0

�� G t t0� y)� �7

3�4�� t .��

� ! "# $exp Lu .� p� 7 .d

t0

t

�(

1 PN0

�� G t t0� y)� �7

C R� (


4

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It is also easy to find that

, .

Consequently, there exists a number depending on the radius of dissipativity and the parameters of the problem such that the value

(5.16)

lies in the ball

for . Therefore, with the help of (5.12) and (2.23) we have that

. (5.17)

Let . Evidently equation (5.11) is valid. Moreover, is positively invariant. The continuity of with recpect to the both variables

in the space (see Exercise 3.8) and attraction property (5.17) imply that is a closed set in . Let us prove that is bounded in . First we note that is bounded in . Indeed, by virtue of the dissipativity we have that

for all and . Since is continuous with respect to the vari-ables , its maximum is attained on the compact . Thus, there exists

such that for all . Let us return to equality (5.16) for and . It is evident that the norm of the right-hand side in the space is bounded by the constant . However, equation (5.14) implies that

, , .

Therefore, equation (5.16) leads to the uniform estimate

, , .

Thus, the set is bounded in . Theorem 5.2 is proved.

Show that for any a bounded set of is attractedby at an exponential rate with respect to the metric of the space

. Thus, we can replace by in (5.11).

Prove that if the hypotheses of Theorem 5.2 hold, then the as-sertion on the asymptotic compactness of solutions to problem (0.1)and (0.2) remains true in the case of nonstationary load

(see also Exercise 5.1).

Prove that the hypotheses of Theorem 5.1 and 5.2 hold forproblem (0.3) and (0.4) for any , provided that and lies in the Sobolev space .

PN0

Sty

�7N0

7S

ty

�R N0

7( ( t t0�

R7 R

Sty 1 P

N0�� U t t0�� S

t0y0� P

N0S

ty 1 P

N0�� G t t0� y)� �

B7 y : y�7

R7(% &

t t0�

dist�

Sty B7�� 1 P

N0�� U t t0�� S

t0y R 3 e

�4�� t t0��

( (

K7 �� B7� St

B7� t 0�C� K7S

ty

t y)� �

K7 � K7 �7K7 � S

ty R(

y B7� t t7 t B7� �� Sty

t y)� 0 t7�* + B78R 0� y R( y K7� t0 0

y0 B7� �7C C 7 R��

1 PN0�� U t 0�� y0 �7

3 R7( t 0� y0 B7�

St y�7

R7( t 0� y B7�

K7 �7

E x e r c i s e 5.2 0 s 7( ( �s

K7�s �

dist�

sdist

E x e r c i s e 5.3

p t� �L� R� �7��

E x e r c i s e 5.4

0 7 1 4�� 0�p x t�� p x� � H0

1 ��


Let us consider the dissipativity properties of smooth solutions (see Section 4) toproblem (0.1) and (0.2).

Theorem 5.3

Let the hypotheses of Theorem Let the hypotheses of Theorem Let the hypotheses of Theorem Let the hypotheses of Theorem 5.1 hold. Assume that hold. Assume that hold. Assume that hold. Assume that

and the initial conditions and the initial conditions and the initial conditions and the initial conditions are such that equations are such that equations are such that equations are such that equations (4.1) (and (and (and (and

hence hence hence hence (4.2)) are valid for the solution ) are valid for the solution ) are valid for the solution ) are valid for the solution .... Then there exists Then there exists Then there exists Then there exists

such that for any initial data such that for any initial data such that for any initial data such that for any initial data possessing the property possessing the property possessing the property possessing the property

,,,, ,,,, (5.18)

the solution the solution the solution the solution admits the estimate admits the estimate admits the estimate admits the estimate

(5.19)

for all for all for all for all as soon as as soon as as soon as as soon as ....

We use induction to prove the theorem. The proof is based on the following asser-tion.

Lemma 5.2

Assume that the hypotheses of Theorem 5.3 hold for . Then the dy-

namical system generated by problem (0.1) and (0.2) in the

space is dissipative.

Proof.

Let be a semitrajectory of the dynamical system and let . If the hypotheses of the lem-

ma hold, then the function is a weak solution to problem (4.15) ob-tained by formal differentiation of (0.1) with respect to (as we have shown inSection 4). By virtue of Theorem 2.1 the energy equality of the form (2.6) holdsfor the function . We rewrite it in the differential form:

, (5.20)

where

(5.21)

and

The dissipativity property of given by Theorem 5.1 leads to the esti-mates

M z� Cl 1� R�� y0 u0 u1)�

u t� u� t� )� St y0Rl 0� y0 u0 u1)�

u k 1�� 0� 2Au k� 0� 2

A2 u k 1�� 0� 2� � �2� k 1 2 ' l� � �

u t� u� t� )� Sty0

u k 1�� t� 2Au k� t� 2

A2 u k 1�� t� 2� � Rl2�

k 1 2 ' l� � � t t0 ��

l 1�1 S

t��

�1 �2 �18

u t� u� t� )� St y0�1 St�� y0 u0 u1)� �1 �2 �18�

w t� u� t� t

w t�

dtd

��F t w t� w� t� �)� � w� t� 2� D u t� w t� ��

F t w w��)� 12�� w�

2Aw

2M A1 2� u t� 2� �� A1 2� w

26� ��

D u t� w t� �� Lw t� w� t� ��

� M = A1 2� u t� 2� �� Au t� u� t� �� A1 2� w t� 2 2 Au t� w� t� �� .

� St

��


4

C

h

a

p

t

e

r

,

for all with the property

and for all large enough. Hereinafter is the radius of dissipativityof the system . These estimates imply that for we have

(5.22)

and

, (5.23)

where the constants depend on . Here . Similarly, we use Lem-ma 5.1 to find that

for . Consequently, the function

possesses the properties

, ,

and

for and for small enough. This implies that

(5.24)

for , provided that

. (5.25)

If we use (5.4)–(5.6), then it is easy to find that

, ,

under condition (5.25). Using the energy equality for the weak solutions toproblem (4.15) we conclude that

M A1 2� u t� 2� �� A1 2� w

2CR Aw t� (

D u t� w t� �� Lw w� CR

Aw t� w� t� �� (

u0 u1)�

Au02

u12� �2(

t t0 �� R

� St

�� t t0 ��

14�� w�

2Aw

2�� 11� F t w w��)� 12 w t� 2

w� t� 2�� 13�( (

dtd

��F t w w��)� �2�� w�

2� 2 A@w2 14 Aw 15� �(

1j 0� R @ 1�

dtd

�� w t� w� t� �� 2�� w t� 2�

� ! "# $

w� t� 2 12�� Aw t� 2

CR

��(

t t0 ��

V t� F t w w��)� A w w�� 2�� w

2�� ! "# $

�

Vdtd

�� EV� CR

( E 0�

14�� w

� 2Aw

2�� a1� V a2 w

2w�

2�� a3�( (

t t0 �� A 0�

w� t� 2Aw t� 2� C1 w� t0� 2

Aw t0� 2�� e E t� C2�(

t t0� t0 ��

Au02

u12� �2(

Au t� 2u� t� 2� C� R�( t 0�


.

Therefore, standard reasoning in which Gronwall’s lemma is used leads to

,

provided that equation (5.25) is valid. Here and are some positive con-stants depending on and . This and equation (5.24) imply that

, (5.26)

where depends on only. Since

,

provided that

,

equation (5.26) gives us the estimate

, .

This easily implies the dissipativity of the dynamical system . Thus,Lemma 5.2 is proved.

Prove that the dynamical system generated byproblem (0.1) and (0.2) with the initial data

is asymptotically compact provided that equations (5.10)hold.

In order to complete the proof of Theorem 5.3, we should note first that Lemma 5.2coincides with the assertion of the theorem for and second we should use thefact that the derivatives are weak solutions to the problem obtained by dif-ferentiation of the original equation. The main steps of the reasoning are given in thefollowing exercises.

Assume that the hypotheses of Theorem 5.3 hold for and its assertion is valid for . Show that isa weak solution to the problem of the form

where

for all .

dtd

�� Aw t� 2w� t� 2��

� � C1 � R�� Aw w�6 6 C2 � R�� (

Aw t� 2w� t� 2� Aw 0� 2 w� 0� 2

a� �� eb t(

a b

� R

w� t� 2Aw t� 2� C1 � R�� 1 w� 0� 2�� Aw 0� 2��

� � e E t� C2�(

C2 R

w� 0� 2Aw 0� 2� C�( u1

2Au0

2��1�� 2

(

Au12

A2u02� �2(

u�� t� 2Au� t� 2� bR

2( t t 0 ��

�1 St

��

E x e r c i s e 5.5 �1 St�� y0 u0 u1)� �1 �

�2 �18

l 1u k� t�

E x e r c i s e 5.6 l n 1�l n w t� u n 1�� t�

w�� t� �w� t� A2 w t� M A1 2� u2

Aw Lw� � � � Gn 1� t� ,

w 0� u n 1�� 0� w� 0� u n 2�� 0� ,��4!4#

Gn 1� t� C R

n� ( t t0 ��


4

C

h

a

p

t

e

r

Use the result of Exercise 5.6 and the method given in theproof of Lemma 5.2 to prove that can be estima-ted as follows:

(5.27)

where , the numbers depend on and , , andthe constant depends on only.

Use the induction assumption and equation (5.27) to provethe assertion of Theorem 5.3 for .

§ 6 Global Attractor and § 6 Global Attractor and § 6 Global Attractor and § 6 Global Attractor and Inertial SetsInertial SetsInertial SetsInertial Sets

The above given properties of the evolutionary operator generated by problem(0.1) and (0.2) in the case of stationary load enable us to apply the gene-ral assertions proved in Chapter 1 (see also [10]).

Theorem 6.1

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (3.2), , , , (5.2), , , , (5.3), and , and , and , and (5.10) are fulfilled. Then are fulfilled. Then are fulfilled. Then are fulfilled. Then

the dynamical system the dynamical system the dynamical system the dynamical system generated by problem generated by problem generated by problem generated by problem (0.1) andandandand (0.2) pos- pos- pos- pos-

sesses a global attractor sesses a global attractor sesses a global attractor sesses a global attractor of a finite fractal dimension. This attractor is of a finite fractal dimension. This attractor is of a finite fractal dimension. This attractor is of a finite fractal dimension. This attractor is

a connected compact set in and is bounded in the space a connected compact set in and is bounded in the space a connected compact set in and is bounded in the space a connected compact set in and is bounded in the space ,,,,

where where where where is defined by condition is defined by condition is defined by condition is defined by condition (5.10)....

Proof.

By virtue of Theorems 5.1, 5.2, and 1.5.1 we should prove only the finite dimen-sionality of the attractor. The corresponding reasoning is based on Theorem 1.8.1and the following assertions.

Lemma 6.1.

Assume that conditions (3.2), (5.2), and (5.3) are fulfilled. Let .

Then for any pair of semitrajectories , posses-

sing the property for all the estimate

, , (6.1)

holds with the constant depending on .

E x e r c i s e 5.7

w t� u n 1�� t�

w� t� 2Aw t� 2�

C1 w� 0� 2Aw 0� 2

C2� �� e E t� C3 ,�

(

(

E 0� Cj

� Rn

j 1 2�C3 R

n

E x e r c i s e 5.8

l n 1�

St

p t� p�

� St

��

� �7 �1 7� �787 0�

p H�S

ty

j: t 0�% & j 1 2 ,�

Sty

jR( t 0�

Sty1 S

ty2�

�a0 t� exp y1 y2�

�( t 0�

a0 R

G l o b a l A t t r a c t o r a n d I n e r t i a l S e t s 255

Proof.

If and are solutions to problem (0.1) and (0.2) with the initialconditions and , then the function

satisfies the equation

, (6.2)

where

.

It is evident that the estimate

holds, provided that . Therefore, (2.20)implies that

.

Gronwall’s lemma gives us equation (6.1).

Lemma 6.2

Assume that the hypotheses of Theorem 6.1 hold. Let be the compact

positively invariant set constructed in Theorem 5.2. Then for any

the inequality

(6.3)

is valid, where , , the orthoprojector and the

number are defined as in (5.14), and are positive constants

which depend on the parameters of the problem.

Proof.

It is evident that

,

where is the orthoprojector onto the closure of the span of elements in . Moreover, the function

is a solution to the equation

,

where

.

u1 t� u2 t� y1 u01 u11)� y2 u02 u12)� v t�

u1 t� u2 t� �

v�� v� A2 v� � � u1 u2 t� ��

� u1 u2 t� �� M A1 2� u2 t� 2� �� Au2 t� M A1 2� u1 t� 2

� �� Au1 t� � L v t� �

� u1 u2 t� �� CR A u1 t� u2 t� �� 6(

yi

t� �

2u� i t� 2

ui

t� 12� R2(

Sty1 S

ty2�

�

2y1 y2�

�

2a1 S

ty1 S

ty2�

�

2 .d

0

t

��(

K7

y1 y2� K7�

QN

Sty1 S

ty2��

�a2 e

�4�� t�

1L7

�N 1�7�� e

a3 t��

y1 y2��

(

QN 1 PN� N N0� PN

N0 L7 ai

QN

Sty

iq

Nu

it� q

Nu� i t� )�

qN

ek

: %k N 1� N 2� '� � & �0 H w

Nt� q

Nu1 t� u2 t� ��

w��N �w�N A2 wN M A1 2� u12

� �� AwN� � � BN u1 u2 t� ��

BN

u1 u2 t� �� M A1 2� u12

� �� M A1 2� u2

2� �� q

NAu2� q

NL u1 u2��


4

C

h

a

p

t

e

r

Let us estimate the value . Since for (see the proofof Theorem 5.2), we have

.

Using equation (5.10) we similarly obtain that

.

Therefore,

.

Consequently,

. (6.4)

Using equation (2.27) we obtain that

,

where is the evolutionary operator of homogeneous problem (2.1) with and . Therefore, (2.23) and (6.4) imply that

(6.5)

We substitute (6.1) in this equation to obtain (6.3). Lemma 6.2 is proved.

Let us choose and such that

, , .

Then Lemmata 6.1 and 6.2 enable us to state that

and

,

where and the elements and lie in the global attractor . Hence,we can use Theorem 1.8.1 with , , and . Therefore, the fractaldimension of the attractor is finite. Thus, Theorem 6.1 is proved.

BN

y�7

R7( y K7�

qN

Au2 qN

u2 1 �

N 1�7�

qN

u2 1 7� �N 1�7�

R7( (

qN

Lv �N 1�7�

A7Lv C�N 1�7�

Av( (

qN L u1 t� u2 t� �� C�N 1�7�

St y1 St y2��

(

BN

u1 u2 t� �� C R7�

�N 1�7�� S

ty1 S

ty2�

�(

QN St y1 St y2�� U t 0�� QN y1 y2�� U t .�� 0 BN .� )� .d

0

t

��

U t .�� b t� M A1 2� u1 t� 2� h t� 0

QN

Sty1 S

ty2��

�

3 e

�4�� t�

y1 y2��

�C R7� �

N 1�7�� e

�4�� .

S.y1 S.y2� .d

0

t

�� 4 4! "4 4# $

.

(

(

t0 N N0�

a2�4�� t0�

� ! "# $

exp 02�� L7�N 1�

7�a3 t0% &exp 1( 0 1�

St0y1 St0

y2��

l y1 y2��

(

QN St0y1 St0

y2��

y1 y2��

0(

l ea0 t0 y1 y2 �

M � V St0

P PN

�


Theorem 5.2 and Lemmata 6.1 and 6.2 enable us to use Theorem 1.9.2 to obtain anassertion on the existence of the inertial set (fractal exponential attractor) for thedynamical system generated by problem (0.1) and (0.2).

Theorem 6.2

Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem 6.1 hold. Then there exists hold. Then there exists hold. Then there exists hold. Then there exists

a compact positively invariant set a compact positively invariant set a compact positively invariant set a compact positively invariant set of the finite fractal dimen- of the finite fractal dimen- of the finite fractal dimen- of the finite fractal dimen-

sion such thatsion such thatsion such thatsion such that

for any bounded set for any bounded set for any bounded set for any bounded set in in in in and and and and .... Here Here Here Here and and and and are positive num-are positive num-are positive num-are positive num-

bers. The inertial set bers. The inertial set bers. The inertial set bers. The inertial set is bounded in the space is bounded in the space is bounded in the space is bounded in the space ....

To prove the theorem we should only note that relations (5.11), (6.1), and (6.3)coincide with conditions (9.12)–(9.14) of Theorem 1.9.2.

Using (1.8.3) and (1.9.18) we can obtain estimates (involving the parameters of theproblem) for the dimensions of the attractor and the inertial set by an accurate ob-serving of the constants in the proof of Theorems 5.1 and 5.2 and Lemmata 6.1 and6.2. However, as far as problem (0.3) and (0.4) is concerned, it is rather difficultto evaluate these estimates for the values of parameters that are very interestingfrom the point of view of applications. Moreover, these estimates appear to be quiteoverstated. Therefore, the assertions on the finite dimensionality of an attractor andinertial set should be considered as qualitative results in this case. In particular, thisassertions mean that the nonlinear flutter of a plate is an essentially finite-dimen-sional phenomenon. The study of oscillations caused by the flutter can be reduced tothe study of the structure of the global attractor of the system and the propertiesof inertial sets.

Prove that the global attractor of the dynamical system gene-rated by problem (0.1) and (0.2) is a uniformly asymptotically stableset (Hint: see Theorem 1.7.1).

We note that theorems analogous to Theorems 6.1 and 6.2 also hold for a class of re-tarded perturbations of problem (0.1) and (0.2). For example, instead of (0.1) and(0.2) we can consider (cf. [11–13]) the following problem

,

.

� St

��

�exp ��

St y �exp��

dist : y B�� ! "# $sup C e

A t tB

�� (

B � t tB

� C A�exp �7

E x e r c i s e 6.1

u�� u� A2 u M A1 2� u2

� �� Au L u q u

t� � � � � � p

ut 0 u0 , u�

t 0 u1 , ut r� 0�� 5 t�


4

C

h

a

p

t

e

r

Here , , and are the same as in Theorems 6.1 and 6.2, the symbol denotesthe function on which is given by the equality for

, the parameter is a delay value, and is a linear mapping from into possessing the property

for small enough and for all , where is a positive number. Sucha formulation of the problem corresponds to the case when we use the model of thelinearized potential gas flow (see [11–14]) to take into account the aerodynamicpressure in problem (0.3) and (0.4).

The following assertion gives the time smoothness of trajectories lying in the attrac-tor of problem (0.1) and (0.2).

Theorem 6.3

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (3.2) and and and and (5.2) are fulfilled and the linear ope- are fulfilled and the linear ope- are fulfilled and the linear ope- are fulfilled and the linear ope-

rator rator rator rator possesses the property possesses the property possesses the property possesses the property

,,,, (6.6)

for allfor allfor allfor all .... Let Let Let Let .... Then the assertions of Theorem Then the assertions of Theorem Then the assertions of Theorem Then the assertions of Theorem 6.1are valid for any are valid for any are valid for any are valid for any .... Moreover, if Moreover, if Moreover, if Moreover, if for some for some for some for some

, then the trajectories , then the trajectories , then the trajectories , then the trajectories lying in the global attractor lying in the global attractor lying in the global attractor lying in the global attractor

of the system of the system of the system of the system generated by problem generated by problem generated by problem generated by problem (0.1) andandandand (0.2) for for for for pos- pos- pos- pos-

sess the propertysess the propertysess the propertysess the property

(6.7)

for allfor allfor allfor all ,,,, ,,,, where where where where is a constant depending onis a constant depending onis a constant depending onis a constant depending on

the problem parameters onlythe problem parameters onlythe problem parameters onlythe problem parameters only....

Proof.

It is evident that conditions (5.3) and (5.10) follow from (6.6). Therefore, wecan apply Theorem 6.1 which guarantees the existence of a global attractor . Letus assume that , . Let be a trajectory in

, . We consider a function , where is the ortho-projector onto the span of the basis vectors in for large enough.It is clear that and satisfies the equation

. (6.8)

Equation (6.6) for implies that is a continuously differentiablefunction. It is also evident that . Therefore, we differenti-ate equation (6.8) with respect to to obtain the equation

A M L ut

r� 0�* + ut7� u t 7�� 7 �

r� 0�* +� r q .� L2 r� 0 �1 1�)�� H

A1q v� 2c0 A1 1� v 7� 2 7d

r�

0

�(

c0 1 0 10�* +� 10

L

A7Lu C A7 1 2�� u( u D A� �

7 1 2�� 1 2�� *� p �1 2��7 0 1 2�� M z� Cl 1� R��

l 1� y u t� u� t� ��

� St

�� 0�

u k 1�� t� 2Au k� t� 2

A2 u k 1�� t� 2� � Rl2(

�� t �� k 1 2 ' l� � � Rl

�

M z� Cl 1� R�� l 1� y t� u t� u� t� �� t �� u

mt� p

mu t� p

m

e1 ' em

� �% & �0 m

um

t� C2 R �2)� �

u��m �u�m A2 um

M A1 2� u t� 2

� �� Au

mp

mL u� � � � p

mp

7 1 2�� pm L u

M A1 2� u t� 2� C1 R� �t


for the function . Here

.

Since any trajectory lying in the attractor possesses the property

, , (6.9)

it is clear that

, . (6.10)

Relation (6.9) also implies that the function

(6.11)

possesses the property

, .

Therefore, as in the proof of Theorem 2.2, we find that there exists such that

(6.12)

for all real , where , is the orthoprojector onto

,

, and is the evolutionary operator of the problem

with of the form (6.10). Moreover, can be presentedin the form

. (6.13)

Then for we have

w��m �w� m A2wm

M A1 2� u t� 2

� �� Aw

m� � � p

mLu� t� � F

mt� �

wm t� u�m t� pmu� t�

Fm t� 2� M = A1 2� u t� 2

� �� A u t� u� t� �� A um t�

y u t� u� t� ��

u� t� 2Au t� 2� R0

2( �� t ��

Fm

t� CR0

( �� t ��

b t� M A1 2� u t� 2� ��

b t� b�

t� � CR0

( �� t ��

N0

1 PN

�� U t .�� y�

S

C 1 PN

�� y�

S

e

�4�� t .��

(

s �s �1 s� �s8 PN

LN

Lin ek

0�� 0 ek

�� ) : k 1 2 ' N� � �% &

N N0� U t .��

u�� u� A2 u b t� Au� � � 0 ,

ut 0 u0 , u�

t 0 u1 ,�!#

b t� zm t� wm t� w� m t� ��

zm

t� U t t0�� zm

t0� U t .�� 0 pm

Lu� .� � Fm

.� �)� .d

t0

t

��

m N N0� �

1 PN�� zm t� � 1 2��

C e

�4�� t t0��

zm t0� � 1 2��

C e

�4�� t .��

pm

Lu� .� Fm

.� � 1 2�� . .d

t0

t

�

�

�

(


4

C

h

a

p

t

e

r

Therefore, we use (6.9), (6.10), and (6.6) for to obtain that

.

We tend in this inequality to find that

,

where does not depend on . It further follows from (6.9) and equation (0.1)that

,

where the constant can depend on . Hence,

.

We tend to find that any trajectory lying in the attrac-tor possesses the property

, .

By virtue of (6.6) we have

, .

Therefore, we reason as above to find that equation (6.13) implies

.

Similarly we get

for all . Consequently, using equation (0.1) we obtain estimate (6.7) for. In order to prove (6.7) for the other values of we should use induction with

respect to and similar arguments. We offer the reader to make an independent de-tailed study as an exercise.

In addition to the hypotheses of Theorem 6.3 we assume that and . Prove that the global attractor of

the system lies in .

7 1 2��

1 PN

�� zm

t� � 1 2��

CR01 m

e

�4�� t t0��

CR0

e

�4�� t .��

.d

t0

t

��(

t0 ��-

1 PN�� zm t� � 1 2��t R�

sup C(

C 0� m

PN

zm

t� � 1 2��t R�

sup C(

C N

pm A1 2� u� t� 2

A 1� 2� pm u�� t� 2

��

t R�sup C(

m �- y t� u t� u� t� ��

u� t� 1 2� A1 2� u� t� CR0

( �� t ��

Lu� t� CR0

( �� t ��

1 PN�� zm t� �

CR0 m e

�4�� t t0��

CR0�(

u�� t� 2Au� t� 2� R1

2(

t �� k 1 k

k

E x e r c i s e 6.2

L 0� � �l� D Al� � �

� St�� l 1� �l8

C o n d i t i o n s o f R e g u l a r i t y o f A t t r a c t o r 261

§ 7 Conditions of Regularity of Attractor§ 7 Conditions of Regularity of Attractor§ 7 Conditions of Regularity of Attractor§ 7 Conditions of Regularity of Attractor

Unfortunately, the structure of the global attractor of problem (0.1) and (0.2) can bedescribed only under additional conditions that guarantee the existence of the Lya-punov function (see Section 1.6). These conditions require that and assumethe stationarity of the transverse load . For the Berger system (0.3) and (0.4)these hypotheses correspond to and , i.e. to the case of plateoscillations in a motionless stationary medium.

Thus, let us assume that the operator is identically equal to zero and in (0.1). Assume that the hypotheses of Theorem 3.1 hold. Then energy

equality (3.6) implies that

, (7.1)

where , the function is a weak solution to problem(0.1) and (0.2) with the initial conditions , and is the energyof the system defined by formula (3.7).

Let us prove that the functional with is a Lyapunov function (for definition see Section 1.6) of the dynamical system

. Indeed, it is evident that the functional is continuous on . By vir-tue of (7.1) it is monotonely increasing. If for some , then

.

Therefore, for , i.e. is a stationary solution to prob-lem (0.1) and (0.2). Hence, is a fixed point of the semigroup .

Therefore, Theorems 1.6.1 and 6.1 give us the following assertion.

Theorem 7.1

Assume that Assume that Assume that Assume that ,,,, ,,,, and and and and for some for some for some for some .... We also assume We also assume We also assume We also assume

that the function that the function that the function that the function satisfies conditions satisfies conditions satisfies conditions satisfies conditions (3.2) and and and and (5.2).... Then the global Then the global Then the global Then the global

attractor attractor attractor attractor of the dynamical system of the dynamical system of the dynamical system of the dynamical system generated by problem generated by problem generated by problem generated by problem (0.1)andandandand (0.2) has the form has the form has the form has the form

,,,, (7.2)

where where where where is the set of fixed points of the semigroup is the set of fixed points of the semigroup is the set of fixed points of the semigroup is the set of fixed points of the semigroup , i.e., i.e., i.e., i.e.

,,,, (7.3)

and is the unstable set emanating from and is the unstable set emanating from and is the unstable set emanating from and is the unstable set emanating from (for definition see Sec-

tion 1.6).

L 0�p t�

� 0 p x t�� p x� �

L

p t� p�

E y t2� � E y t1� � � �� u� .� 2 .d

t1

t2

� p u� .� �� 2 .d

t1

t2

��

y t� St y u t� u� t� �� u t� y u0 u1)� E y�

D y� E y� p u0�� y u0 u1)�

� St�� D y� �

E y t0� � E y� t0 0�

u� .� 2 .d

0

t0

� 0

u� .� 0 . 0 t0�* +� u t� uy u 0)� St

� 0� L 0� p �7� 7 0�M z�

� � St

��

� MMMM� �

St

u 0)� : u �1� A2 u� M A1 2� u2

� �� Au p�

� ! "# $

MMMM+ �


4

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h

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Let . Prove that if the hypotheses of Theorem 7.1 hold,then any fixed point of problem (0.1) and (0.2) either equals to ze-ro, , or has the form , where the constant

is the solution to the equation .

Assume that and . Then problem (0.1)and (0.2) has a unique fixed point for .If , then the number of fixed points is equal to and all of them have the form , ,where

, , .

Show that if the hypotheses of Exercise 7.2 hold, then theenergy of each fixed point has the form

, , ,

for .

Assume that the hypotheses of Theorem 7.1 hold. Show thatif the set

(7.4)

is not empty, then it is a closed positively invariant set of the dyna-mical system generated by weak solutions to problem (0.1)and (0.2).

Assume that the hypotheses of Theorem 7.1 hold and the set defined by equality (7.4) is not empty. Show that the dynamical

system possesses a compact global attractor ,where is the set of fixed points of satisfying the condition

.

Show that if the hypotheses of Theorem 7.1 hold, then theglobal minimal attractor (for definition see Section 1.3)of problem (0.1) and (0.2) coincides with the set of the fixedpoints (see (7.3)).

Further we prove that if the hypotheses of Theorem 7.1 hold, then the attractor of problem (0.1) and (0.2) is regular in generic case. As in Section 2.5, the corre-sponding arguments are based on the results obtained by A. V. Babin and M. I. Vishik(see also Section 1.6). These results prove that in generic case the number of fixedpoints is finite and all of them are hyperbolic.

E x e r c i s e 7.1 p 0�z

z 0 0)� z c ek

6 0)� c M c2�

k� �

k� 0

E x e r c i s e 7.2 p 0� M z� � z�z0 0 0)� �1(

�n �n 1�(� 2n 1�zk wk 0)� k 0 1F ' nF� � �

w0 0 wkF

�k

��k

��F ek

6 k 1 2 ' n� � �

E x e r c i s e 7.3

E zk

� zk

E z0� 0 E zkF� 1

4��

k�� 2 k 1 2 ' n� � �

�n �n 1�(�

E x e r c i s e 7.4

�c

y u0 u1)� �� : D y� E y� p u0�� c(�� ! "# $

� St

��

E x e r c i s e 7.5

�c

�c

St

�� c

MMMM� c

�

cS

t

D z� c(

E x e r c i s e 7.6

�min

�


Lemma 7.1.

Assume that conditions (3.2) and (5.2) are fulfilled. Then the problem

, , (7.5)

possesses a solution for any , where . If is a bounded set

in , then its preimage is bounded in . If

is a compact in , then is a compact in , i.e. the

mapping is proper.

Proof.

We follow the line of arguments given in the proof of Lemma 2.5.3. Let usconsider the continuous functional

(7.6)

on , where is a primitive of the function .Equation (3.2) implies that

(7.7)

Thus, the functional is bounded below. Let us consider it on the subspace, where is the orthoprojector onto as before. Since

as , there exists a minimum point on the subspace. This minimum point evidently satisfies the equation

. (7.8)


with the constants being independent of . Therefore, it follows from (7.8)that , provided . This estimate enables us to pass to thelimit in (7.8) and to prove that if , then equation (7.5) is solvable for any

. Equation (7.5) implies that

for ,

i.e. is bounded in if is bounded. In order to prove that themapping is proper we should reason as in the proof of Lemma 2.5.3. We givethe reader an opportunity to follow these reasonings individually, as an exercise.Lemma 7.1 is proved.

� u* + A2 u M A1 2� u2

� �� Au� p� u D A2 7��

p �7� 7 0�

�7 � 1� � �2 7� D A2 7��

�7 � 1� � D A2 7��

W u� 12�� Au Au�� A1 2� u

2� ��

� ! "# $

p u��

�1 D A� � z� M :� :d0

z

� M z�

W u� 12�� Au

2a A1 2� u

2� b�� A 1� p Au6��

14�� 1 a

�1��

� � Au2 b

2�� 1 a

�1��

� � 1�A 1� p

2 .��

W u� pm�1 pm Lin e1 ' em� �% &W u� ��- Au �- um

pm�1

A2 um

M A1 2� um

2� �� Au

m� p

mp

Aum

2c1 c2

W u� : u pm�1�

� ! "# $

inf c3 A 1� p2

� �(

m

A2 um CR( p R(7 0

p �7�

A2 7� um

CR

( p 7 R(

� 1� � D A2 7��

�


4

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Lemma 7.2

Let . Then the operator defined by the formula

(7.9)

with the domain is selfadjoint and

.

Proof.

It is clear that is a symmetric operator on . Moreover, it iseasy to verify that

, , (7.10)

i.e. is a relatively compact perturbation of the operator . Therefore, is selfadjoint. It is further evident that

.

However, due to (7.10) the operator is compact. Therefore,. Lemma 7.2 is proved.

Prove that for any the operator is boundedbelow and has a discrete spectrum, i.e. there exists an orthonormalbasis in such that

, , .

Assume that , where is a constant and is anelement of the basis of eigenfunctions of the operator . Showthat for all , where

.

Here for and for .

As in Section 2.5, Lemmata 7.1 and 7.2 enable us to use the Sard-Smale theorem(see, e.g., the book by A. V. Babin and M. I. Vishik [10]) and to state that the set

for all

of regular values of the operator is an open everywhere dense set in for.

Show that the set of solutions to equation (7.5) is finite for (Hint: see the proof of Lemma 2.5.5).

u �1� � = u* +

� = u* +w A2 w 2 M = A1 2� u2

� �� Au w�� Au M A1 2� u

2� �� Aw� �

D � = u* +� D A2� Ker� = u* +dim ��

� = u* + D A2�

� = u* +w A2 w� C u� Aw6( w D A2� �

� = W* + A2

� = u* +

Ker� = u* + Ker I A 2� � = u* + A2��

A 2� � = u* + A2�� Ker� = u* +dim ��

E x e r c i s e 7.7 u �1� � = u* +

fk% & �

� = u* + fk Gk fk k 1 2 '� � G1 G2 '( ( Gnn �-

lim �

E x e r c i s e 7.8 u c0 ek0 c0 ek0e

k% & A

� = u* + ek

Gk

ek

k 1 2 '� �

Gk

�k2 1 20

k k0c0

2M = c0

2�k0

� � M c02�

k0� �

k�

0k k0

1 k k0 0k k0

0 k k0H

�7 h �7� : � = u* +* + 1�I�!#

u � 1� h* +� "$

� �77 0�

E x e r c i s e 7.9

p �7�


Let us consider the linearization of problem (0.1) and (0.2) on a solution to problem (7.5):

(7.11)

Here is given by formula (7.9).

Prove that problem (7.11) has a unique weak solution on anysegment if , , and the function

possesses property (3.2).

Thus, problem (7.11) defines a strongly continuous linear evolutionary semigroup in the space by the formula

, (7.12)

where is a weak solution to problem (7.11).

Let be the orthonormal basis of eigenelements of theoperator and let be the corresponding eigenvalues. Theneach subspace

is invariant with respect to . The eigenvalues of the restrictionof the operator onto the subspace have the form

.

Lemma 7.3

Let . Assume that possesses property (3.2). Then

the evolutionary operator of problem (0.1) and (0.2) is Frechét dif-

ferentiable at each fixed point . Moreover, ,

where is defined by equality (7.12).

Proof.

Let

,

where , , and is a solution to equation (7.5).It is clear that , where is a weak so-lution to problem

(7.13)

u D A2� �

w�� w� � = u* +w� � 0 ,

wt 0 w0 w�

t 0 w1 .�

� = u* +


0 T�* + w0 �1� w1 �0� M z� �C2 R��

Tt

u* + � �1 �08

Tt

u* + w0 w1)� w t� w� t� )�

w t�

E x e r c i s e 7.11 fk% &� = u* + Gk

�k Lin fk 0)� 0 fk)� �% & ��

Tt

u* +T

tu* + �

k

�2��

�2

4�� G

k�F� �

� � t�� ! "# $

exp

L 0� M z� C2 R�� St

y u 0)� St= u* + Tt u* +

Tt u* +

z t� St

y h�* + y� Tt

u* +h�

h h0 h1)� �� y u 0)� u

z t� v t� v� t� )� v t� u t� u� w t� ��

v�� v A2 v� � F u t� u w t� � �� ,

vt 0 0 , v�

t 0 0 .�!#


4

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Here

where is a weak solution to problem (0.1) and (0.2) with the initial condi-tions and is a solution to problem (7.11) with

and . It is evident that

, (7.14)

where

.

It is also evident that the value can be estimated in the following way

for and for , where the constants and depend on ,, and . This implies that

. (7.15)

Let us rewrite the value in the form

Consequently, the estimate

(7.16)

holds for and for . Therefore, equations (7.14)–(7.16) giveus that

on any segment . Here and . We use conti-nuity property (3.20) of a solution to problem (0.1) and (0.2) with respect tothe initial conditions to obtain that

F u t� u w t� � �� M A1 2� u2

� �� Au M A1 2� u t� 2

� �� Au t� ��

�M A1 2� u2

� �� Aw t� 2 M = A1 2� u� �

� � Au w t� �� Au ,�

u t� y0 y h� u h0� h1)� � w t�

w0 h0 w1 h1

F u t� u w t� � �� M A1 2� u2

� �� Av F1 t� M = A1 2� u

2� �� F2 t� ��

F1 t� M A1 2� u t� 2� �� M A1 2� u

2� ��

�!#

�

M = A1 2� u2

� �� A1 2� u t� �

2A1 2� u

2��

"$

Au t� ,�

F2 t� A1 2� u t� 2A1 2� u

2�� Au t� 2 Au w t� �� Au�

F1 t�

F1 t� c1 M == z� : z 0 c2�* +�� ! "# $

max A1 2� u t� 2A1 2� u

2�2

6(

t 0 T�* +� h�

R( c1 c2 T

R u

F1 t� C T R u� �� A u t� u�� 2(

F2 t�

F2 t� u t� u� A u t� u�� A u t� u�� 6

A1 2� u t� u�� 2Au 2 u Av t� �� Au .

�

� �

F2 t� C1 T R u� �� A u t� u�� 2C2 u� Av t� �(

t 0 T�* +� h�

R(

F u t� u w t� � �� C1 Av t� C2 A u t� u�� 2�(

0 T�* + C1 C1 u� C2 C2 T R u� ��


, , .

Therefore,

for and for . Hence, the energy equality for the solutionsto problem (7.13) gives us that

.

Therefore, Gronwall’s lemma implies that

, .

This equation can be rewritten in the form

.


Use the arguments given in the proof of Lemma 7.3 to verifythat under condition (3.2) for the evolutionaryoperator of problem (0.1) and (0.2) in belongs to the class and

for any and .

Use the results of Exercises 7.7 and 7.11 to prove that fora regular value of the mapping the spectrum of the opera-tor does not intersect the unit circumference while the eigen-subspace which corresponds to the spectrum outside the unitdisk does not depend on and is finite-dimensional.

The results presented above enable us to prove the following assertion (see Chap-ter V of the book by A. V. Babin and M. I. Vishik [10]).

Theorem 7.2

Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem 7.1 hold. Then there exists an hold. Then there exists an hold. Then there exists an hold. Then there exists an

open dense set open dense set open dense set open dense set in in in in such that the dynamical system such that the dynamical system such that the dynamical system such that the dynamical system possesses possesses possesses possesses

a regular global attractor for every a regular global attractor for every a regular global attractor for every a regular global attractor for every , i. e., i. e., i. e., i. e.

,,,,

where is the unstable manifold of the evolutionary operator ema-where is the unstable manifold of the evolutionary operator ema-where is the unstable manifold of the evolutionary operator ema-where is the unstable manifold of the evolutionary operator ema-

nating from the fixed point . Moreover, each set is a finite-dimen-nating from the fixed point . Moreover, each set is a finite-dimen-nating from the fixed point . Moreover, each set is a finite-dimen-nating from the fixed point . Moreover, each set is a finite-dimen-

sional surface of the class sional surface of the class sional surface of the class sional surface of the class ....

A u t� u�� CT R� h�

( t 0 T�* +� h�

R(

F u t� u w t� � �� C1 Av t� C2 h�

2�(

t 0 T�* +� h�

R(

dtd

�� Av t� 2v� t� 2��

� � a Av t� 2v� t� 2��

� � C h4�(

Av t� 2v� t� 2� C h

4( t 0 T�* +�

St

y h�* + y� Tt

u* +h��

C h2(


M z� C2 R�� S

t� C1

St= y1* + S

t= y2* +�

� � �� C y1 y2�

�(

t 0� yj

��


p � u* +T

tu* +

E�t

�7 �7 � St

�� p �7�

� MMMM+ zj

� j 1

N

C

MMMM+ zj� St

zj MMMM+ zj� C1


4

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In the case of a zero transverse load Theorem 7.2 is not applicable in gene-ral. However, this case can be studied by using the structure of the problem. For exam-ple, we can guarantee finiteness of the set of fixed points if we assume (see Exer-cise 7.1) that the equation , first, is solvable with respect to only for a finite number of the eigenvalues and, second, possesses not more thana finite number of solutions for every . The solutions to equation (7.5) are either

, or , where and satisfy . The eigen-values of the operator have the form

if

and

if .

Therefore, the result of Exercise 7.11 implies that the fixed points are hyperbolicif all the numbers are nonzero, i.e. if

, ; , ;

for all and such that . In particular, if ,then for any real there exists a finite number of fixed points (see Exercise 7.2)and all of them are hyperbolic, provided that for all and the eigenvalues

satisfying the condition are simple. Moreover, we can prove that for the unstable manifold , , emanating

from the fixed point (see Exercise 7.2) possesses the property

, .

§ 8 On Singular Limit in the Problem§ 8 On Singular Limit in the Problem§ 8 On Singular Limit in the Problem§ 8 On Singular Limit in the Problem

of Oscillations of a Plateof Oscillations of a Plateof Oscillations of a Plateof Oscillations of a Plate

In this section we consider problem (0.1) and (0.2) in the following form:

, (8.1)

. (8.2)

Equation (8.1) differs from equation (0.1) in that the parameter is intro-duced. It stands for the mass density of the plate material. The introduction of a newtime transforms equation (8.1) into (0.1) with the medium resistance pa-rameter instead of . Therefore, all the above results mentioned aboveremain true for problem (8.1) and (8.2) as well.

p 0��

M c2�k

� �k

� 0 c

�k

k

u 0� u c0 ek0 c0 k0 M c0

2�k0

� �k0

� 0� = u* +

Gk

�k2

M 0� �k

� u 0�

Gk �k �k �k020k k0

�k0c0

2M = c0

2�k0� �� u c0 ek0

Gk

M 0� �k

�H k 1 2 '� � �k

�k0

H k k0H M c02�

k0� 0H

c0 k0 M c02 �k0

� �k0� 0 M z� z��

�kH k

�j �j ��n �n 1�� MMMM+ zk� k 0 1F ' nF� � �

zk

MMMM� z0� dim n MMMM� zk� dim k 1�

Gu�� u� A2 u M A1 2� u2

� �� Au Lu� � � � p t 0��

ut 0 u0 u�

t 0 u1�

G 0�

t = t G�� = � G� �

O n S i n g u l a r L i m i t i n t h e P r o b l e m o f O s c i l l a t i o n s o f a P l a t e 269

The main question discussed in this section is the asymptotic behaviour of thesolution to problem (8.1) and (8.2) for the case when the inertial forces are smallwith respect to the medium resistance forces . Formally, this assumptionleads to a quasistatic statement of problem (8.1) and (8.2):

, (8.3)

. (8.4)

Here we prove that the global attractor of problem (8.1) and (8.2) is close to the glo-bal attractor of the dynamical system generated by equations (8.3) and (8.4)in some sense.

Without loss of generality we further assume that . We also note thatproblem (8.3) and (8.4) belongs to the class of equations considered in Chapter 2.

Assume that conditions (3.2) and (3.3) are fulfilled and. Show that problem (8.3) and (8.4) has a unique mild

(in ) solution on any segment , i.e. there existsa unique function such that

(Hint: see Theorem 2.2.4 and Exercise 2.2.10).

Let us consider the Galerkin approximations of problem (8.3) and (8.4):

, (8.5)

, (8.6)

where is the orthoprojector onto the first eigenvectors of the operator and.

Assume that conditions (3.2) and (3.3) are fulfilled and. Then problem (8.5) and (8.6) is solvable on any seg-

ment and

, . (8.7)

Theorem 8.1

Let Let Let Let and assume that conditions and assume that conditions and assume that conditions and assume that conditions (3.2),,,, (5.2),,,, and and and and (5.3) are ful- are ful- are ful- are ful-

filled. Then the dynamical system filled. Then the dynamical system filled. Then the dynamical system filled. Then the dynamical system generated by weak solutions to generated by weak solutions to generated by weak solutions to generated by weak solutions to

problem problem problem problem (8.3) andandandand (8.4) possesses a compact connected global attractor possesses a compact connected global attractor possesses a compact connected global attractor possesses a compact connected global attractor ....

This attractor is a bounded set in This attractor is a bounded set in This attractor is a bounded set in This attractor is a bounded set in for for for for and has a finite frac- and has a finite frac- and has a finite frac- and has a finite frac-

tal dimensiontal dimensiontal dimensiontal dimension....

G �J

�u� A2 u M A1 2� u2

� �� Au L u� � � p t 0��

ut 0 u0

� 1

E x e r c i s e 8.1

p �0� H�1 D A� 0 T�* +

u t� C 0 T� �1)� �

u t� e A2 t� u0 �

e A2 t .�� M A1 2� u .� 2� �� Au .� Lu .� p��

� ! "# $

. .d

0

t

��

u�m t� A2 um

t� M A1 2� um

t� 2� �� Au

mp

mL u

mt� � � � p

mp

um

0� pm

u0

pm

m A

um

t� Lin e1 ' em

� �% &�

E x e r c i s e 8.2

p �0� H0 T�* +

A u t� um

t� �� 0 T�* +max 0- m �-

p H��1 S

t��

�

�1 2� 0 2 1�(


4

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Proof.

First we prove that the system is dissipative. To do that we considerthe Galerkin approximations (8.5) and (8.6). We multiply (8.5) by scalarwiseand find that

Using equation (5.2) we obtain that

We use equation (5.3) and reason in the same way as in the proof of Theorem 5.1 tofind that

(8.8)

with some positive constants , . Multiplying equation (8.5) by we obtain that

,

where

.

It follows that

. (8.9)

If we summarize (8.8) and (8.9), then it is easy to find that

.

This implies that

.

We use (8.7) to pass to the limit as and to obtain that

.

This implies the dissipativity of the dynamical system generated by prob-lem (8.3) and (8.4). In order to complete the proof of the theorem we use Theorem2.4.1.

�1 St�� um t�

12��

dtd

�� um t� 2Aum t� 2

M A1 2� um t� 2� �� A1 2� um t� 2 +

Lum t� um t� ��

� �

� p um t� �� .

12��

dtd

�� um t� 2Aum t� 2

a1� A1 2� um t� 2� �� (

a3( a2 A1 2� um t� 2 21�Lum t� um t� �� p um t� �� .�

12��

dtd

�� um2

b0 Aum2

� A1 2� um2

� ��

� � b1 A1 2� um2 21�� b2(

bj

j 0 1 2� � u�m t�

12��

dtd

�� K um

t� � u�m t� 2Lu

mt� u�m t� �� p u�m t� ��

K u� Au2

� A1 2� u2

� ��

dtd

�� K um t� � u�m t� 2� 2 p2

C A@um t� 2�(

dtd

�� um

2 K um

� �� ! "# $

b0 um

2 K um

� �� ! "# $

� C(

um t� 2 K um t� � � pmu02 K pmu0� �

� ! "# $

eb0 t�

C�(

m �-

u t� 2 K u t� � � u02 K u0� �

� ! "# $

eb0 t�

C�(

�1 St

��


We note that the dissipativity also implies that the dynamical system pos-sesses a fractal exponential attractor (see Theorem 2.4.2).

Assume that the hypotheses of Theorem 8.1 hold and .Show that for generic the attractor of the dynamical system

generated by equations (8.3) and (8.4) is regular (see thedefinition in the statement of Theorem 7.2). Hint: see Section 2.5.

We assume that and conditions (3.2), (5.2), (5.3), and (5.10) arefulfilled. Let us consider the dynamical system generated by problem(8.1) and (8.2) in the space . Lemma 5.2 and Exer-cise 5.5 imply that possesses a compact global attractor for any

.The main result of this section is the following assertion on the closeness of

attractors of problem (8.1) and (8.2) and problem (8.3) and (8.4) for small .

Theorem 8.2

Assume that Assume that Assume that Assume that and conditions and conditions and conditions and conditions (3.2),,,, (5.2),,,, (5.3),,,, andandandand

(5.10) concerning concerning concerning concerning ,,,, ,,,, and are fulfilled and are fulfilled and are fulfilled and are fulfilled.... Then the equationThen the equationThen the equationThen the equation

(8.10)

is valid, where is valid, where is valid, where is valid, where is a global attractor of the dynamical system is a global attractor of the dynamical system is a global attractor of the dynamical system is a global attractor of the dynamical system

generated by problem generated by problem generated by problem generated by problem (8.1) andandandand (8.2),,,,

....

Here Here Here Here is a global attractor of problem is a global attractor of problem is a global attractor of problem is a global attractor of problem (8.3) andandandand (8.4) in in in in and and and and

is the distance between the element is the distance between the element is the distance between the element is the distance between the element and the set and the set and the set and the set in the in the in the in the

space space space space . We remind that . We remind that . We remind that . We remind that in equations in equations in equations in equations (8.1) and and and and (8.3)....

The proof of the theorem is based on the following lemmata.

Lemma 8.1

The dynamical system is uniformly dissipative in with

respect to for some , i.e. there exists and

such that for any set which is bounded in we have

(8.11)

for all , .

�1 St

��

E x e r c i s e 8.3 L 0�p H�

�1 St

��

M z� C2 R�� 1 St

G�� 1 �2 �18 D A2� D A� 8�

�1 StG�� G

G 0�

G 0�

M z� C2 R�� M z� L p

y �*�� dist : y �G�% &supG 0-lim 0

�G �1 StG��

�* z0 z1)� : z0 �� , z1 A2 z0� M A1 2� z02

� �� Az0� Lz0 p��

� ! "# $

� �1y A��

�dist y A

� �1 �08 � 1

�1 StG��

G 0 G0� +�� G0 0� G0 0�R 0� B �1� �

StG

B y u0 u1)� : G u12

Au02� R2(

� ! "# $

�

t t B G�� G 0 G0� +��


4

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Proof.

We use the arguments from the proof of Theorem 5.1 slightly modifyingthem. Let

, ,

where

and

.

As in the proof of Theorem 5.1 it is easy to find that the inequalities

(8.12)

and

(8.13)

are valid for . Here is an arbitrary number,the constants and do not depend on . Moreover, it is also evident that

(8.14)

for and for any . Here and do not depend on and . Equations (8.12)–(8.14) lead us to the inequality

where the constant does not depend on . If we choose smallenough, then we can take and independent of andsuch that

, (8.15)

where does not depend on . Moreover, we can assume(due to the choice of ) that

, (8.16)

where and do not depend on . Using equations (8.14)–(8.16)we obtain the assertion of Lemma 8.1.

V y� E y� A B y� � y u0 u1)�

E y� 12�� G u1

2Au0

2� A1 2� u0

2� ��

� �

B y� G u0 u1�� 12�� u0

2�

dtd

�� E y t� � 12�� u

2� /2�� Au

2C/ A1 2� u

2C1� � �(

dtd

��B y t� � G u�2 1

2�� Au

2� a1� A1 2� u2

� ��

a2 A1 2� u2 21�

C2 u2

C3� ��

�(

y t� u t� u� t� )� StG

y0 / 0�aj cj G

V y� 12��20 G u1

2Au0

2� A1 2� u0

2� ��

� � 21�(

G 0 G0� +�� G0 20 21 G �0 G0� +�� A 1(

dtd

��V y t� � 0V y t� � � 12�� 1 020 A�� G�� u�

2� �(

12�� A 020� /�� Au

2� Aa1020

2��

� � � A1 2� u2

� �� ,��

� G 0 G0� +�� G00 0� A 0� G 0 G0� +��

dtd

��V y t� � 0V y t� � � �1(

�1 0� G 0 G0� +��G0

V y t� � 23 G u� t� 2Au t� 2��

� � C��

23 C G 0 G0� +��


Lemma 8.2

Let be a solution to problem (8.1) and (8.2) such that

for all . Then the estimate

is valid for , where is a positive constant such that .

Proof.

It is evident that the estimate

holds, provided . Therefore, equaiton (8.1) easily implies the esti-mate

for the solution . We multiply this inequality by . Then by virtueof the fact that we have

for . We integrate this equation from to to obtain the assertionof the lemma.

Lemma 8.3

Let be a solution to problem (8.1) and (8.2) with the initial condi-

tions and such that for . Then

the estimate

(8.17)

is valid for the function . Here , is small

enough, , and the numbers and do not depend on

.

Proof.

Let us consider the function

for . It is clear that

u t� Au t� R(t 0�

12�� u� .� 2

e 2� t .�� .d

0

t

� G u� 0� 2Au 0� 2��

� � e 2� t C R 2�� (

t 0� 2 2G 1 2�(

M A1 2� u t� 2� �� Au t� L u t� p�� CR(

Au t� R(

12��

dtd

�� G u� t� 2Au t� 2��

� � 12�� u� t� 2� C

R(

u t� 2 2 t� expAu t� R(

dtd

�� e2 t G u� t� 2Au t� 2��

� � 12�� u� t� 2

e2 t� CR 2� e2 t(

G2 1 2�( 0 t

u t� u0 u1)� �1 �2 �18� Au t� R( t 0�

G w� t� 2Aw t� 2� C1 1 w� 0� 2� Aw 0� 2��

� � e20� t

C2�(

w t� u� t� G 0 G0� +�� G020 0� C1 C2

G 0 G0� +��

W t� 12�� G w� t� 2

Aw t� 2�� A G w� w�� 1

2�� w

2��

A 0�


4

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h

a

p

t

e

r(8.18)

Since the function is a weak solution to the equation obtained by the dif-ferentiation of (8.1) with respect to (cf. (4.15)):

,

where

,

then we have that

It follows that

.

We take and choose small enough to obtain with the helpof (8.18) that

, , ,

where the constants and do not depend on . Consequently,

.

Therefore, estimate (8.17) follows from equation (8.18) and Lemma 8.2. Thus,Lemma 8.3 is proved.

Lemma 8.3 and equations (8.11) imply the existence of a constant such that forany bounded set in there exists such that

, , (8.19)

where is a solution to problem (8.1) and (8.2) with the initial conditions from. However, due to (8.1) equations (8.11) and (8.19) imply that for

. Thus, there exists such that

12��G 1 AG�� w�

2 12�� Aw

2� W t�

12�� G 1 AG�� w�

2 12�� A�1

2�� Aw

2 .�

( (

(

w t� t

Gw�� t� w� t� A2 w t� M A1 2� u t� 2� �� Aw t� L w t� � � � � F t�

F t� 2� M = A1 2� u t� 2� �� Au t� w t� �� Au t�

dtd

��W t� 1 AG�� w� 2� M A1 2� u t� 2� A w w

�� L w w�� F w

��

A A2 w 2 M A1 2� u t� 2� A1 2� w 2 L w w�� F w�� ! "# $

.�

��

dtd

��W t� 12�� AG�� w�

2� 12�� A C

R

1� �� Aw2� AC

R

2� w

2�(

A CR

1� 2� G0

dtd

��W t� 20W t� � C w t� 2( t 0� G 0 G0� +��

20 0� C G

W t� W 0� e20� t

C u� .� 2e

20� t .�� .d

0

t

��(

R1B �1 t0 t0 B G��

G u�� t� 2Au� t� 2� R1

2( G 0 G0��

u t� B A2u t� C(t t0 B G�� R2 0�


, , (8.20)

where is a solution to system (8.1) and (8.2) with the initial conditions from thebounded set in , does not depend on , and is smallenough. Equation (8.20) and the invariance property of the attractor imply theestimate

(8.21)

for any trajectory lying in for all .Let us prove (8.10). It is evident that there exists an element

from such that

.

Let be a trajectory of system (8.1) and (8.2) lying in the at-tractor and such that . Equation (8.21) implies that there exista subsequence and an element such that for any segment the sequence converges to in the

weak topology of the space as . Equation (8.21) gives usthat the subsequence is uniformly continuous and uniformly boundedin . Therefore (cf. Exercise 1.14),

(8.22)

for any . However, it follows from (8.21) that as . There-fore, we pass to the limit in equation (8.1) and obtain that the function is a bounded (on the whole axis) solution to problem (8.3) and (8.4). Hence, it lies inthe attractor of the system . With the help of (8.21) and (8.22) it is easyto find that

, ,

where

.


G u�� t� 2Au� t� 2 A2u t� 2� � R2

2( t t0 B G��

u t� B �1 R2 G 0 G0�� G0

�G

G u��G t� 2Au� G t� 2

A2uG t� 2� � R22(

StG

y0 uG t� u� G t� )� �G t �� yG u0G u1G)�

�G

d yG� yG �*��

dist� y �*�� dist : y �G�% &sup

yG t� uG t� u� G t� )� �G yG 0� yG

yGnt� % & y t� u t� u� t� )� L� �� 1)� �

a b�* + yGnt� y t�

* - L� a b� �1)� Gn 0-AuGn

t� % &H

A uGn

t� u t� �� t a b�* +�

maxG

n0-

lim 0

a b� G u��G t� 0- G 0-G 0- u t�

� �1 St��

d yGn

� yGn

y0��

( 0- Gn

0-

y0 u 0� Au 0� � M A1 2� u 0� 2� �� Au0� L u0� p�)� �

� � �*�


4

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§ 9 On Inertial and § 9 On Inertial and § 9 On Inertial and § 9 On Inertial and ApproximateApproximateApproximateApproximate

Inertial ManifoldsInertial ManifoldsInertial ManifoldsInertial Manifolds

The considerations of this section are based on the results presented in Sections 3.7,3.8, and 3.9. For the sake of simplicity we further assume that .

Theorem 9.1

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (3.2), , , , (5.2), and , and , and , and (5.3) are fulfilled. We also as- are fulfilled. We also as- are fulfilled. We also as- are fulfilled. We also as-

sume that eigenvalues of the operator sume that eigenvalues of the operator sume that eigenvalues of the operator sume that eigenvalues of the operator possess the properties possess the properties possess the properties possess the properties

andandandand ,,,, ,,,, ,,,, (9.1)

for some sequence for some sequence for some sequence for some sequence .... Then there exist numbers Then there exist numbers Then there exist numbers Then there exist numbers and and and and

such that the conditionssuch that the conditionssuch that the conditionssuch that the conditions

andandandand (9.2)

imply that the dynamical system imply that the dynamical system imply that the dynamical system imply that the dynamical system generated by problem generated by problem generated by problem generated by problem (0.1) andandandand

(0.2) possesses a possesses a possesses a possesses a local inertial manifold, i.e. there exists a finite-dimen-local inertial manifold, i.e. there exists a finite-dimen-local inertial manifold, i.e. there exists a finite-dimen-local inertial manifold, i.e. there exists a finite-dimen-

sional manifold sional manifold sional manifold sional manifold in in in in of the form of the form of the form of the form

,,,, (9.3)

where where where where is a Lipschitzian mapping from is a Lipschitzian mapping from is a Lipschitzian mapping from is a Lipschitzian mapping from into into into into and and and and is a is a is a is a

finite-dimensional projector in finite-dimensional projector in finite-dimensional projector in finite-dimensional projector in .... This manifold possesses the properties: This manifold possesses the properties: This manifold possesses the properties: This manifold possesses the properties:

1) for any bounded set for any bounded set for any bounded set for any bounded set in in in in and for and for and for and for

;;;; (9.4)

2) there exists there exists there exists there exists such that the conditions such that the conditions such that the conditions such that the conditions and and and and for for for for

imply that imply that imply that imply that for for for for ;;;;

3) if the global attractor of the system if the global attractor of the system if the global attractor of the system if the global attractor of the system exists, then the set exists, then the set exists, then the set exists, then the set

contains it contains it contains it contains it (see Theorem 6.1)....

Proof.

Conditions (3.2), (5.2), and (5.3) imply (see Theorem 5.1) that the dynamicalsystem is dissipative, i.e. there exists such that

, , (9.5)

for any bounded set . This enables us to use the dynamical system generated by an equation of the type

(9.6)

to describe the asymptotic behaviour of solutions to problem (0.1) and (0.2). Here

p t� g� H�

A

�N

�N 1�

��N

inf 0� �N k� 1� c0 k

� 1 o 1� �� 0� k �-

N k� % & �- �0 0� k0 0�

� �0� �N k� 1�2 �

N k� 2� k0�N k� 1��

� St

��

� � �1 �08� y w B w� � : w P�� , B w� 1 P�� % &

B .� P� 1 P�� P

�

B � t t0 B� �

St y �� dist : y �% &sup C 2 t t0 BBBB� �� % &exp(

R 0� y �� Sty

�R(

t 0 t0�* +� Sty �� t 0 t0�* +�

� St

��

� St�� R 0�S

ty

�R( y B� t t0 B� �

B �� S�

t��

u�� u� A2 u� � BR

u� ,

ut 0 u0 , u�

t 0 u1 ,�!#

BR

u� , 2R� 1�Au� �

� � g M A1 2� u2

� �� Au� Lu�

� ! "# $6

O n I n e r t i a l a n d A p p r o x i m a t e I n e r t i a l M a n i f o l d s 277

and is an infinitely differentiable function on possessing the properties

; ;

, ; , .

It is easy to find that there exists a constant such that

and

.

Therefore, we can apply Theorem 3.7.2 to the dynamical system generatedby equation (9.6). This theorem guarantees the existence of an inertial manifold ofthe system if the hypotheses of Theorem 9.1 hold. However, inside the dissi-pativity ball problem (9.6) coincides with problem (0.1) and (0.2).This easily implies the assertion of Theorem 9.1.

Show that the hypotheses of Theorem 9.1 hold for the prob-lem on oscillations of an infinite panel in a supersonic flow of gas:

Here and are real parameters and .

It is evident that the most essential assumption of Theorem 9.1 that restricts its ap-plication is condition (9.2). In this connection the following assertion concerning thecase when problem (0.1) and (0.2) possesses a regular attractor is of some interest.

Theorem 9.2

Assume that in equation Assume that in equation Assume that in equation Assume that in equation (0.1) we have we have we have we have and and and and

for some for some for some for some .... We also assume that conditions We also assume that conditions We also assume that conditions We also assume that conditions (3.2) and and and and

(5.2) are fulfilled. Then there exists are fulfilled. Then there exists are fulfilled. Then there exists are fulfilled. Then there exists such that for all such that for all such that for all such that for all the the the the

subspacesubspacesubspacesubspace

(9.7)

is an invariant and exponentially attracting set of the dynamical systemis an invariant and exponentially attracting set of the dynamical systemis an invariant and exponentially attracting set of the dynamical systemis an invariant and exponentially attracting set of the dynamical system

generated by problem generated by problem generated by problem generated by problem (0.1) andandandand (0.2)::::

,,,, (9.8)

for for for for and for any bounded set in . Here is the orthoprojectorand for any bounded set in . Here is the orthoprojectorand for any bounded set in . Here is the orthoprojectorand for any bounded set in . Here is the orthoprojector

onto onto onto onto ....

, s� R�

0 , s� 1( ( ,= s� 2(

, s� 1 0 s 1( ( , s� 0 s 2�

CR

BR

u� CR

(

BR

u1� BR

u2� � CR

A u1 u2�� (

� S�

t��

� S�

t�� y : y

�R(% &

E x e r c i s e 9.1

�t2u ��

tu �

x4u �

xu x t�� 2

xd

0

3

��

�x2u ��

xu� � � � g x� , x 0 3�� t 0 ,��

ux 0 x 3� �

x2u

x 0 x 3� 0 , ut 0 u0 x� � , �

tu

t 0u1 x� � .�

44!44#

� g x� L2 0 3��

L 0� p t� g� �Lin e1 ' eN0

� �% &� N0N1 N0� N N1�

�N Lin ek 0)� 0 ek)� � : k 1 2 ' N� � �% &

� St

��

Sty �

N�� dist C

B1 P

N�� y

�e

�4�� t t0 B� ��

( y B�

t t0 B� � B � PN

�N


4

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h

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p

t

e

r

Proof.

Since for , where is the orthoprojector onto the span of, the uniqueness theorem implies the invariance of . Let us prove

attraction property (9.8). It is sufficient to consider a trajectory lying inthe ball of dissipativity . Evidently the function satisfies the equation

(9.9)

It is also clear that the conditions

and

hold in the ball of dissipativity. This fact enables us to use Theorem 2.2 with . In particular, equation (2.23) guarantees the existence of a num-

ber which depends on , , and and such that

, ,

for all , where is the orthoprojector onto and .This implies estimate (9.8). Theorem 9.2 is proved.

Assume that the hypotheses of Theorem 9.2 hold. Show thatfor any semitrajectory there exists an induced trajectory in

, i.e. there exists such that

for and for some .

Write down an inertial form of problem (0.1) and (0.2) in thesubspace , provided the hypotheses of Theorem 9.2 hold. Provethat the inertial form coincides with the Galerkin approximation ofthe order of problem (0.1) and (0.2).

Show that if the hypotheses of Theorem 9.2 hold, then theglobal attractor of problem (0.1) and (0.2) coincides with the globalattractor of its Galerkin approximation of a sufficiently large order.

Let us now turn to the question on the construction of approximate inertial mani-folds for problem (0.1) and (0.2). In this case we can use the results of Section 3.8and the theorems on the regularity proved in Sections 4 and 5.

pN g g N N0� pN

e1 ' eN0� �% & �N

u t� u� t� �� y : y

�R(% & v t� 1 pN�� u t�

v�� v� A2v M A1 2� u t� 2� �� Av� � � 0 ,

vt 0 1 p

N�� u0 , v�

t 0 1 pN

�� u1 .�4!4#

M A1 2� u t� 2� �� b0 R� ( d

td��M A1 2� u t� 2

� �� b1 R� (

b t� M A1 2� u t� 2�

N1 N0� � b0 R� b1 R�

y t� �

1 PN

�� y t� �

3 1 PN

�� y 0� e

�4�� t�

( t 0�

N N1� PN

�N

y t� v t� v� t� ��

E x e r c i s e 9.2

Sty

�N

y �N

�

Sty S

ty�

�C

Re

�4�� t t0��

(

t t0� t0 t0 y�

�

E x e r c i s e 9.3

�N

N

E x e r c i s e 9.4

O n I n e r t i a l a n d A p p r o x i m a t e I n e r t i a l M a n i f o l d s 279

Assume that , and

,

where and . Show that the mapping has the Frechét derivatives up to the order inclusive. Moreo-ver, the estimates

(9.10)

and

(9.11)

are valid, where , , , and. Here is the value of the

Frechét derivative on the elements .

We consider equations (9.10) and (9.11) as well as Theorem 5.3 which guaranteesnonemptiness of the classes corresponding to the problem considered when

is large enough. They enable us to apply the results of Section 3.8.Let be the orthoprojector onto the span of elements in and

let . We define the sequences and of map-pings from into by the formulae

, (9.12)

(9.13)

(9.14)

Here , and are the Frechét de-rivatives with respect to the corresponding variables, , , where

is a stationary transverse load in (0.1), , the numbers are chosen to fulfil the inequality .

Evaluate the functions and .

E x e r c i s e 9.5 M z� Cm 1� R�� m 1�

B u� p M A1 2� u2

� �� Au� Lu�

p H� u � A� � �1 B .� B k� l

B k� u* + w1 ' wk

� �)L M CR

Awj

j 1

k

N(

B k� u* + B k� u** +� w1 ' wk

� �)L M

CR

A u u*�� Awj

j 1

k

N

(

(

k 0 1 ' m� � � Au R( Au* R(wj � A� � B k� u* + w1 ' wk� �)L M

w1 ' wk� �

Lm R�

R 0�P e1 ' e

N� �% & H

Q 1 P� hn

p p�� % &n 0�

ln

p p�� % &n 0�

PH PH8 QH

h0 p p�� l0 p p�� 0

A2 hk

p p�� a0 Mk 1� p p�� Ah

k 1�� QL p hk 1�� lA k� ��

0plk 1� p�)L M 0

p�lk 1� �p� A2p b0� M

k 1� p p�� Ap PL p hk 1�� )L M ,��

lk

p p�� 0p

hk 1� p�)L M �

0p�

hk 1� �p� A2 p b0� M

k 1� p p�� Ap PL p hk 1�� )L M .�

Mk

p p�� M A1 2� p2

A1 2� hk

p p�� 2�� 0

p0

p�

a0 Q g b0 Pgg p t� � k 1 2 ' m� � �A k� � k 1� A k� k( (

E x e r c i s e 9.6 h1 p p�� l1 p p��


4

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Theorem 3.8.2 implies the following assertion.

Theorem 9.3

Assume that Assume that Assume that Assume that , , , , , , , , , and conditions, and conditions, and conditions, and conditions

(3.2), , , , (5.2),,,, andandandand (5.3) are fulfilled. Then for all the collection are fulfilled. Then for all the collection are fulfilled. Then for all the collection are fulfilled. Then for all the collection of of of of

mappings mappings mappings mappings given by equalities given by equalities given by equalities given by equalities (9.12)––––(9.14) possesses the properties possesses the properties possesses the properties possesses the properties

1) there exist constants there exist constants there exist constants there exist constants and and and and , , , , such that such that such that such that

,,,, ,,,,

,,,,

,,,,

for all for all for all for all and and and and from from from from and such that and such that and such that and such that

,,,, , ;;;;

2) for any solution for any solution for any solution for any solution to problem to problem to problem to problem (0.1) andandandand (0.2) which satisfies com- which satisfies com- which satisfies com- which satisfies com-

patibility conditions patibility conditions patibility conditions patibility conditions (4.3) with with with with the estimate the estimate the estimate the estimate

is valid for is valid for is valid for is valid for and for and for and for and for large enough. Here large enough. Here large enough. Here large enough. Here

,,,,

,,,,

is the is the is the is the -th eigenvalue of the operator -th eigenvalue of the operator -th eigenvalue of the operator -th eigenvalue of the operator and the constant and the constant and the constant and the constant de- de- de- de-

pends on the radius of dissipativity.pends on the radius of dissipativity.pends on the radius of dissipativity.pends on the radius of dissipativity.

In particular, Theorem 9.3 means that the manifold

attracts sufficiently smooth trajectories of the dynamical system generatedby problem (0.1) and (0.2) into a small vicinity (of the order ) of .

Assume that the hypotheses of Theorem 6.3 hold (this theo-rem guarantees the existence of the global attractor consisting ofsmooth trajectories of problem (0.1) and (0.2)). Prove that

for all (the number is defined by the condition).

Prove the analogue of Theorem 3.9.1 on properties of the non-linear Galerkin method for problem (0.1) and (0.2).

p t� g� H� M z� Cm 1� R�� m 2�k 1 ' m� �

hn ln�� Mj Mj n �� Lj n �� j 1 2�

A2 hn

p0 p� 0�� M1( Aln

p0 p� 0�� M2(

A2 hn

p1 p� 1�� hn

p2 p� 2�� L1 A2 p1 p2�� A p� 1 p� 2�� (

A ln

p1 p� 1�� ln

p2 p� 2�� L2 A2 p1 p2�� A p� 1 p� 2�� (

pj p� j PH

A2pj

2Ap� j

2� �( j 0 1 2� � � 0�

u t� l m

A2 u t� un

t� �� 2A u t� u

nt� �� 2�

� ! "# $1 2�

Cn�

N 1�n�(

n m 1�( t

un

t� p t� hn

p t� p� t� ��

un

t� p� t� ln

p t� p� t� ��

�N

N A Cn

�n p hn p p�� p� ln p p�� )� : p p�� PH�% &

� St

�� C

n�

N 1�n�

�n

E x e r c i s e 9.7

�

y �n

�� dist : y ��% &sup Cn�

N 1�n�(

n m 1�( m

M z� Cm 1� R��

E x e r c i s e 9.8


1. BERGER M. A new approach to the large deflection of plate // J. Appl. Mech.–1955. –Vol. 22. –P. 465–472.

2. ZH. N. DMITRIEVA. On the theory of nonlinear oscillations of thin rectangular

plate // J. of Soviet Mathematics. –1978. –Vol. 22. –P. 48–51.3. BOLOTIN V. V. Nonconservative problems of elastic stability. –Oxford: Pergamon

Press, 1963.4. DOWELL E. H. Flutter of a buckled plate as an example of chaotic motion of de-

terministic autonomous system // J. Sound Vibr. –1982. –Vol. 85. –P. 333–344.5. YOSIDA K. Functional analysis. –Berlin: Springer, 1965.6. LADYZHENSKAYA O. A. Boundary-value problems of mathematical physics.

–Moscow: Nauka, 1973 (in Russian).7. MIKHAJLOV V. P. Differential equations in partial derivatives. –Moscow:

Nauka, 1983 (in Russian).8. LIONS J.-L. Quelques methodes de resolution des problemes aus limites

non linearies. –Paris: Dunod, 1969.9. GHIDAGLIA J.-M., TEMAM R. Regularity of the solutions of second order evolution

equations and their attractors // Ann. della Scuola Norm. Sup. Pisa. –1987. –Vol. 14. –P. 485–511.


11. CHUESHOV I. D. On a certain system of equations with delay, occurring

in aeroelasticity // J. of Soviet Mathematics. –1992. –Vol. 58. –P. 385–390.12. CHUESHOV I. D., REZOUNENKO A. V. Global attractors for a class of retarded qua-

silinear partial differential equations // Math. Phys., Anal. Geometry. –1995. –Vol. 2. –# 3/4. –P. 363–383.

13. BOUTET DE MONVEL L., CHUESHOV I. D., REZOUNENKO A. V. Long-time behavior of

strong solutions of retarded nonlinear P.D.E.s // Commun. Partial Dif. Eqs.–1997. –Vol. 22. –P. 1453–1474.

14. BOUTET DE MONVEL L., CHUESHOV I. D. Non-linear oscillations of a plate in

a flow of gas // C. R. Acad. Sci. –Paris. –1996. –Ser. I. –Vol. 322. –P. 1001–1006.

C h a p t e r 5

Theory of FunctionalsTheory of FunctionalsTheory of FunctionalsTheory of Functionals

that Uniquely Determine Long-Time Dynamicsthat Uniquely Determine Long-Time Dynamicsthat Uniquely Determine Long-Time Dynamicsthat Uniquely Determine Long-Time Dynamics

C o n t e n t s

. . . . § 1 Concept of a Set of Determining Functionals . . . . . . . . . . . 285

. . . . § 2 Completeness Defect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

. . . . § 3 Estimates of Completeness Defect in Sobolev Spaces . . . . 306

. . . . § 4 Determining Functionals for Abstract SemilinearParabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

. . . . § 5 Determining Functionals for Reaction-Diffusion Systems . 328

. . . . § 6 Determining Functionals in the Problemof Nerve Impulse Transmission . . . . . . . . . . . . . . . . . . . . . . 339

. . . . § 7 Determining Functionals for Second Orderin Time Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

. . . . § 8 On Boundary Determining Functionals . . . . . . . . . . . . . . . . 358

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

The results presented in previous chapters show that in many cases the asymp-totic behaviour of infinite-dimensional dissipative systems can be described by a fi-nite-dimensional global attractor. However, a detailed study of the structure ofattractor has been carried out only for a very limited number of problems. In this re-gard it is of importance to search for minimal (or close to minimal) sets of naturalparameters of the problem that uniquely determine the long-time behaviour of a sys-tem. This problem was first discussed by Foias and Prodi [1] and by Ladyzhenskaya[2] for the 2D Navier-Stokes equations. They have proved that the long-time beha-viour of solutions is completely determined by the dynamics of the first Fouriermodes if is sufficiently large. Later on, similar results have been obtained forother parameters and equations. The concepts of determining nodes and deter-mining local volume averages have been introduced. A general approach to the prob-lem on the existence of a finite number of determining parameters has beendiscussed (see survey [3]).

In this chapter we develop a general theory of determining functionals. Thistheory enables us, first, to cover all the results mentioned above from a unified pointof view and, second, to suggest rather simple conditions under which a set of func-tionals on the phase space uniquely determines the asymptotic behaviour of the sys-tem by its values on the trajectories. The approach presented here relies on theconcept of completeness defect of a set of functionals and involves some ideas andresults from the approximation theory of infinite-dimensional spaces.

§ 1 Concept of a Set of Determining§ 1 Concept of a Set of Determining§ 1 Concept of a Set of Determining§ 1 Concept of a Set of Determining

FunctionalsFunctionalsFunctionalsFunctionals

Let us consider a nonautonomous differential equation in a real reflexive Banachspace of the type

. (1.1)

Let be a class of solutions to (1.1) defined on the semiaxis suchthat for any there exists a point of time such that

, (1.2)

where is a reflexive Banach space which is continuously embedded into . Here-inafter is the space of strongly continuous functions on with thevalues in and has a similar meaning. The symbols and stand for the norms in the spaces and , .

N

N

H

udtd

�� F u t�� t 0 ut 0�� u0� �

� R+ t : t 0�� u t� � � t0 0�

u t� � C t0 +� H�� Lloc2

t0 +� V��

V H

C a b X�� a b�� X Lloc

2 a b X�� .H

.V

H V .H

.V

�

286 T h e o r y o f F u n c t i o n a l s t h a t U n i q u e l y D e t e r m i n e L o n g - T i m e D y n a m i c s

C

h

a

p

t

e

r

5

The following definition is based on the property established in [1] for the Fou-rier modes of solutions to the 2D Navier-Stokes system with periodic boundary con-ditions.

Let be a set of continuous linear functionals on .Then is said to be a set of asymptotically set of asymptotically set of asymptotically set of asymptotically -determining func--determining func--determining func--determining func-

tionals tionals tionals tionals (or elements) for problem (1.1) if for any two solutions the con-dition

for (1.3)

implies that

. (1.4)

Thus, if is a set of asymptotically determining functionals for problem (1.1), theasymptotic behaviour of a solution is completely determined by the behaviourof a finite number of scalar values . Further, if no ambi-guity results, we will sometimes omit the spaces , , and in the description ofdetermining functionals.

Show that condition (1.3) is equivalent to

,

where is a seminorm in defined by the equation

.

Let and be stationary (time-independent) solutions toproblem (1.1) lying in the class . Let be aset of asymptotically determining functionals. Show that condition

for all implies that .

The following theorem forms the basis for all assertions known to date on the exis-tence of finite sets of asymptotically determining functionals.

Theorem 1.1.

Let be a family of continuous linear functionalsLet be a family of continuous linear functionalsLet be a family of continuous linear functionalsLet be a family of continuous linear functionals

on . Son . Son . Son . Suuuuppose that there exists a continuous function on ppose that there exists a continuous function on ppose that there exists a continuous function on ppose that there exists a continuous function on

with the values in which possesses the following properties:with the values in which possesses the following properties:with the values in which possesses the following properties:with the values in which possesses the following properties:

a) there exist positive numbers and such thatthere exist positive numbers and such thatthere exist positive numbers and such thatthere exist positive numbers and such that

for allfor allfor allfor all ;;;; (1.5)

� lj: j 1 � N� �� V

� V H �� u v ��

lj

u �� lj

v �� 2 �d

t

t 1�

�t ��lim 0� j 1 � N� ��

u t� � v t� �� Ht ��lim 0�

�

u t� �lj

u t� �� : j 1 2 � N� � �� V H �

E x e r c i s e 1.1

��

u �� v ��

t

t 1�

�t ��lim

2 �d 0�

��

u� � H

��

u� � l u� �l �max�

E x e r c i s e 1.2 u1 u2� � l

j: j 1 � N� ��

lj

u1� � lj

u2� �� j 1 2 � N� � �� u1 u2�

� lj: j 1 � N� ��

V � u t�� H R+�R+

� �

� u t�� u �� u H t R+�

C o n c e p t o f a S e t o f D e t e r m i n i n g F u n c t i o n a l s 287

b) for any two solutions to problem for any two solutions to problem for any two solutions to problem for any two solutions to problem (1.1) there exist there exist there exist there exist (i)a point of time , a point of time , a point of time , a point of time , (ii) a function that is locally integrable a function that is locally integrable a function that is locally integrable a function that is locally integrable

over the half-interval and such thatover the half-interval and such thatover the half-interval and such thatover the half-interval and such that

(1.6)

andandandand

(1.7)

for some , and for some , and for some , and for some , and (iii) a positive constant such that for all a positive constant such that for all a positive constant such that for all a positive constant such that for all

we have we have we have we have

(1.8)

Then is a set of asymptotically -determining functionals forThen is a set of asymptotically -determining functionals forThen is a set of asymptotically -determining functionals forThen is a set of asymptotically -determining functionals for

problem problem problem problem (1.1)....

It is evident that the proof of this theorem follows from a version of Gronwall's lemmastated below.

Lemma 1.1.

Let and be two functions that are locally integrable over

some half-interval . Assume that (1.6) and (1.7) hold and

is nonnegative and possesses the property

, . (1.9)

Suppose that is a nonnegative continuous function satisfying the

inequality

(1.10)

for all . Then as .

It should be noted that this version of Gronwall's lemma has been used by many au-thors (see the references in the survey [3]).

u t� � v t� �� t0 0� � t� �

t0 ��

��+ � �� d

t

t a�

�t ��lim� 0�

��+ 0 � �� max �d

t

t a�

�t ��lim ��

a 0� C

t s t0� �

� u t� � v t� � t�� u �� v �� d

s

t

��

� u s� � v s� � s�� C lj

u �� lj

v �� 2

j 1 � N� ��max � .d

s

t

��

�

�

� V H ��

� t� � g t� �t0 �� g t� �

g �� d

t

t a�

�t ��lim 0� a 0�

w t� �

w t� � � �� w �� d

s

t

�� w s� � g �� d

s

t

��

t s t0� � w t� � 0� t ��


C

h

a

p

t

e

r

5

Proof.

Let us first show that equation (1.10) implies the inequality

(1.11)

for all . It follows from (1.10) that the function is absolutelycontinuous on any finite interval and therefore possesses a derivative al-most everywhere. Therewith, equation (1.10) gives us

(1.12)

for almost all . Multiplying this inequality by

,

we find that

almost everywhere. Integration gives us equation (1.11).Let us choose the value such that

, (1.13)

and

, (1.14)

for all . It is evident that if and , where is the in-teger part of a number, then

Thus, for all

w t� � w s� � � �� d

s

t

�� ! !" #! !$ %

g �� d

�

t

�� ! !" #! !$ %

exp �d

s

t

��exp�

t s t0� � w t� �w� t� �

w� t� � � t� �w t� �� g t� ��

t

e t� � � �� d

s

t

�� ! !" #! !$ %

exp�

dtd�� w t� � e t� �� g t� � e t� ��

s

� �� 0�� max � � 1��d

�

� a�

� � ��+

� �� 2��d

�

� a�

� � ��+

� s� t � s� � kt ��

a�� .� �

� �� d

�

t

� � �� d

�

ka ��

� � ��

�2�� k � �� 0�� max �d

ka ��

k 1�� a ��

��2�� k � 1�� .�

�

� �

d

ka ��

t

��

t � s� �


.

Consequently, equation (1.11) gives us that

,

where . Therefore,

. (1.15)

It is evident that

where is the integer part of the number . Therefore,

Since , this implies that

(1.16)

� �� d

�

t

� �2 a�� t �� 1 �

�2�� & '

( )��

w t� � C � �� w s� ��

2a�� t s��

� " #$ %

g r� ��

2a�� t ��

� " #$ %

exp �d

s

t

��exp�

C � �� 1 ��2��

� " #$ %

exp�

w t� �t ��lim C � �� g ��

s

t

�t ��lim

�2a�� t ��

� " #$ %

�dexp��

G t s�� g ��

2a�� t ��

� " #$ %

exp �d

s

t

�

g ��

2a�� t ��

� " #$ %

exp � ,d

s ka�

s k 1�� a�

�k 0�

N

*

�

�

Nt s�

a�� t s�

a��

G t s�� g �� d

�

� a�

�� s�sup

�2a�� t ��

� " #$ %

exp �d

s ka�

s k 1�� a�

�k 0�

N

*��

g ��

2a�� t ��

� " #$ %

exp �d

s

s N 1�� a�

�d

�

� a�

�� s�sup

2a�� g ��

�2a�� t s��

� " #$ %

exp�d

�

� a�

�� s�sup e

�2�� N 1��

1� .��

� �

�

Nt s�

a��

G t s�� t ��lim C a �� g �� d

�

� a�

�� s�sup��


C

h

a

p

t

e

r

5

for any such that equations (1.13) and (1.14) hold. Hence, equations (1.15)and (1.16) give us that

.

If we tend , then with the help of (1.9) we obtain

.

This implies the assertion of Lemma 1.1.

In cases when problem (1.1) is the Cauchy problem for a quasilinear partial differen-tial equation, we usually take some norm of the phase space as the function when we try to prove the existence of a finite set of asymptotically determining func-tionals. For example, the next assertion which follows from Theorem 1.1 is oftenused for parabolic problems.

Corollary 1.1.

Let and be reflexive Banach spaces such that is continuously

and densely embedded into . Assume that for any two solutions

to problem (1.1) we have

(1.17)

for , where is a constant and the function depends on

and in general and possesses properties (1.6) and (1.7).

Assume that the family on possesses the pro-

perty

(1.18)

for any , where and are positive constants depending

on . Then is a set of asymptotically determining functionals for

problem (1.1), provided

. (1.19)

s

w t� �t ��lim C � � a� �� g �� d

�

� a�

�� s�sup��

s ��

w t� �t ��lim 0�

� u t��

V H V

H

u1 t� � u2 t� ��

u1 t� � u2 t� ��V

2 � �� u1 �� u2 �� V

2 �d

s

t

��

u1 s� � u2 s� ��V

2K u1 u2�

H

2 �d

s

t

��

�

�

t s t0� � K � t� �u1 t� � u2 t� �

� lj : j 1 � N� �� V

vH

C lj

v� �j 1 � N�

max +�

vV

��

v V C +�

� �

+�

2 1K�� 1

a�� d

t

t a�

��t ��lim��

+a 1� K 1�


Proof.

Using the obvious inequality

, ,

we find from equation (1.18) that

(1.20)

for any . Therefore, equation (1.17) implies that

where . Consequently, if for some the function

possesses properties (1.6) and (1.7) with some constants and , thenTheorem 1.1 is applicable. A simple verification shows that it is sufficient to re-quire that equation (1.19) be fulfilled. Thus, Corollary 1.1 is proved.

Another variant of Corollary 1.1 useful for applications can be formulated as follows.

Corollary 1.2.


embedded into . Assume that for any two solutions to

problem (1.1) there exists a moment such that for all

the equation

(1.21)

holds. Here and the positive function is locally integrable

over the half-interval and satisfies the relation

(1.22)

a b�� 2 1 ,�� a2 1 1,��& '

( ) b2�� , 0�

v H2 1 ,�� +

�

2 v V2

C, lj v� � 2

j 1 � N�max��

, 0�

u1 t� � u2 t� ��V

2 � �� 1 ,�� K +�

2�& '( ) u1 �� u2 ��

V

2 �d

s

t

��

u1 s� � u2 s� ��V

2C �

�u1 �� u2 �� 2 � ,d

s

t

��

�

�

��

v� � lj

v� �j 1 � N�

max� , 0�

�� t� � � t� � 1 ,�� K+�

2��

� � 0�

V H V

H u t� � v t� � ��t0 0� t s t0� �

u t� � v t� ��H2 - u �� v ��

V2 �d

s

t

��

u s� � v s� ��H2 . �� u �� v ��

H2� �d

s

t

��

- 0� . t� �t0 ��

1a�� . �� d

t

t a�

�t ��lim R�


C

h

a

p

t

e

r

5

for some , where the constant is independent of and

. Let be a family of continuous linear func-

tionals on possessing the property

for any . Here and are positive constants. Then is a set

of asymptotically -determining functionals for problem (1.1),

provided that .

Proof.


for any . Therefore, (1.21) implies that

where . Using (1.22) and applying Theorem 1.1with , we complete the proof of Corollary 1.2.

Other approaches of introduction of the concept of determining functionals are alsopossible. The definition below is an extension to a more general situation of the pro-perty proved by O.A. Ladyzhenskaya [2] for trajectories lying in the global attractorof the 2D Navier-Stokes equations.

Let be a class of solutions to problem (1.1) on the real axis such that. A family of continuous linear

functionals on is said to be a set of set of set of set of -determining functionals -determining functionals -determining functionals -determining functionals (orelements) for problem (1.1) if for any two solutions the condition

for and almost all (1.23)

implies that .

It is easy to establish the following analogue of Theorem 1.1.

Theorem 1.2.

Let be a family of continuous linear functionalsLet be a family of continuous linear functionalsLet be a family of continuous linear functionalsLet be a family of continuous linear functionals

on on on on .... Let be a class of solutions to problem Let be a class of solutions to problem Let be a class of solutions to problem Let be a class of solutions to problem (1.1) on the real axis such on the real axis such on the real axis such on the real axis such

thatthatthatthat

a 0� R 0� u t� �v t� � � l

j: j 1 � N� ��

V

w H C�

lj w� �j 1 � N� ��

max +�

w V��

w V C�

+�

�

V H �� +�

- R/�

wV2 1 ,�� 1� +

�

2�w

H2� � C

� ,� lj

w� � 2

j 1 � N� ��max��

, 0�

u t� � v t� ��H2 � �� u �� v ��

H2� �d

s

t

��

u s� � v s� ��H2 -C

� ,� lj

u �� v �� 2

j 1 � N� ��max � ,d

s

t

��

�

�

� t� � - 1 ,�� 1� +�

2� . t� �� u t�� u

2�

� R� Lloc

2 � +� V�� 0 � lj: j 1 � N� ��

V V �� u v ��

lj u t� �� lj v t� �� j 1 � N� �� t R

u t� � v t� �

� lj : j 1 � N� �� V � R


.... (1.24)

Assume that there exists a continuous function on with theAssume that there exists a continuous function on with theAssume that there exists a continuous function on with theAssume that there exists a continuous function on with the

values in which possesses the following properties:values in which possesses the following properties:values in which possesses the following properties:values in which possesses the following properties:

a) there exist positive numbers and such thatthere exist positive numbers and such thatthere exist positive numbers and such thatthere exist positive numbers and such that

for all , ;for all , ;for all , ;for all , ; (1.25)

b) for any for any for any for any

;;;; (1.26)

c) for any two solutions to problem for any two solutions to problem for any two solutions to problem for any two solutions to problem (1.1) there exist there exist there exist there exist (i)a function locally integrable over the axis with the propertiesa function locally integrable over the axis with the propertiesa function locally integrable over the axis with the propertiesa function locally integrable over the axis with the properties

(1.27)

andandandand

(1.28)

for some , and for some , and for some , and for some , and (ii) a positive constant such that equation a positive constant such that equation a positive constant such that equation a positive constant such that equation

(1.8) holds for all holds for all holds for all holds for all .... Then is a set of -determining func- Then is a set of -determining func- Then is a set of -determining func- Then is a set of -determining func-

tionals for problem tionals for problem tionals for problem tionals for problem (1.1)....

Proof.

It follows from (1.23), (1.8), and (1.11) that the function satisfies the inequality

(1.29)

for all . Using properties (1.27) and (1.28) it is easy to find that there existnumbers , , and such that

, .

This equation and boundedness property (1.26) enable us to pass to the limitin (1.29) for fixed as and to obtain the required assertion.

Using Theorem 1.2 with as above, we obtain the following assertion.

� C –� +� H�� Lloc2 –� +� V�� 0

� u t�� H R�R

� �

� u t�� u�� u H t R

u t� � v t� ��

� u t� � v t� � t�� t Rsup ��

u t� � v t� �� t� � R

�� d

t

t a�

�t ��lim 0�

��

0 � �� max �d

t

t a�

�t ��lim ��

a 0� C

t s� � V ��

w t� � � u t� � v t� � t��

w t� � w s� � � �� d

s

t

�� ! !" #! !$ %

exp��

t s�s* a0 0� b0 0�

� �� d

s1

s2

� a0 s2 s1�� b0�� s1 s2 s*� �

t s ��

� u t�� u2�


C

h

a

p

t

e

r

5

Corollary 1.3.


embedded into . Let be a class of solutions to problem (1.1) on the

real axis possessing property (1.24) and such that

for all . (1.30)

Assume that for any and for all real equation

(1.21) holds with and a positive function locally integrable

over the axis and satisfying the condition

(1.31)

for some . Here is a constant independent of and .

Let be a family of continuous linear functionals

on possessing property (1.18) with . Then is a set of

asymptotically -determining functionals for problem (1.1).

Proof.

As in the proof of Corollary 1.2 equations (1.20) and (1.21) imply that

for all , where and is an arbitrary positivenumber. Hence

(1.32)

for all . Using (1.31) it is easy to find that for any there exists such that

for all . This equation and boundedness property (1.30) enableus to pass to the limit as in (1.32), provided , and to obtainthe required assertion.

We now give one more general result on the finiteness of the number of determiningfunctionals. This result does not use Lemma 1.1 and requires only the convergenceof functionals on a certain sequence of moments of time.

V H V

H �

R

u t� � Ht Rsup �� u t� � �

u t� � v t� �� t s�- 0� . t� �

R

1a�� . �� d

s

s a�

�s ��lim R�

a 0� R 0� u t� � v t� �� l

j: j 1 � N� ��

V +�

- R/� �

V ��

u t� � v t� ��H

� �� u �� v �� H2 �d

s

t

�� u s� � v s� ��H2�

t s� � �� - 1 ,�� 1� +�

2� . �� ,

u t� � v t� ��H2

u s� � v s� ��H2 � �� d

s

t

�� ! !" #! !$ %

exp�

t s� 1 0�M1 0�

. �� d

s1

s2

� R 1�� s2 s1� a��

s1 s2 M1�� s �� +

�- R/�


Theorem 1.3.

Let and be reflexive Banach spaces such that is continuouslyLet and be reflexive Banach spaces such that is continuouslyLet and be reflexive Banach spaces such that is continuouslyLet and be reflexive Banach spaces such that is continuously

embedded into . Assume that is a class of solutions to problem embedded into . Assume that is a class of solutions to problem embedded into . Assume that is a class of solutions to problem embedded into . Assume that is a class of solutions to problem (1.1)possessing property possessing property possessing property possessing property (1.2). Ass. Ass. Ass. Assumumumume that there exist constants ,e that there exist constants ,e that there exist constants ,e that there exist constants ,

, and such that for any pair of solutions and, and such that for any pair of solutions and, and such that for any pair of solutions and, and such that for any pair of solutions and

from we have from we have from we have from we have

,,,, ,,,, (1.33)

andandandand

,,,, (1.34)

for large enough. Let be a finite set of continuous linear functionalsfor large enough. Let be a finite set of continuous linear functionalsfor large enough. Let be a finite set of continuous linear functionalsfor large enough. Let be a finite set of continuous linear functionals

on possessing property on possessing property on possessing property on possessing property (1.18) with with with with .... Assume that is a Assume that is a Assume that is a Assume that is a

sequence of positive numbers such that and sequence of positive numbers such that and sequence of positive numbers such that and sequence of positive numbers such that and ....

Assume thatAssume thatAssume thatAssume that

,,,, .... (1.35)

ThenThenThenThen

asasasas .... (1.36)

It should be noted that relations like (1.33) and (1.34) can be obtained for a wideclass of equations (see, e.g., Sections 1.9, 2.2, and 4.6).

Proof.

Let . Then equations (1.34) and (1.18) give us

,

where . Therefore, after iterations we obtain that

.

Hence, equation (1.35) implies that as . Therefore, (1.36) fol-lows from equation (1.33). Theorem 1.3 is proved.

Application of Corollaries 1.1–1.3 and Theorem 1.3 to the proof of finiteness of a set of determining elements requires that the inequalities of the type (1.17) and

(1.21), or (1.30) and (1.33), or (1.33) and (1.34), as well as (1.18) with the constant small enough be fulfilled. As the analysis of particular examples shows, the fulfil-

ment of estimates (1.17), (1.21), (1.30), (1.31), (1.33), and (1.34) is mainly con-nected with the dissipativity properties of the system. Methods for obtaining themare rather well-developed (see Chapters 1 and 2 and the references therein) and inmany cases the corresponding constants , , , and either are close to opti-

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mal or can be estimated explicitly in terms of the parameters of equations. There-fore, the problem of description of finite families of functionals that asymptoticallydetermine the dynamics of the process can be reduced to the study of sets of func-tionals for which estimate (1.18) holds with small enough. It is convenient tobase this study on the concept of completeness defect of a family of functionals withrespect to a pair of spaces.

§ 2 Completeness Defect§ 2 Completeness Defect§ 2 Completeness Defect§ 2 Completeness Defect

Let and be reflexive Banach spaces such that is continuously and denselyembedded into . The completeness defect completeness defect completeness defect completeness defect of a set of linear functionalson with respect to is defined as

. (2.1)

It should be noted that the finite dimensionality of is not assumed here.

Prove that the value can also be defined by oneof the following formulae:

, (2.2)

, (2.3)

. (2.4)

Let be two sets in the space of linear functionalson . Show that .

Let and let be a weakly closed span of the set in the space . Show that .

The following fact explains the name of the value . We remind that a set of functionals on is said to be complete if the condition for all im-plies that .

Show that for a set of functionals on to be complete it isnecessary and sufficient that .

+�

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1�� " #$ %

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V H��

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: w V , l w� � 0� , wV

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E x e r c i s e 2.4 � V

+�

V H�� 0�

C o m p l e t e n e s s D e f e c t 297

The following assertion plays an important role in the construction of a set of deter-mining functionals.

Theorem 2.1.

Let be the completeness defect of a set of linear func-Let be the completeness defect of a set of linear func-Let be the completeness defect of a set of linear func-Let be the completeness defect of a set of linear func-

tionals on with respect to tionals on with respect to tionals on with respect to tionals on with respect to .... Then there exists a constant such Then there exists a constant such Then there exists a constant such Then there exists a constant such

thatthatthatthat

(2.5)

for any element for any element for any element for any element ,,,, where is a weakly closed span of the set where is a weakly closed span of the set where is a weakly closed span of the set where is a weakly closed span of the set in in in in

....

Proof.

Let

(2.6)

be the annihilator of . If , then it is evident that for all .Therefore, equation (2.4) implies that

for all , (2.7)

i.e. for equation (2.5) is valid.Assume that . Since is a subspace in , it is easy to verify that there

exists an element such that

. (2.8)

Indeed, let the sequence be such that

.

It is clear that is a bounded sequence in . Therefore, by virtue of the reflexivi-ty of the space , there exist an element from and a subsequence such that weakly converges to as , i.e. for any functional theequation

holds. It follows that

.

Therefore, we use the reflexivity of once again to find that

.

However, . Hence, . Thus, equation (2.8) holds.

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*1�

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V *

�4 v V : l v� � 0� , l ��

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u V� u �4

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w �4

u w�V

u �4�� Vdist u v�V

: v �4� " #$ %

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d u �4�� Vdist u vn

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vn

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f u w�� u vnk

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sup d��

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Equation (2.7) and the continuity of the embedding of into imply that

.

It is clear that

.

Therefore,

. (2.9)

Let us now prove that there exists a continuous linear functional on the space possessing the properties

, , for . (2.10)

To do that, we define the functional by the formula

, ,

on the subspace

.

It is clear that is a linear functional on and for . Let us cal-culate its norm. Evidently

, .

Since , equation (2.8) implies that

, , .

Consequently, for any

.

This implies that has a unit norm as a functional on . By virtue of the Hahn-Ba-nach theorem the functional can be extended on without increase of the norm.Therefore, there exists a functional on possessing properties (2.10). Therewith

lies in a weakly closed span of the set . Indeed, if , then using the re-flexivity of and reasoning as in the construction of the functional it is easy toverify that there exists an element such that and for all

. It is impossible due to (2.10) .In order to complete the proof of Theorem 2.1 we use equations (2.9) and

(2.10). As a result, we obtain that

.

However, , , and . Therefore, equation (2.5)holds. Theorem 2.1 is proved.

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l�

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l�

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l0 V

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l �

u H +�

u V +�

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l0 �� l0 u w�� l0 u� �� l0 *

1�


Assume that is a finite set in .Show that there exists a constant such that

(2.11)

for all .

In particular, if the hypotheses of Corollaries 1.1 and 1.3 hold, then Theorem 2.1 andequation (2.11) enable us to get rid of assumption (1.18) by replacing it with the cor-responding assumption on the smallness of the completeness defect .

The following assertion provides a way of calculating the completeness defect whenwe are dealing with Hilbert spaces.

Theorem 2.2.

Let and be separable Hilbert spaces such that is compactly andLet and be separable Hilbert spaces such that is compactly andLet and be separable Hilbert spaces such that is compactly andLet and be separable Hilbert spaces such that is compactly and

densely embedded into densely embedded into densely embedded into densely embedded into .... Let be a selfadjoint positive compact operator Let be a selfadjoint positive compact operator Let be a selfadjoint positive compact operator Let be a selfadjoint positive compact operator

in the space defined by the equalityin the space defined by the equalityin the space defined by the equalityin the space defined by the equality

,,,, ....

Then the completeness defect of a set of functionals on can be evalua-Then the completeness defect of a set of functionals on can be evalua-Then the completeness defect of a set of functionals on can be evalua-Then the completeness defect of a set of functionals on can be evalua-

ted by the formulated by the formulated by the formulated by the formula

,,,, (2.12)

where is the orthoprojector in the space onto the annihilatorwhere is the orthoprojector in the space onto the annihilatorwhere is the orthoprojector in the space onto the annihilatorwhere is the orthoprojector in the space onto the annihilator

and is the maximal eigenvalue of the operator and is the maximal eigenvalue of the operator and is the maximal eigenvalue of the operator and is the maximal eigenvalue of the operator ....

Proof.

It follows from definition (2.1) that

where is the unit ball in . Due to the compactnessof the embedding of into , the set is compact in . Therefore, there existsan element such that

.

Therewith is the maximum point of the function on the set .Hence, for any and we have

.

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C�

uH

+�

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lj

u� �j 1 � N� ��

max��

u V

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It follows that

.

It is also clear that . Therefore,

for all . This implies that

for any . If we take instead of in this equation, then we obtain the op-posite inequality. Therefore,

, .

Consequently,

,

i.e. is an eigenvalue of the operator . It is evidentthat this eigenvalue is maximal. Thus, Theorem 2.2 is proved.

Corollary 2.1.

Assume that the hypotheses of Theorem 2.2 hold. Let be an ortho-

normal basis in the space that consists of eigenvectors of the opera-

tor :

, , , .

Then the completeness defect of the system of functionals

is given by the formula .

To prove this assertion, it is just sufficient to note that is the orthoprojectoronto the closure of the span of elements and that commuteswith .

Let be a positive operator with discrete spectrum in thespace :

, , , ,

and let , , be a scale of spaces generated by theoperator (see Section 2.1). Assume that

. (2.13)

Prove that for all .

It should be noted that the functionals in Exercise 2.6 are often called modes modes modes modes.

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Let us give several more facts on general properties of the completeness defect.

Theorem 2.3.

Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem 2.2 hold. Assume that is a set hold. Assume that is a set hold. Assume that is a set hold. Assume that is a set

of linear functionals on and is a family of linear bounded operatorsof linear functionals on and is a family of linear bounded operatorsof linear functionals on and is a family of linear bounded operatorsof linear functionals on and is a family of linear bounded operators

that map into and are such that for all . Let that map into and are such that for all . Let that map into and are such that for all . Let that map into and are such that for all . Let

(2.14)

be the global approximation error in arising from the approximationbe the global approximation error in arising from the approximationbe the global approximation error in arising from the approximationbe the global approximation error in arising from the approximation

of elements by elements of elements by elements of elements by elements of elements by elements .... Then Then Then Then

.... (2.15)

Proof.

Let . Equation (2.14) implies that

, .

Therefore, for we have , i.e. for all. Let us show that there exists an operator such that

. Equation (2.12) implies that

,

where is the orthoprojector in the space onto and is thenorm of the operator in the space of bounded linear operators in .Therefore, the definition of the operator implies that

. (2.16)

It is evident that the orthoprojector belongs to (it projects ontothe subspace that is orthogonal to in ). Theorem 2.3 is proved.

Assume that is a finite set. Show thatthe family consists of finite-dimensional operators of theform

, ,

where is an arbitrary collection of elementsof the space (they do not need to be distinct). How should thechoice of elements be made for the operator from theproof of Theorem 2.3?

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Theorem 2.3 will be used further (see Section 3) to obtain upper estimates ofthe completeness defect for some specific sets of functionals. The simplest situationis presented in the following example.

E x a m p l e 2.1

Let and let . As usual, here isthe Sobolev space of the order and is the closure of the set in . We define the norms in and by the equalities

, .

Let , , . Consider a set of functionals

on . Assume that is a transformation that maps a function into itslinear interpolating spline

.

Here for and for . We apply Theo-rem 2.3 and obtain

.

We use an easy verifiable equation

, ,

to obtain the estimate

.

This implies that .

The assertion on the interdependence of the completeness defect and the Kol-mogorov -width made below enables us to obtain effective lower estimatesfor .

Let and be separable Hilbert spaces such that is continuously anddensely embedded into . Then the Kolmogorov Kolmogorov Kolmogorov Kolmogorov -width of the embedding-width of the embedding-width of the embedding-width of the embedding

of into of into of into of into is defined by the relation

H L2 0 l�� V H2 H01 � � 0 l�� Hs 0 l��

s H0s 0 l�� C0

� 0 l�� Hs 0 l�� H V

uH2

u2

u x� �� 2 xd

0

l

�� uV2

u99 2�

h l N/� xj

j h� j 1 � N� 1��

� l u� � u xj� � : j 1 � N� 1��

V R u V

s x� � u xj� � : x

h�� j�& '

( )

j 1�

N 1�

*�

: x� � 1 x�� x 1� : x� � 0� x 1�

+�

V H�� u s� : u V , u99 1�� " #$ %

sup�

u x� � s x� �� 1h�� ; u99 y� � yd

�

;

�d

xj

xj 1�

�d

xj

x

�� x xj xj 1��

u s� h2

3�� u99�

+�

H V�� h2 3/�

+�

N

+�

V H�� V H V

H N

V H


, (2.17)

where is the family of all -dimensional subspaces of the space and

is the global error of approximation of elements in by elements of the sub-space . Here

.

In other words, the Kolmogorov -width of the embedding of into is theminimal possible global error of approximation of elements of in by elementsof some -dimensional subspace.

Theorem 2.4.

Let and be separable Hilbert spaces such that is continuouslyLet and be separable Hilbert spaces such that is continuouslyLet and be separable Hilbert spaces such that is continuouslyLet and be separable Hilbert spaces such that is continuously

and densely embedded into and densely embedded into and densely embedded into and densely embedded into .... Then Then Then Then

,,,, (2.18)

where is the nonincreasing sequence of eigenvalues of the operator where is the nonincreasing sequence of eigenvalues of the operator where is the nonincreasing sequence of eigenvalues of the operator where is the nonincreasing sequence of eigenvalues of the operator

defined by the equality defined by the equality defined by the equality defined by the equality ....

The proof of the theorem is based on the lemma given below as well as on the factthat , where is a compact positive operator in the space (see Theorem 2.2). Further the notation stands for the proper basis of theoperator in the space while the notation stands for the corresponding eigen-values:

, , , .

It is evident that is an orthonormalized basis in the space .

Lemma 2.1.

Assume that the hypotheses of Theorem 2.4 hold. Then

. (2.19)

<N

<N

V H�� eV

HF� � : F �

N

� " #$ %

inf� �

�N

N F V

eV

HF� � v F��

H: dist v

V1�

� " #$ %

sup�

v V H

F

distH v F�� v f� H : f F� " #$ %

inf�

N <N

V H

V H

N

V H V

H

<N

V H�� +�

V H�� : � V * Lin �dim N��0� " #$ %

min 6N 1��

6j

� � K

u v�� H

Ku v�� V

�

u v�� H Ku v�� V� K V

ej� �j 1��

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K ej 6j ej� 61 62 �� 6n 0� ei ej�� V

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H

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Proof.

By virtue of Corollary 2.1 it is sufficient to verify that

for all , where are linearly independent functionalson . Definition (2.1) implies that

(2.20)

for all such that and , . Let us substi-tute in (2.20) the vector

,

where the constants are choosen such that for and. Therewith equation (2.20) implies that

.


We now prove that . Let us use equation (2.16)

. (2.21)

Here is the orthoprojector onto the subspace orthogonal to in . It isevident that is isomorphic to . Therefore, . Hence, equa-tion (2.21) gives us that

(2.22)

for all such that . Conversely, let be an -dimensionalsubspace in and let be a orthonormalized basis in the space .Assume that

.

Let be the orthoprojector in the space onto . It is clear that

.

Therefore, if , then . It is clear that is a bounded operatorfrom into . Using Theorem 2.3 we find that

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j

V

+�

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ju e

j��

V

2

j 1�

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*�

u V u V 1� lj u� � 0� j 1 � N� ��

u cj

ej

j 1�

N 1�

*�

cj

lj

u� � 0� j 1 � N� ��u

V1�

+�

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j2

j 1�

N 1�

* 6N 1� c

j2

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N 1�

*� � 6N 1� u

V2 6

N 1��

<N

+�

� �min�

+�

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u�H

: u V 1�� " #$ %

sup�

Q�

Q�

V �4 V

Q�

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Vdim N�

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V H�� u Q�

V�� H

dist : uV

1�� " #$ %

sup <N

� �

� V * Lin�dim N� F N

V fj: j 1 � N� �� H

�F lj : lj v� � fj v�� H

� : j 1 � N� ��

QH F� H F

QH F� u u f

j��

Hfj

j 1�

N

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.

However, . Hence,

(2.23)

for any -dimensional subspace in . Equations (2.22) and (2.23) imply that

.

This equation together with Lemma 2.1 completes the proof of Theorem 2.4.

Let and be reflexive Banach spaces such that is con-tinuously and densely embedded into , let be a set of linearfunctionals on , . Assume that

,

where

, .

Prove that

.

Use Lemma 2.1 and Corollary 2.1 to calculate the Kolmogorov-width of the embedding of the space into

for , where is a positive operator with discrete spectrum.

Show that in Example 2.1 .Prove that .

Assume that there are three reflexive Banach spaces such that all embeddings are dense and continuous.

Let be a set of functionals on . Prove that (Hint: see (2.3)).

In addition to the hypotheses of Exercise 2.11, assume thatthe inequality

, ,

holds for some constants and . Show that

.

+�

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HQH F�� u QH F� u�

H: u V 1�

� " #$ %

sup�

u QH F� u�

Hu F��

Hdist�

+�

� �min +�

FV H�� e

VH

F� ��

N F V

<N

V H�� +�

V H�� : � V * Lin�dim N��0� " #$ %

min�

E x e r c i s e 2.8 Vk

Hk

Vk

Hk

�k

Vk

k 1 2��

� �1 �2= V1 V2�� *0�

�k l V1 V2�� * : l v1 v2�� l vk� �� , l �k� " #$ %

� k 1 2��

+�

V1 V2� H1 H2�� +�1

V1 H1�� +�2

V2 H2�� " #$ %

max�

E x e r c i s e 2.9

N �s D As� �� D A�� s �� A

E x e r c i s e 2.10 <N

V H�� l2 > N 1�� 2��> 2� h2 +

�V H�� h2 3/� �

E x e r c i s e 2.11 V0W H0 0� W +

�V H��

+�

V W�� +�

W H��


u W a? u H?

u V1 ?�� u V

a? 0� ? 0 1��

a?1� +

�V W��

1?��

+�

V H�� a? +� W H�� 1

1 ?��

� �


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§ 3 Estimates of Completeness Defect§ 3 Estimates of Completeness Defect§ 3 Estimates of Completeness Defect§ 3 Estimates of Completeness Defect

in Sobolev Spacesin Sobolev Spacesin Sobolev Spacesin Sobolev Spaces

In this section we consider several families of functionals on Sobolev spaces that areimportant from the point of view of applications. We also give estimates of the cor-responding completeness defects. The exposition is quite brief here. We recommendthat the reader who does not master the theory of Sobolev spaces just get acquain-ted with the statements of Theorems 3.1 and 3.2 and the results of Examples 3.1 and3.2 and Exercises 3.2–3.6.

We remind some definitions (see, e.g., the book by Lions-Magenes [4]). Let be a domain in . The Sobolev space of the order is a set of functions

,

where , , and

. (3.1)

The space is a separable Hilbert space with the inner product

.

Further we also use the space which is the closure (in ) of the set of infinitely differentiable functions with compact support in and the

space which is defined as follows:

,

where , is the Fourier transform of the function ,

,

, and . Evidently this definition coin-cides with the previous one for natural and .

Show that the norms in the spaces possess the pro-perty

, ,

, .

@R- H m @� � m m 0 1 2 ��

H m @� � f L2 @� � : � j f x� � L2 @� � , j m�� " #$ %

�

j j1 � j-� �� jk

0 1 2 �� j j1 � j-� ��

� j f x� �A j f

Ax1j1 Ax2

j2 � Ax-j-�

��

H m @� �

u v�� m

� j u � j v� xd

@�

j m�*�

H0m @� � Hm @� �

C0� @� � @

H s R-� �

H s R-� � u x� � L2 R-� � : 1 y2�� s u� y� � 2

yd

R-

� us2 ��

� " #$ %

�

s 0� u� y� � u x� �

u� y� � ei x y u x� � xd

R-��

y2

y12 � y2

2� �� x y x1 y1 � x-y-� ��s @ R-�

E x e r c i s e 3.1 Hs R-� �

u s ?� � u s1? u s2

1 ?�� s ?� � ?s1 1 ?�� s2��

0 ? 1� � s1 s2� 0�

E s t i m a t e s o f C o m p l e t e n e s s D e f e c t i n S o b o l e v S p a c e s 307

We can also define the space as restriction (to ) of functions from with the norm

and the space as the closure of the set in . The spaces and are separable Hilbert spaces. More detailed information on the Sobolevspaces can be found in textbooks on the theory of such spaces (see, e.g., [4], [5]).

The following version of the Sobolev integral representation will be used fur-ther.

Lemma 3.1.

Let be a domain in and let be a function from such

that

, . (3.2)

Assume that is a star-like domain with respect to the support

of the function . This means that for any point the cone

(3.3)

belongs to the domain . Then for any function the re-

presentation

(3.4)

is valid, where

, (3.5)

, , ,

. (3.6)

Proof.

If we multiply Taylor’s formula

Hs @� � @ Hs R-� �

us @� v

s R-�: v x� � u x� �� in @ , v Hs R-� �

� " #$ %

inf�

H0s @� � C0

� @� � Hs @� � Hs @� �H0

s @� �

@ R- 7 x� � L� R-� �

supp 7 0 @0 7 x� � xd

R-� 1�

@ supp 77 x @

Vx

z �x 1 �� y�� 0 � 1� � y supp7��

@ u x� � Hm @� �

u x� � Pm 1� x u�� m

�B�� x y�� K x y�� u y� � yd

Vx

�� m�*��

Pm 1� x u�� 1�B�� 7 y� � x y�� u y� � yd

@�

� m�*�

� �1 � �-� �� B �1B��-B� z� z1�1� z-

�-�

K x y�� s -� 1� 7 xy x�

s��& '

( ) sd

0

1

��

u x� �x y��

�B�� u y� �� m�*

m�x y��

�B�� sm 1� ��u x s y x�� sd

0

1

�� m�*

��


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by and integrate it over , then after introducing a new variable we obtain the assertion of the lemma.

Integral representation (3.4) enables us to obtain the following generalization of thePoincaré inequality.

Lemma 3.2.

Let the hypotheses of Lemma 3.1 be valid for a bounded domain

and for a function . Then for any function

the inequality

(3.7)

is valid, where , is the surface measure of

the unit sphere in and .

Proof.

We use formula (3.4) for :

. (3.8)

It is clear that when . There-fore,

.

Consequently,

, . (3.9)

Thus, it follows from (3.8) that

(3.10)

Let . Then it is evident that

, (3.11)

7 y� � y

z x s y x��

@ R-0 7 x� � u x� � H1 @� �

u uC D7�L2 @� �

�--�� d- 1� 7

L� @� �Eu

L2 @� ��

uC D7 7 x� �u x� � xd@�� -

R- d @diam x y� : x y @�� sup�

m 1�

u x� � uC D7 K x y�� xj yj�� Au

Ayj

�� y� �� yd

Vx

�j 1�

-

*��

7 x 1 s/� � y x�� 0� s 1� x y� d @diam�

K x y�� s -� 1� 7 x1s�� y x�� & '

( ) sd

d 1� x y�

1

��

K x y�� ,x y� -�� , , 7 -�� d-

-�� 7L� @� �

�

u x� � uC D7� ,Eu y� �

x y� - 1�� yd

@�

,Eu y� � 2

x y� - 1�� yd

@�& '

F G( )1 2�

yd

x y� - 1��

@�& '

F G( )1 2�

.

� �

�

Br

x� � y : x y� r��

yd

x y� - 1��

@� yd

x y� - 1��

Bd x� �� - d��


where is the surface measure of the unit sphere in . Therefore, equation(3.10) implies that

.

After integration with respect to and using (3.11) we obtain (3.7). Lemma 3.2is proved.

Lemma 3.3.

Assume that the hypotheses of Lemma 3.1 are valid for a bounded do-

main from and for a function . Let , where

is a sign of the integer part of a number. Then for any function

we have the inequality

(3.12)

where is defined by formula (3.5), ,

is the surface measure of the unit sphere in , and .

Proof.

Using (3.4) and (3.9) we find that

As above, we obtain that

.

Thus, equation (3.12) is valid. Lemma 3.3 is proved.

�- R-

u x� � uC D7� 2 ,2 �- dEu y� � 2

x y� - 1�� yd

@��

x

@ R- 7 x� � m- - 2/� � 1��.� �

u x� � Hm- @� �

u x� � Pm- 1� x u�� x @max

c--�� d

m--2��

7L� @� �

m-�B�� u

L2 @� � ,

� m-�*

�

�

Pm 1� x u�� c- �- 2 m- -�� /� �1 2��

�- R- d @diam�

u x� � Pm- 1� x u��

m-�B�� x y�

m- K x y�� u y� � yd�

Vx

�� m-�*

,m-�B�� x y�

m- -�� u y� �� yd

@�

� m-�*

,m-�B�� u

L2 @� � x y�2 m- -��

yd

@�& '

F G( )1 2�

.�� m-�*

� �

� �

�

x y� 2 m- -�� yd

@� �- r

2 m- -� 1�rd

0

d

� �-d2 m- -�

2 m- -�� 1��


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Lemma 3.4.

Assume that the hypotheses of Lemma 3.3 hold. Then for any function

and for any the inequality

(3.13)

is valid, where is a constant that depends on only and is the

diameter of the domain .

Proof.

It is evident that

, (3.14)

where

.

The structure of the polynomial implies that

for all . Therefore, estimate (3.13) follows from (3.14) and Lemmata 3.2and 3.3. Lemma 3.4 is proved.

These lemmata enable us to estimate the completeness defect of two families offunctionals that are important from the point of view of applications. We considerthese families of functionals on the Sobolev spaces in the case when the domain isstrongly Lipschitzian, i.e. the domain possesses the property: for every

there exists a vicinity such that

u x� � Hm- @� � x* @

u u x*

� ��L2 @� �

u x� � u x*

� �� 2xd

@�& '

F G( )1 2�

C- d- 7L� @� �

� � d j 1�B�� u

L2 @� �� j�*

j 1�

m-

*� �

�

�

C- - d

@

u u x*

� ��L2 @� �

u uC D7�L2 @� �

d- 2� uC D7 u x*

� ��

uC D7 7 y� � u y� � yd

@��

Pm 1� x u��

uC D7 u x� ��

u x� � Pm- 1� x u�� 1�B�� 7 y� � x y�� u y� � y

u x� � Pm- 1� x u�� d �

�B�� 7L2 @� �

��uL2 @� ��

1 � m- 1�� *�

�

�

d

@�

1 � m- 1�� *�

�

�

x @

@ R-0x A@ U

U @ x x1 � x-� �� : x- f x1 � x- 1��


in some system of Cartesian coordinates, where is a Lipschitzian function.For strongly Lipschitzian domains the space consists of restrictions to of functions from , (see [5] or [6]).

Theorem 3.1.

Assume that a bounded strongly Lipschitzian domain in can beAssume that a bounded strongly Lipschitzian domain in can beAssume that a bounded strongly Lipschitzian domain in can beAssume that a bounded strongly Lipschitzian domain in can be

divided into subdomains such thatdivided into subdomains such thatdivided into subdomains such thatdivided into subdomains such that

,,,, forforforfor .... (3.15)

Here the bar stands for the closure of a set. Assume that is a functionHere the bar stands for the closure of a set. Assume that is a functionHere the bar stands for the closure of a set. Assume that is a functionHere the bar stands for the closure of a set. Assume that is a function

in such thatin such thatin such thatin such that

,,,, (3.16)

and is a star-like domain with respect to and is a star-like domain with respect to and is a star-like domain with respect to and is a star-like domain with respect to .... We define the set We define the set We define the set We define the set

of generalized local volume averages corresponding to the collectionof generalized local volume averages corresponding to the collectionof generalized local volume averages corresponding to the collectionof generalized local volume averages corresponding to the collection

as the family of functionalsas the family of functionalsas the family of functionalsas the family of functionals

.... (3.17)

Then the estimateThen the estimateThen the estimateThen the estimate

(3.18)

holds, whereholds, whereholds, whereholds, where ,,,, ,,,,

,,,,

and are constants. and are constants. and are constants. and are constants.

Proof.

Let us define the interpolation operator for the collection by the formula

, , .

f x� �Hs @� � @

Hs R-� � s 0�

@ R-

@j : j 1 2 � N� � ��

@ @ j

j 1�

N

=� @i

@j

H� i j3

7j

x� �L� @

j� �

supp 7j 0 @j0 7j x� � xd

@j

� 1�

@j

supp7j

�

� @j

7j

�� : j 1 2 � N� � ��

� lj u� � = 7j x� � u x� � xd

@j

� , j 1 2 � N� � �� " #$ %

�

+�

Hs @� � H� @� �� C - s�� Id� �s ��

,,,, s 1 , 0 � s , , , ,� ��

C-I1 �

s��

d s �� ,,,, 0 � s 1 , s 0 , , , ,3� � ��!"!$

�

I dj- 7j L� @� �

: j 1 2 � N� � �� " #$ %

max� d djj

max�

dj

@diam x y� : x y @j

�� sup�

C - s�� C-

�

�

�

u� � x� � lj u� � 7j x� � u x� � xd

@j

�� x @j j 1 2 � N� � ��


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5

It is easy to check that

, .

It further follows from Lemma 3.2 that

.

This implies the estimate

.

Using the fact (see [4, 6]) that

, ,

and the interpolation theorem for operators [4] we find that

for all . Consequently, Theorem 2.3 gives the equation

, . (3.19)

Using the result of Exercise 2.12 and the interpolation inequalities (see, e.g., [4,6])

, , , (3.20)

it is easy to obtain equation (3.18) from (3.19).

Let us illustrate this theorem by the following example.

E x a m p l e 3.1

Let be a cube in with the edge of the length . We constructa collection which defines local volume averages in the following way.Let be the standard unit cube in and let be a measurable setin with the positive Lebesgue measure, . We define the function

on by the formula

Assume that

,

, ,

u �

u�L2 @� �

C-I uL2 @� �

� u L2 @� �

u lj

u� ��L2 @

j� �

�--�� d

j- 1� 7

j L� @j

� �Eu

L2 @j

� ��

u �

u�L2 @� �

�--�� dI u

H1 @� ��

uHs @� �

C uL2 @� �1 s�

uH1 @� �s�� 0 s 1� �

u RT

u�L2 @� �

C- I� �1 s� �--�� dI& '

( )s

uH s @� �

�

0 s 1� �

+�

Hs @� � L2 @� �� C-Ids� 0 s 1� �

uHs ?� � @� �

C uH

s1 @� �

?u

Hs2 @� �

1 ?�� s ?� � s1? s2 1 ?�� 0 ? 1� �

@ 0 l�� -� R- l

�

K 0 1�� -� R- JK Jmes 0�

7 J x�� K

7 J x�� Jmes� � 1� , x J ,

0 , x K \J .�"$

�

@j

h j K�� x x1 � x-� �� : ji

xi

h�� j

i1 , i�� 1 2 � -� � ��

� " #$ %

�

7j

x� � 1h-�� 7 J x

h�� j��& '

( )� x @j


for any multi-index , where , .It is clear that the hypotheses of Theorem 3.1 are valid for the collection

. Moreover,

,

and hence in this case we have

(3.21)

for the set of functionals of the form (3.17) when and .It should be noted that in this case the number of functionals in the set isequal to . Thus, estimate (3.21) can be rewritten as

.

However, one can show (see, e.g., [6]) that the Kolmogorov -width of the em-bedding of into has the same order in , i.e.

.

Thus, it follows from Theorem 2.4 that local volume averages have a complete-ness defect that is close (when the number of functionals is fixed) to the mini-mal. In the example under consideration this fact yields a double inequality

, , (3.22)

where and are positive constants that may depend on , , , and .Similar relations are valid for domains of a more general type.

Another important example of functionals is given in the following assertion.

Theorem 3.2.

Assume the hypotheses of Theorem Assume the hypotheses of Theorem Assume the hypotheses of Theorem Assume the hypotheses of Theorem 3.1.... Let us choose a point (called Let us choose a point (called Let us choose a point (called Let us choose a point (called

a node) in every set and define a set of functionals on ,a node) in every set and define a set of functionals on ,a node) in every set and define a set of functionals on ,a node) in every set and define a set of functionals on ,

, by, by, by, by

.... (3.23)

Then for all and the estimateThen for all and the estimateThen for all and the estimateThen for all and the estimate

(3.24)

is valid for the completeness defect of this set of functionals, whereis valid for the completeness defect of this set of functionals, whereis valid for the completeness defect of this set of functionals, whereis valid for the completeness defect of this set of functionals, where

j j1 � j-� �� j- 0 1 � N 1�� h l N/�

� @j

7j

��

dj

diam @j

- h�� I -- 2�

Jmes��

+�

Hs @� � H� @� �� C- s�h

Jmes��& '

( )s ��

�

� s 1� 0 � s� ��

��

N-�

+�

Hs @� � H� @� �� C- s�lJmes

��

& 'F G( )

s ��1�

�

�� & 'F G( )

s ��-��

��

N�

Hs @� � H� @� � ��

<��

Hs @� � H� @� �� c01

N�

��

& 'F G( )

s ��-��

�

c1 hs �� +L Hs @� � H� @� �� c2 hs �� s�

c1 c2 s - J @

xj

@j

Hm @� �m - 2/� � 1��

� lj

u� � u xj

� �� : xj

@j, j 1 � N� ��

� " #$ %

�

s m� 0 � s� �

+�

Hs @� � H� @� �� C - s�� dI� �s ��


C

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,,,, ,,,,

....

Proof.

Let and let , . Then using(3.13) for and we obtain that

,

where

.

It follows that

for all such that , . Using interpola-tion inequality (3.20) we find that

. (3.25)

By virtue of the inequality

, , ,

we get

for and for all . We substitute these inequalities in (3.25)to obtain that

(3.26)

d dj: j 1 2 � N� � ��

� " #$ %

max� dj

@diam�

I dj- 7

j L� @� �: j 1 2 � N� � ��

� " #$ %

max�

u H m @� � lj u� � u xj� � 0� � j 1 2 � N� � ��@ @j� x

*xj�

uL2 @

j� �

C- dj- 7

j L� @j

� �& '( ) d

jl

ul @

j�

l 1�

m

*�

u l @j

�1�B�� u

L2 @j

� �� l�*�

uL2 @� �

C -� � I dl uHl @� �

l 1�

m

*� ��

u H m @� � lj u� � u xj� � 0� � j 1 2 � N� � ��

uL2 @� �

C- I dl uL2 @� �

1 l

m��

uHm @� �l m��

l 1�

m

*� ��

x yx p

p��

yq

q�� 1

p�� 1

q�� 1� x y� 0�

I dl uL2 @� �

1 l

m��

uHm @� �l m�� u

L2

1 l

m��

dmIm

l��

uHm

& 'F G( )

l

m��

�

1 lm�� ,

m

m l��

uL2 @� �

lm�� ,

m

l��

d mIm

l��

uHm @� �

��

�

�

�

l 1 2 � m 1�� , 0�

uL2 @� �

C- 1 lm�� ,

m

m l��

uL2 @� �

�l 1�

m 1�

*�

C-l

m�� ,

m

l��

Im

l��

l 1�

m 1�

* I�& 'F G( )

d m uHm @� �

.

�

�


We choose such that

.

Then equation (3.26) gives us that

for all such that , . Hence, the estimate

is valid. Since , this implies inequality (3.24) for and . As in Theorem 3.1 further arguments rely on Lemma 2.1 and interpola-

tion inequalities (3.20). Theorem 3.2 is proved.

E x a m p l e 3.2

We return to the case described in Example 3.1. Let us choose nodes and assume that . Then for a set of functionals of the form (3.23) wehave

for all , and for any location of the nodes inside the. In the case under consideration double estimate (3.22) is preserved.

In the exercises below several one-dimensional situations are given.

Prove that

for

.

Verify that

for

.

, , - m��

C- 1 lm��& '

( ) ,m

m l��

l 1�

m 1�

* 12��

uL2 @� �

C -� � lm��I

m

l��

dm uHm @� �

�l 1�

m

*��

u Hm @� � u xj� � 0� j 1 2 � N� � ��

+�

Hm @� � L2 @� �� C -� � dm l

m��I

m

l��

l 1�

m

*��

I 1� � 0� s m� �- 2/� � 1��

xj

@j

J K� �

+�

Hs @� � ; H� @� �� C- s� hs ��

s - 2/� � 1�� 0 � s� � xj

@j

E x e r c i s e 3.2

+�

H1 0 l�� L2 0 l�� <N

H1 0 l�� L2 0 l�� 1 >N

l��& '( )2

�12��

� �

� ljc

u� � u x� � j>l�� xcos x , jd

0

l

� 0 1 � N 1�� " #$ %

�

E x e r c i s e 3.3

+�

H01 0 l�� L2 0 l�� <

N 1� H01 0 l�� L2 0 l�� 1 >N

l��& '( )2

�12��

� �

� ljs u� � u x� � j>x

l��sin x j�d

0

l

� 1 2 � N 1�� " #$ %

�


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Let

.

Show that

where consists of functionals and for (the functionals and are defined in Exercises 3.2 and 3.3).

Consider the functionals

,

, , ,

on the space . Assume that an interpolation operator maps an element into a step-function equal to on the segment . Show that

.

Prove the estimate

.

Consider a set of functionals

, , , ,

on the space . Assume that an interpolation operator maps an element into a step-function equal to on the segment . Show that

.

Prove the estimate

.

E x e r c i s e 3.4

Hper1 0 l�� u H1 0 l�� : u 0� � u l� ��

+�

Hper1 0 l�� L2 0 l��

<2 N 1� Hper1 0 l�� L2 0 l�� 1 2>N

l��& '( )2

�12��

,

�

� �

� l2 kc

l2 ks

k 0 1 � N 1�� ljc

ljs

E x e r c i s e 3.5

lj u� � 1h�� u xj �� d

0

h

��

xj

j h� hl

N�� j 0 1 � N 1��

L2 0 l�� Rh

u L2 0 l�� lj

u� �x

jx

j 1��

u Rh

u�L2 0 l��

h u9L2 0 l��

�

h

>2 h2�� +

�H1 0 l�� L2 0 l�� h� �

E x e r c i s e 3.6 �

lj

u� � u xj

� �� xj

j h� hl

N�� j 0 1 � N 1��

H1 0 l�� Rh

u H1 0 l�� lj u� �xj xj 1��

u Rhu�L2 0 l��

h

2�� u9

L2 0 l��

h

>2 h2�� +

�H1 0 l�� L2 0 l�� h

2��

D e t e r m i n i n g F u n c t i o n a l s f o r A b s t r a c t S e m i l i n e a r P a r a b o l i c E q u a t i o n s 317

§ 4 Determining Functionals for Abstract§ 4 Determining Functionals for Abstract§ 4 Determining Functionals for Abstract§ 4 Determining Functionals for Abstract

Semilinear Parabolic EquationsSemilinear Parabolic EquationsSemilinear Parabolic EquationsSemilinear Parabolic Equations

In this section we prove a number of assertions on the existence and properties ofdetermining functionals for processes generated in some separable Hilbert space by an equation of the form

, , . (4.1)

Here is a positive operator with discrete spectrum (for definition see Section 2.1)and is a continuous mapping from into possessing the pro-perties

, (4.2)

for all and for all such that , where is an arbitrarypositive number, and are positive numbers.

Assume that problem (4.1) is uniquely solvable in the class of functions

and is pointwise dissipative, i.e. there exists such that

when (4.3)

for all . Examples of problems of the type (4.1) with the properties listedabove can be found in Chapter 2, for example.

The results obtained in Sections 1 and 2 enable us to establish the following as-sertion.

Theorem 4.1.

For the set of linear functionals on the spaceFor the set of linear functionals on the spaceFor the set of linear functionals on the spaceFor the set of linear functionals on the space

with the norm to be -asymptotically with the norm to be -asymptotically with the norm to be -asymptotically with the norm to be -asymptotically

determining for problem determining for problem determining for problem determining for problem (4.1) under conditions under conditions under conditions under conditions (4.2) and and and and (4.3), it is suffi- it is suffi- it is suffi- it is suffi-

cient that the completeness defect satisfies the inequalitycient that the completeness defect satisfies the inequalitycient that the completeness defect satisfies the inequalitycient that the completeness defect satisfies the inequality

,,,, (4.4)

where and are the same as in where and are the same as in where and are the same as in where and are the same as in (4.2) and and and and (4.3)....

Proof.

We consider two solutions and to problem (4.1) that lie in .By virtue of dissipativity property (4.3) we can suppose that

, , . (4.5)

H

udtd

�� Au� B u t�� t 0� ut 0� u0�

A

B u t�� D A1 2�� R� H

B 0 t�� M0� B u1 t�� B u2 t�� M K� � A1 2� u1 u2��

t uj

D A1 2�� A1 2� uj

K� KM0 M K� �

� C 0 +�� H�� C 0 +�� D A1 2��

R 0�

A1 2� u t� � R� t t0 u� ��

u t� � �

� lj: j 1 2 � N� � ��

V D A1 2�� . V A1 2� .

H� V V ��

+�

V H��

+�

+�

V H�� M R� � 1��

M K� � R

u1 t� � u2 t� � �

A1 2� uj

t� � R� t 0� j 1 2��


C

h

a

p

t

e

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5

Let . If we consider as a solution to the linear problem

,

then it is easy to find that

for all . We use (2.11) to obtain that

for any , where

.

Therefore,

. (4.6)

Using (4.4) we can choose the parameter such that .Thus, we can apply Theorem 1.1 and find that under condition (4.4) equation

implies the equality

. (4.7)

In order to complete the proof of the theorem we should obtain

(4.8)

from (4.7). To prove (4.8) it should be first noted that

(4.9)

for any . Indeed, the interpolation inequality (see Exercise 2.1.12)

, ,

and dissipativity property (4.5) enable us to obtain (4.9) from equation (4.7). Nowwe use the integral representation of a weak solution (see (2.2.3)) and the methodapplied in the proof of Lemma 2.4.1 to show (do it yourself) that

u t� � u1 t� � u2 t� �� u t� �

udtd

�� Au� f t� � B u1 t� � t�� B u2 t� � t��

12�� u t� � 2 Au �� u �� d

s

t

�� 12�� u s� � 2 M R� � A1 2� u �� u �� d

s

t

��

t s 0� �

M R� � A1 2� u u� � +�

M R� � A1 2� u2� � A1 2� u

2, C R � ,� �� N u� �� 2� ��

, 0�

N u� � lj

u� � : j 1 2 � N� � �� max�

u t� � 2 2 1 ,� +�

M R� �� A1 2� u �� 2 �d

s

t

��

u s� � 2C R � ,� �� N u �� 2 �d

s

t

��

, 0� 1 ,� +�

M R� �� 0�

N u1 �� u2 �� 2 �d

t

t 1�

�t ��lim 0�

u1 t� � u2 t� ��t ��lim 0�

A1 2� u1 t� � u2 t� �� t ��lim 0�

A? u1 t� � u2 t� �� t ��lim 0�

0 ? 1 2/��

A?u u1 2?�

A1 2� u2?�� 0 ? 1 2/��


, , .

Therefore, using the interpolation inequality

, ,

we obtain (4.8) from (4.9). Theorem 4.1 is proved.

Show that if the hypotheses of Theorem 4.1 hold, then equa-tion (4.9) is valid for all .

The reasonings in the proof of Theorem 4.1 lead us to the following assertion.

Corollary 4.1.

Assume that the hypotheses of Theorem 4.1 hold. Then for any two weak

(in ) solutions and to problem (4.1) that are boun-

ded on the whole axis,

, , (4.10)

the condition for and implies that

.

Proof.

In the situation considered equation (4.6) implies that

for all and some . It follows that

, .

Therefore, if we tend , then using (4.10) we obtain that for all , i.e. .

It should be noted that Corollary 4.1 means that solutions to problem (4.1) that arebounded on the whole axis are uniquely determined by their values on the functionals

. It was this property of the functionals which was used by Ladyzhenskaya [2]to define the notion of determining modes for the two-dimensional Navier-Stokessystem. We also note that a more general variant of Theorem 4.1 can be found in [3].

Assume that problem (4.1) is autonomous, i.e. . Let be a global attractor of the dynamical system

generated by weak (in ) solutions to problem (4.1) andassume that a set of functionals possesses

A2uj

t� � C R M0�� 12�� 2 1� � t t0�

A1 2� u A1 2/ ,� u A1 2/ ,� u�� 0 , 12��

E x e r c i s e 4.1

0 ? 1��

D A1 2�� u1 t� � u2 t� �

A1 2� ui

t� � : �� t �� " #$ %

sup R� i 1 2��

lj

u1 t� �� lj

u2 t� �� lj

� t Ru1 t� � u2 t� �

u t� � 2 2�

A1 2� u �� 2 �d

s

t

�� u s� � 2�

t s� 2�

0�

u t� � e2�

t s�� u s� �� t s�

s �� u t� � 0�t R u1 t� � u2 t� �

lj lj� �

E x e r c i s e 4.2 B u t�� B u� � V S

t��

V D A1 2�� l

j: j 1 � N� ��


C

h

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p

t

e

r

5

property (4.4). Then for any pair of trajectories and lying in the attractor the condition impliesthat for all and .

Theorems 4.1 and 2.4 enable us to obtain conditions on the existence of deter-mining functionals.

Corollary 4.2.

Assume that the Kolmogorov -width of the embedding of the space

into possesses the property . Then

there exists a set of asymptotically -determining functionals

for problem (4.1) consisting of elements.

Theorem 2.4, Corollary 2.1, and Exercise 2.6 imply that if the hypotheses of Theorem4.1 hold, then the family of functionals given by equation (2.13) is a -determining set for problem (4.1), provided . Here and arethe constants from (4.2) and (4.3). It should be noted that the set of the form(2.13) for problem (4.1) is often called a set of determining modes determining modes determining modes determining modes.... Thus, Theo-rem 4.1 and Exercise 2.6 imply that semilinear parabolic equation (4.1) possessesa finite number of determining modes.

When condition (4.2) holds uniformly with respect to , we can omit the re-quirement of dissipativity (4.3) in Theorem 4.1. Then the following assertion is valid.

Theorem 4.2.

Assume that a continuous mapping from into Assume that a continuous mapping from into Assume that a continuous mapping from into Assume that a continuous mapping from into

possesses the propertiespossesses the propertiespossesses the propertiespossesses the properties

,,,, (4.11)

for all for all for all for all .... Then a set of linear functionals Then a set of linear functionals Then a set of linear functionals Then a set of linear functionals

on is asymptotically -determining for problem on is asymptotically -determining for problem on is asymptotically -determining for problem on is asymptotically -determining for problem (4.1),,,,

provided provided provided provided ....

Proof.

If we reason as in the proof of Theorem 4.1, we obtain that if , thenfor an arbitrary pair of solutions and emanating from the points and

at a moment the equation (see (4.6))

(4.12)

is valid. Here , and are positive con-stants, and

.

u1 t� � u2 t� � l

ju1 t� �� l

ju2 t� ��

u1 t� � u2 t� � t R j 1 2 � N� � ��

N

N

V D A1 2�� H <N V H�� M R� � 1��V V ��

N

� V V �� 7

N 1� M R� �2� M R� � R

�

K

B u t�� D A1 2�� R� H

B 0 t�� M0� B u1 t�� B u2 t�� M A1 2� u1 u2��

uj

D A1 2�� lj: j 1 2 � N� � ��

V D A1 2�� V V �� +�

+�

V H�� M 1��

+�

M 1��u1 t� � u2 t� � u1

u2 s

u t� � 2 2 A1 2� u �� 2 �d

s

t

�� u s� � 2C N u �� 2 �d

s

t

��

u t� � u1 t� � u2 t� �� 2 2 +�

M�� C � M��

N u� � lj

u� � : j 1 2 � N� � �� max�


Therefore, using Theorem 1.1 we conclude that the condition

(4.13)

implies that

. (4.14)

Since is a solution to the linear equation

, (4.15)

it is easy to verify that

(4.16)

for . It should be noted that equation (4.16) can be obtained with the helpof formal multiplication of (4.15) by with subsequent integration. This conver-sion can be grounded using the Galerkin approximations. If we integrate equation(4.16) with respect to from to , then it is easy to see that

.

Using the structure of the function and inequality (4.11), we obtain that

.

Consequently, (4.14) gives us that

.

Therefore, Theorem 4.2 is proved.

Further considerations in this section are related to the problems possessing inertialmanifolds (see Chapter 3). In order to cover a wider class of problems, it is conve-nient to introduce the notion of a process.

Let be a real reflexive Banach space. A two-parameter family of continuous mappings acting in is said to be evolutionaryevolutionaryevolutionaryevolutionary ,

if the following conditions hold:(a) , , .(b) is a strongly continuous function of the variable .

N u �� 2 �d

t

t 1�

�t ��lim 0�

u t� � 2A1 2� u �� 2 �d

t

t 1�

�� " #$ %

t ��lim 0�

u t� � u1 t� � u2 t� ��

udtd

�� Au� f t� � B u1 t� � t�� B u2 t� � t��

A1 2� u t� � 2A1 2� u s� � 2 1

2�� f �� 2 �d

s

t

��

t s�u� t� �

s t 1� t

A1 2� u t� � A1 2� u s� � 2sd

t 1�

t

� 12�� f �� 2 �d

t 1�

t

��

f t� �

A1 2� u t� � 1 M2

2��& '

( ) A1 2� u �� 2 �d

t 1�

t

��

A1 2� u1 t� � u2 t� �� t ��lim 0�

H S t �� t � � t R�� H

S t s�� S s �� S t �� t s �� S t t�� I�S t s�� u0 t


C

h

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5

A pair with being an evolutionary family in is often calleda processprocessprocessprocess . Therewith the space is said to be a phase space and the familyof mappings is called an evolutionary operator. A curve

is said to be a trajectory of the process emanating from the point at the moment .It is evident that every dynamical system is a process. However, main

examples of processes are given by evolutionary equations of the form (1.1). There-with the evolutionary operator is defined by the obvious formula

,

where is the solution to problem (1.1) with the initial condition at the moment .

Assume that the conditions of Section 2.2 and the hypothesesof Theorem 2.2.3 hold for problem (4.1). Show that weak solutionsto problem (4.1) generate a process in .

Similar to the definitions of Chapter 3 we will say that a process actingin a separable Hilbert space possesses an asymptotically complete finite-dimen-sional inertial manifold if there exist a finite-dimensional orthoprojector inthe space and a continuous function such that

(a) (4.17)

for all , , where is a positive constant;(b) the surface

(4.18)

is invariant: ;(c) the condition of asymptotical completeness holds: for any and

there exists such that

, , (4.19)

where and are positive constants which may depend on and.

Show that for any two elements , the fol-lowing inequality holds

.

Using equation (4.19) prove that

for .

H S t �� S t �� H

H

S t ��

�s

u0� � u t� � S t s s�� u0 : t 0��

u0 s

H St��

S t s�� u0 u t s u0��

u t� � u t s u0�� u0 s

E x e r c i s e 4.3

H

V S t �� V

Mt

� � P

V L p t�� : PV R 1 P�� V

L p1 t�� L p2 t�� V

L p1 p2�V

�

pj P V t R L

Mt

p L p t�� : p P V�� V0

S t �� M� Mt

0s R

u0 V u0* M

s

S t s�� u0 S t s�� u0*�

VC e � t s�� t s�

C � u0s R

E x e r c i s e 4.4 u1 u2� Ms

s R

1 L2�� 1 2��u1 u2�

VP u1 u2��

Vu1 u2�

V� �

E x e r c i s e 4.5

1 P�� S t s�� u0 L P S t s�� u0 t�� V

1 L�� C e � t s��

t s�


Show that for any two trajectories , ,of the process the condition

implies that

.

In particular, the results of these exercises mean that is homeomorphic to a sub-set in , , for every . The corresponding homeomorphism

can be defined by the equality , where is a ba-sis in . Therewith the set of functionals appearsto be asymptotically determining for the process. The following theorem containsa sufficient condition of the fact that a set of functionals possesses the proper-ties mentioned above.

Theorem 4.3.

Assume that and Assume that and Assume that and Assume that and are are are are separable Hilbert spaces such that is con-separable Hilbert spaces such that is con-separable Hilbert spaces such that is con-separable Hilbert spaces such that is con-

tinuously and densely embedded into tinuously and densely embedded into tinuously and densely embedded into tinuously and densely embedded into .... Let a process possess Let a process possess Let a process possess Let a process possess

an asymptotically complete finite-dimensional inertial manifold an asymptotically complete finite-dimensional inertial manifold an asymptotically complete finite-dimensional inertial manifold an asymptotically complete finite-dimensional inertial manifold ....

Assume that the orthoprojector from the definition of can be conti-Assume that the orthoprojector from the definition of can be conti-Assume that the orthoprojector from the definition of can be conti-Assume that the orthoprojector from the definition of can be conti-

nuously extended to the mapping from into nuously extended to the mapping from into nuously extended to the mapping from into nuously extended to the mapping from into ,,,, i.e. there exists a constant i.e. there exists a constant i.e. there exists a constant i.e. there exists a constant

such that such that such that such that

,,,, .... (4.20)

If If If If is ais ais ais a set of linear functionals on such that set of linear functionals on such that set of linear functionals on such that set of linear functionals on such that

,,,, (4.21)

then the following conditions are valid:then the following conditions are valid:then the following conditions are valid:then the following conditions are valid:

1) there exist positive constants and depending on such thatthere exist positive constants and depending on such thatthere exist positive constants and depending on such thatthere exist positive constants and depending on such that

(4.22)

for all for all for all for all ,,,, ; i.e. the mapping acting from into ; i.e. the mapping acting from into ; i.e. the mapping acting from into ; i.e. the mapping acting from into

according to the formula is a Lipschitzian ho- according to the formula is a Lipschitzian ho- according to the formula is a Lipschitzian ho- according to the formula is a Lipschitzian ho-

meomorphism from into for every ;meomorphism from into for every ;meomorphism from into for every ;meomorphism from into for every ;

2) the set of functionals is determining for the process inthe set of functionals is determining for the process inthe set of functionals is determining for the process inthe set of functionals is determining for the process in

the sense that for any two trajectories the conditionthe sense that for any two trajectories the conditionthe sense that for any two trajectories the conditionthe sense that for any two trajectories the condition

implies that implies that implies that implies that

....

E x e r c i s e 4.6 uj

t� � S t s�� uj

� j 1 2��V S t ��

P u1 t� � u2 t� �� Vt ��

lim 0�

u1 t� � u2 t� ��Vt ��

lim 0�

Ms

RN N Pdim� s Rr : V RN� r u u .

j��

V� �

j 1�N� .

j� �

PV lj

u� � u 8j

�� V

: j 1 � N� ��

lj

� �

V H V

H V S t �� M

t� �

P Mt

� �H V

I I P� � 0��Pv

VI v

H�� v V

� lj : j 1 � N� �� V

+�

V H�� 1 L2�� 1 2�� I 1��

c1 c2 �

c1 u1 u2�H

lj

u1 u2�� : j 1 � N� �� max c2 u1 u2�H

� �

u1 u2� Mt

t R r�

V

RN r�

u lj

u� �� j 1�N�

Mt

RN t R� V S t ��

uj


�

lj

u1 t� �� lj

u2 t� �� t ��lim 0�

u1 t� � u2 t� ��Vt ��

lim 0�


C

h

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5

Proof.

Let . Then

, .

Therewith equation (4.17) gives us that

, . (4.23)

Consequently, using Theorem 2.1 and inequality (4.20) we obtain that

,

where and . Therefore, equa-tion (4.21) implies that

,

where .

On the other hand, (4.20) and (4.23) give us that

. (4.25)

Equations (4.24) and (4.25) imply estimate (4.22). Hence, assertion 1 of the theo-rem is proved.

Let us prove the second assertion of the theorem. Let , ,be trajectories of the process. Since

,

using (4.17) it is easy to find that

The property of asymptotical completeness (4.19) implies (see Exercise 4.5) that

, .

Therefore, equation (4.20) gives us the estimate

. (4.26)

It follows from Theorem 2.1 that

.

Therefore, provided (4.21) holds, equation (4.26) implies that

, ,

u1 u2� Mt

uj

Puj

L Puj

t�� j 1 2��

u1 u2�V

1 L2�� 1 2�Pu1 Pu2�

V� uj Mt

u1 u2�H

C�

N�

u1 u2�� +�

1 L2�� 1 2� I u1 u2�H

��

N�

u� � lj

u� � : j 1 � N� �� max� +�

+�

V H��

u1 u2�H

C1 �� N�

u1 u2��

C1 �� C�

1 +�

1 L2�� 1 2� I�� 1��(4.24)

lj

u1 u2�� C�

u1 u2�V

C�

1 L2�� 1 2� I u1 u2�H

� �

uj


� t s�

uj

t� � Puj

t� � L Puj

t� � t�� 1 P�� uj

t� � L Puj

t� � t��

u1 t� � u2 t� ��V

1 L2�� 1 2�P u1 t� � u2 t� ��

V��

1 P�� uj

t� � L Puj

t� � t�� V

.j 1 2��*�

1 P�� uj

t� � L Puj

t� � t�� C e � t s�� t s�

u1 t� � u2 t� ��V

1 L2�� 1 2� I u1 t� � u2 t� ��H

� C e � t s��

u1 t� � u2 t� ��H

C�

N�

u1 t� � u2 t� �� +�

u1 t� � u2 t� ��V

��

u1 t� � u2 t� ��V

A�

N�

u1 t� � u2 t� �� B�

e � t s�� t s�


where and are positive numbers. Hence, the condition

implies that

.


Assume that the hypotheses of Theorem 4.3 hold. Let be such that for all . Show that

for .

Prove that if the hypotheses of Theorem 4.3 hold, then in-equality (4.22) as well as the equation

is valid for any and , where are con-stants depending on .

Let us return to problem (4.1). Assume that is a continuous mapping from into , , possessing the properties

,

for all , . Assume that the spectral gap condition

holds for some and . Here are the eigenvalues of the opera-tor indexed in the increasing order and is a constant defined by (3.1.7). Underthese conditions there exists (see Chapter 2) a process generatedby problem (4.1). By virtue of Theorems 3.2.1 and 3.3.1 this process possesses anasymptotically complete finite-dimensional inertial manifold and the corres-ponding orthoprojector is a projector onto the span of the first eigenvectorsof the operator . Therefore,

, .

Therewith the Lipschitz constant for can be estimated by the value. Thus, if is a set of functionals on , then in order to apply

Theorem 4.3 with , , it is sufficient to require that

.

A�

B�

N�

u1 t� � u2 t� �� t ��lim 0�

u1 t� � u2 t� ��Vt ��

lim 0�

E x e r c i s e 4.7

u1 u2 Ms

� lj

u1� � lj

u2� �� lj

�

S t s�� u1 S t s�� u2 t s�

E x e r c i s e 4.8

c1 u1 u2�V

lj

u1 u2�� : j 1 2 � N� � �� max c2 u1 u2�V

� �

u1 u2� Mt t R c1 c2� 0��

B u t�� D A?� � R� H 0 ? 1��

B u0 t�� M 1 A?u0�& '( )� B u1 t�� B u2 t�� M A? u1 u2��

uj

D A?� � 0 ? 1��

7n 1� 7n� 2M

q�� 1 k�� 7n 1�

? 7n?��

n 0 q 2 2�� 7n

� �A k

D A?� � S t s��

Mt

� �P n

A

A? Pu 7n? s�

Asu� � s ?� ��

L L p t�� q 1 q�� / � V D A?� ��

H D As� �� s ?� ��

+�

D A?� � D As� �� 1 q�

2 2 q� 2 q2�� 1

7n? s��


C

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5

Due to Theorem 2.4 this estimate can be rewritten as follows:

,

where is the Kolmogorov -width of the embedding of into , , .

It should be noted that the assertion similar to Theorem 4.3 was first estab-lished for the Kuramoto-Sivashinsky equation

, , ,

with the periodic boundary conditions on in the case when is a set ofuniformly distributed nodes on , i.e.

, where , .

For the references and discussion of general case see survey [3].

In conclusion of this section we give one more theorem on the existence of deter-mining functionals for problem (4.1). The theorem shows that in some cases we canrequire that the values of functionals on the difference of two solutions tend to zeroonly on a sequence of moments of time (cf. Theorem 1.3).

Theorem 4.4.

As before, assume that is a positive operator with discrete spectrum:As before, assume that is a positive operator with discrete spectrum:As before, assume that is a positive operator with discrete spectrum:As before, assume that is a positive operator with discrete spectrum:

,,,, ,,,, ....

Assume that Assume that Assume that Assume that is a continuous mapping from into foris a continuous mapping from into foris a continuous mapping from into foris a continuous mapping from into for

some and the estimatesome and the estimatesome and the estimatesome and the estimate

,,,, ,,,,

holds. Let be a finite set of linear functionals on holds. Let be a finite set of linear functionals on holds. Let be a finite set of linear functionals on holds. Let be a finite set of linear functionals on .... Then for Then for Then for Then for

any there exists such that the conditionany there exists such that the conditionany there exists such that the conditionany there exists such that the condition

implies that the set of functionals is determining for problem implies that the set of functionals is determining for problem implies that the set of functionals is determining for problem implies that the set of functionals is determining for problem (4.1)in the sense that if for some pair of solutions and for some se-in the sense that if for some pair of solutions and for some se-in the sense that if for some pair of solutions and for some se-in the sense that if for some pair of solutions and for some se-

quence such thatquence such thatquence such thatquence such that

,,,, ,,,, ,,,,

the conditionthe conditionthe conditionthe condition

,,,, ,,,,

holds, thenholds, thenholds, thenholds, then

.... (4.27)

+�

D A?� � D As� �� 1 q�

2 2 q� 2 q2��

7n 1�7n

��& '( )? s�

<n

D A?� � D As� ��

<n

D A?� � D As� �� n D A?� �D As� � � s ?� �� 0 ? 1��

ut

ux x x x

ux x

ux

u� � � 0� x 0 L�� t 0�

0 L�� lj

� �0 L��

lj

u� � u j h� �� h LN�� j 0 1 � N� 1��

A

Aek 7k ek� 0 71 72 �� 7kk ��lim ��

B u t�� D A?� � R� H

0 ? 1� �

B u1 t�� B u2 t�� M A? u1 u2�� u1 u2� D A?� �

� lj

� �� D A?� �0 � 2� � n n � 2 ? 71 M� � � ��

+�

+�

D A?� � H�� 12�� 7

n?��

�

u1 t� � u2 t� �� tk� �

tk

k ��lim �� t

k 1� tk

� 2� � k 1�

lj

u1 tk

� � u2 tk

� �� k ��lim 0� l

j�

A? u1 t� � u2 t� �� t ��lim 0�


Proof.

Let . Then the results of Chapter 2 (see Theorem 2.2.3 andExercise 2.2.7) imply that

, (4.28)

(4.29)

for , where , , and are positive numbers depending on , and and is the orthoprojector onto . It follows form (4.29) that

, , (4.30)

where

.

Let us choose such that . Then equation (4.30)gives us that

, .

This inequality as well as estimate (4.28) enables us to use Theorem 1.3 with and to complete the proof of Theorem 4.4.

Assume that is chosen such that in the proofof Theorem 4.4. Show that the condition as implies (4.27).

The results presented in this section can also be proved for semilinear retardedequations. For example, we can consider a retarded perturbation of problem (4.1)of the following form

where, as usual (see Section 2.8), is an element of defined with the helpof by the equality , , and is a continuous map-ping from into possessing the property

for any . The corresponding scheme of reasoning is similar to the me-thod used in [3], where the second order in time retarded equations are considered.

u t� � u1 t� � u2 t� ��

A?u t� � a1 ea2 t s��

A?u s� ��

1 Pn�� A?u t� � e7

n 1�� t s�� a3

7n 1�1 ?�� e

a2 t s�� " #$ %

A?u s� ��

t s 0� � a1 a2 a3 ? 71�M P

nLin e1 � e

n� ��

1 Pn

�� A?u t� � q� 2� A?u s� �� s �� t s 2��

q� 2� e7

n 1�� a3

7n 1�1 ?�� e

a22��

n n � 2 ? 71 M� � � �� q� 2� 1 2/�

A?u t� � 7n?

u t� � 12�� A?u s� �� s �� t s 2��

V �D A?� ��

E x e r c i s e 4.9 n q� 2� 1�P

nu1 t

k� � u2 t

k� �� 0�

k ��

udtd

�� A u� B u t�� Q ut

t�� ,��

ut r 0�� 8 t� � C

rC r� 0�� D A1 2/� �� ,��

!"!$

ut

Cr

u t� � ut?� � u t ?�� ? r� 0�� Q

Cr

R� H

Q v1 t�� Q v2 t�� 2M1 A1 2� v1 ?� � v2 ?� ��

2 ?d

r�

0

��

v1 v2� Cr


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§ 5 Determining Functionals§ 5 Determining Functionals§ 5 Determining Functionals§ 5 Determining Functionals

for for for for Reaction-Diffusion SystemsReaction-Diffusion SystemsReaction-Diffusion SystemsReaction-Diffusion Systems

In this section we consider systems of parabolic equations of the reaction-diffusiontype and find conditions under which a finite set of linear functionals given on thephase space uniquely determines the asymptotic behaviour of solutions. In particu-lar, the results obtained enable us to prove the existence of finite collections of de-termining modes, nodes, and local volume averages for the class of systems underconsideration. It also appears that in some cases determining functionals can be gi-ven only on a part of components of the state vector. As an example, we considera system of equations which describes the Belousov-Zhabotinsky reaction and theNavier-Stokes equations.

Assume that is a smooth bounded domain in , , is theSobolev space of the order on , and is the closure (in ) of the setof infinitely differentiable functions with compact support in . Let be a normin and let and be a norm and an inner product in , respec-tively. Further we also use the spaces

, .

Notations and have a similar meaning. We denote the norms and the innerproducts in and as in and .

We consider the following system of equations

, , , (5.1)

, ,

as the main model. Here , is the Laplace ope-rator, , and is an -by- matrix with the ele-ments from such that for all and

, . (5.2)

We also assume that the continuous function

is such that problem (5.1) has solutions which belong to a class of functionson with the following properties:

a) for any there exists such that

, , (5.3)

where is the space of strongly continuous functions on with the values in ;

@ Rn n 1� Hs @� �s @ H0

s @� � Hs @� �@ .

s

Hs @� � . . . �� L2 @� �

HHHHs Hs @� �� m Hs @� � � Hs @� �� m 1�

LLLL2 HHHH0s

LLLL2 HHHHs L2 @� � Hs @� �

Atu a x t�� uM f x u Eu t�� x @ t 0�

u A@ 0� u x 0�� u0 x� ��

u x t�� u1 x t�� um

x t�� MEu

kAx1u

k� Axn

uk

� �� a x t�� m m

L� @ R+�� x @ t R+

a+ x t�� 12�� a a*�� 60 I�� 60 0�

f f1 � fm

�� : @ R n 1�� m R+ Rm��

�

R+u � t0 0�

u t� � C t0 +� HHHH2 HHHH01 �� A

tu t� � C t0 +� LLLL2��

C a b X�� a b�� X

D e t e r m i n i n g F u n c t i o n a l s f o r R e a c t i o n - D i f f u s i o n S y s t e m s 329

b) there exists a constant such that for any there exists such that for we have

. (5.4)

It should be noted that if is a diagonal matrix with the elements from and is a continuously differentiable mapping such that

(5.5)

for all , , , and , then under natural compa-tibility conditions problem (5.1) has a unique classical solution [7] which evidentlypossesses properties (5.3) and (5.4). In cases when the dynamical system generatedby equations (5.1) is dissipative, the global Lipschitz condition (5.5) can beweakened. For example (see [8]), if is a constant matrix and

,

where , , and is continuously differentiable and satisfies the conditions

, , ,

, , ,

where and for , then any solutionto problem (5.1) with the initial condition from is unique and possesses proper-ties (5.3) and (5.4).

Let us formulate our main assertion.

Theorem 5.1.

A set A set A set A set of linearly independent continuous linear of linearly independent continuous linear of linearly independent continuous linear of linearly independent continuous linear

functionals on functionals on functionals on functionals on is an asymptotically determining set with respect is an asymptotically determining set with respect is an asymptotically determining set with respect is an asymptotically determining set with respect

to the space to the space to the space to the space for problem for problem for problem for problem (5.1) in the class in the class in the class in the class ifififif

,,,, (5.6)

where , and and arewhere , and and arewhere , and and arewhere , and and are

constants from constants from constants from constants from (5.2) and and and and (5.4).... This means that if inequality This means that if inequality This means that if inequality This means that if inequality (5.6) holds, holds, holds, holds,

then for some pair of solutions then for some pair of solutions then for some pair of solutions then for some pair of solutions the equationthe equationthe equationthe equation

,,,, ,,,, (5.7)

k 0� u v� �t1 0� t t1�

f u Eu t�� f v Ev t�� k u v� Eu Ev��& '( )��

a x t�� C2 @ R+�� f

f x u p t�� f x v q� t�� k u v� p q��

u v� Rm p q� Rn m x @ t R+

a

f x u Eu t�� f x u�� bj

x� � Ax

ju g x� ��

j 1�

n

*��

bj

bj1 � b

jm� �� LLLL�� g g1 � g

m� �� LLLL2� f f1 � fm� ��

f x u�� u 61 up0� f x u�� 62 u

p0 1�C�� p0 2�

fkAu

jA��

C 1 up1�� 1 k m� � 1 j n� �

61 62� 0� p1 4 n/ 2 n 2�� /�� min� n 2�LLLL2

� lj: j 1 � N� ��

HHHH2 HHHH01

HHHH01

�

+�

HHHH2 HHHH01 L2��

c060

k 2�� 1 27

4��

k60��& '

( ) 2��& '

( ) 1 2�� p k 60��

c01� w 2 : w HHHH2 HHHH0

1 � � @� � , wM 1�� sup� 60 k

u v� �

lj u �� lj v �� 2 �d

t

t 1�

�t ��lim 0� j 1 � N� ��


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implies their asymptotic closeness in the space implies their asymptotic closeness in the space implies their asymptotic closeness in the space implies their asymptotic closeness in the space ::::

.... (5.8)

Proof.

Let . Then equation (5.1) for gives us that

.

If we multiply this by in scalarwise and use equation (5.4), then we find that

for large enough. Therefore, the inequality enables us to obtain the estimate

. (5.9)

Theorem 2.1 implies that

(5.10)

for all and for any , where is a constant and . Consequently, estimate (5.9) gives us that

,

where is defined by equation (5.6). It follows that if estimate (5.6) is valid,then there exists such that

for all , where is large enough. Therefore, equation (5.7) implies (5.8).Thus, Theorem 5.1 is proved.

Assume that and are two solutions to equation (5.1)defined for all . Let (5.3) and (5.4) hold for every andlet

.

Prove that if the hypotheses of Theorem 5.1 hold, then equalities for and for some im-

ply that for all .

HHHH1

u t� � v t� �� 1t ��lim 0�

u v� � w u v��

Atw a x t�� wM f x u Eu t�� f x v Ev t��

wM LLLL2

12��

dtd

�� w t� �M 2� 60 w t� �M 2� k w t� � wE� � t� ��& '( ) w t� �M��

t 0� Ew2

w wM�� w wM��

dtd

�� Ew2 60 wM 2� 2 k2

60�� 1 27

4�� k

60��& '

( )2�& '

( ) w2�

w2

C N ,�� lj

w� � 2 1 ,�� +�

2w 2

2� ��j

max�

w HHHH2 HHHH01 , 0� C N ,�� 0� +

�

+�

HHHH2 HHHH01 L2��

dtd

�� Ew t� � 2 60 11 ,�� +

�

2

p k 60�� 2��

& 'F G( )

w t� �M 2� �� C lj w� � 2

jmax��

p k 60�� 2 0�

Ew t� � 2e

2 t t0�� Ew t0� � 2C e 2 t �� lj w �� 2

jmax �d

t0

t

��

t t0� t0

E x e r c i s e 5.1 u t� � v t� �t R t0 R

Eu t� � Ev t� ��& '( )

t 0�sup ��

lj

u t� �� lj

v t� �� j 1 � N� �� t s� s ��u t� � v t� � t s�


Let us give several examples of sets of determining functionals for problem (5.1).

E x a m p l e 5.1 (determining modes, , , , )

Let be eigenelements of the operator in with the Dirichlet bounda-ry conditions on and let be the corresponding eigenva-lues. Then the completeness defect of the set

can be easily estimated as follows: (see Exer-cise 2.6). Thus, if possesses the property , then is a set of asymptotically -determining functionals for prob-lem (5.1).

Considerations of Section 5.3 also enable us to give the following examples.

E x a m p l e 5.2 (determining generalized local volume averages)

Assume that the domain is divided into local Lipschitzian subdomains, with diameters not exceeding some given number .

Assume that on every domain a function is given suchthat the domain is star-like with respect to and the conditions

, ,

hold, where the constant does not depend on and . Theorem 3.1 im-plies that

for the set of functionals

.

Therewith the reasonings in the proof of Theorem 3.1 imply that ,where is the area of the unit sphere in . For every we define thefunctionals on by the formula

, , . (5.11)

Let

. (5.12)

m 1�

ek

� � M� LLLL2

A@ 0 71 72 ��

� lj : lj w� � w x� � ej x� �� xd

@�� , j 1 � N� ��

� " #$ %

�

+�

HHHH2 HHHH01 LLLL2�� n 7N 1�

1��N 7N 1� n p k 60�� 1��

HHHH2 HHHH01 HHHH0

1��

@@

j: j 1 � N� �� h 0�

@j

7j

x� � L� @j

� �@

jsupp 7

j

7j x� � xd

@j

� 1� ess 7j x� �x @

jsup I

hn��

I 0� h j

+�

hH2 @� � H0

1 @� � L2 @� �� cn

h2 I2��

�h lj u� � 7j x� � u x� � xd

@j

�� , j 1 2 � N� � �� " #$ %

�

cn

�n2 n 2��

�n

Rn lj

�h

lj

k� � HHHH2

lj

k� �w� � l

jw

k� �� w w1 � w

m� �� HHHH2� k 1 2 � m� � ��

�h

m� � lj

k� �u� � : k 1 2 � m� � �� , l

j�

h� ��


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One can check (see Exercise 2.8) that

.

Therefore, if is small enough, then is a set of asymptotically deter-mining functionals for problem (5.1).

E x a m p l e 5.3 (determining nodes, )

Let be a convex smooth domain in , . Let and let

,

where . Let us choose a point in every subdomain and define the set of nodes

, . (5.13)

Theorem 3.2 enables us to state that

,

where is an absolute constant. Therefore, the set of functionals definedby formulae (5.11) and (5.12) with given by equality (5.13) possesses theproperty

.

Consequently, is a set of asymptotically determining functionals for prob-lem (5.1) in the class , provided that is small enough.

It is also clear that the result of Exercise 2.8 enables us to construct mixed determin-ing functionals: they are determining nodes or local volume averages dependingon the components of a state vector. Other variants are also possible.

However, it is possible that not all the components of the solution vector appear to be essential for the asymptotic behaviour to be uniquely determined.A theorem below shows when this situation can occur.

Let be a subset of . Let us introduce the spaces

, .

We identify these spaces with , where is the number of elements ofthe set . Notations and have the similar meaning. The set of linearfunctionals on is said to be determining if is determining, where is thenatural projection of onto .

+�

h

m� � HHHH2 HHHH01

LLLL2� � � cm

h2 I2� ��

h �h

m� �

n 3�

@ Rnn 3� h 0�

@j @ x x1 � xn� �� : jk h xk jk 1�� h , k 1 2 3� ��

j j1 � jn

� �� Zn @ � xj

@j

�h

lj

u� � u xj

� �� : j Zn @ � �� n 3�

+�

hH2 @� � H0

1 @� � L2 @� �� c h2��

c �h

m� �

�h

+�

h

m� � HHHH2 HHHH01

LLLL2� � � c h2��

�h

m� �

� h

u x t��

I 1 � m� ��

HHHHIs

w w1 � wm� �� HHHHs : wk 0 k I5�� s R

Hs @� �� I I

I LLLLI2

HHHH0 I�s

�

HHHHI2

pI*� p

I

HHHHs HHHHIs


Theorem 5.2.

Let Let Let Let be a diagonal matrix with constant elementsbe a diagonal matrix with constant elementsbe a diagonal matrix with constant elementsbe a diagonal matrix with constant elements

and let be a partition of the set into two disjoint subsets.and let be a partition of the set into two disjoint subsets.and let be a partition of the set into two disjoint subsets.and let be a partition of the set into two disjoint subsets.

Assume that there exist positive constants , where Assume that there exist positive constants , where Assume that there exist positive constants , where Assume that there exist positive constants , where ,,,,

such that for any pair of solutions the following inequality holdssuch that for any pair of solutions the following inequality holdssuch that for any pair of solutions the following inequality holdssuch that for any pair of solutions the following inequality holds

(hereinafter )(hereinafter )(hereinafter )(hereinafter )::::

(5.14)

Then a set of linearly independent continuous linearThen a set of linearly independent continuous linearThen a set of linearly independent continuous linearThen a set of linearly independent continuous linear

functionals on is an asymptotically determining set with respectfunctionals on is an asymptotically determining set with respectfunctionals on is an asymptotically determining set with respectfunctionals on is an asymptotically determining set with respect

to the space for problem to the space for problem to the space for problem to the space for problem (5.1) in the class if in the class if in the class if in the class if

,,,, (5.15)

where is defined as in where is defined as in where is defined as in where is defined as in (5.6).... This means that if two solutions This means that if two solutions This means that if two solutions This means that if two solutions

possess the property possess the property possess the property possess the property

forforforfor ,,,, (5.16)

where is the natural projection where is the natural projection where is the natural projection where is the natural projection of of of of onto , then equation onto , then equation onto , then equation onto , then equation (5.8)holds.holds.holds.holds.

Proof.

Let and . Then

(5.17)

for . In we scalarwise multiply equations (5.17) by for and by for and summarize the results. Using inequality (5.14)we find that

,

where

.

a diag d1 � dm

� �� I I 9�� 1 � m� ��

J k* ?i

� � i 1 � m� ��u v� �

w u v��

?i

di

2�� w

iM 2� f

iu Eu t�� f

iv Ev t�� w

iM��

� " #$ %

i I*

?i di Ewi2� fi u Eu t�� fi v Ev t�� wi��

� " #$ %

i I 9*

�

�

J wi

2k* w

i2 .

i I*�

i I 9*�

�

�

lj : j 340 1� � � N� ��HHHHI

2 HHHH0 I�1

HHHH01

�

+�

+�

HHHHI2

HHHH0 I�1 LLLLI

2�� c0

di?

i

2 k*��

i Imin� �

c0 0�u v� �

lj pI u �� lj pI v �� 2 �d

t

t 1�

�t ��lim 0� j 1 � N� ��

pI

HHHHs HHHHI

s

u v� � w u v��

Atw

id

iw

iM f

ix u Eu t�� f

ix v Ev t��

i 1 � m� �� L2 @� � ?i

wi

M�i I ?

iw

ii I 9

12��

dtd

�� L w t� �� 12�� d

i?

iw

iM 2

i I* J w

i2

i I 9*� � k* w

i2

i I*�

L w� � ?iEw

i2

i I* ?

iw

i2

i I 9*��


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As in the proof of Theorem 5.1 (see (5.10)) we have

for every . Therefore, provided (5.15) holds, we obtain that

with some constant . Equation (5.16) implies that as .Hence,

. (5.18)

However, equation (5.9) and the inequality

, ,

imply that

for all with large enough and for . Therefore, equation (5.8) followsfrom (5.18). Theorem 5.2 is proved.

The abstract form of Theorem 5.2 can be found in [3].

As an application of Theorem 5.2 we consider a system of equations which describethe Belousov-Zhabotinsky reaction. This system (see [9], [10], and the references there-in) can be obtained from (5.1) if we take , , and

,

where

,

, .

Here and are positive numbers. The theorem on the existence of classi-cal solutions can be proved without any difficulty (see, e.g., [7]). It is well-known[10] that if and , then the domain

is invariant (if the initial condition vector lies in for all , then for and ). Let be a set of classical solutions the initial condi-tions of which have the values in . It is clear that assumptions (5.3) and (5.4) arevalid for . Simple calculations show that the numbers , and can be

wi2

i I* C N ,�� lj pI w� � 2 1 ,�

c02

�� +�

2wiM 2

i I*��

jmax�

, 0�

dtd

�� L w t� �� 2 L w t� �� C lj pI w t� �� 2

jmax��

2 0� L w t� �� 0� t 0�

w t� �t ��lim 0�

Ew C wM�� w HHHH2 HHHH01

Ew t� � 2e

2 t t0�� Ew t0� � 2C e

2 t t0�� w �� 2 �d

t0

t

��

t t0� t0 2 0�

n 3� m 3� a x t�� diag d1 d2 d3� ��

f x u Eu t�� f u� � f1 u� � f2 u� � f3 u� ��

f1 u� � � u2 u1u2� u1 2u12��

f2 u� � 1�� u3 u2� u1u2�� f3 u� � , u1 u3��

� 2 �� ,

a3 a1 1 2 1�� max� � a2 �a3�

D u u1 u2 u3� �� : 0 uj

aj, j = 1 2 3� �� R30�

D x @ u x t�� Dx @ t 0� � �

D

D

� J k* ?2� � ?3


chosen such that equation (5.14) holds for , , and . In-deed, let smooth functions and be such that for all

and let . Then it is evident that there exist constants such that

,

,

.

Consequently, for any we have

It follows that there is a possibility to choose the parameters and such that

with positive constants and . This enables us to prove (5.14) and, hence, thevalidity of the assertions of Theorem 5.2 for the system of Belousov-Zhabotinskyequations. Therefore, if is a set of linear functionals on

such that is small enough, then thecondition

, ,

for some pair of solutions and which lie in implies that

as .

In particular, this means that the asymptotic behaviour of solutions to the Belousov-Zhabotinsky system is uniquely determined by the behaviour of one of the compo-nents of the state vector. A similar effect for the other equations is discussed in thesections to follow.

The approach presented above can also be used in the study of the Navier-Stokessystem. As an example, let us consider equations that describe the dynamics of a vis-cous incompressible fluid in the domain with periodicboundary conditions:

I 1� �� I 9 2 3�� ?1 1�u x� � v x� � u x� � v x� �� D

x @ w x� � u x� � v x� ��C

j0�

L1 f1 u� � f1 v� � w1M�� d12�� w1M 2

C1 w12

w22�& '

( )��

L2 f2 u� � f2 v� � w2�� 12�� w2

2C2 w1

2w3

2�& '( )��

L3 f3 u� � f3 v� � w3�� ,2�� w3

2C3 w1

2��

?2 ?3� 0�

L1 ?2L2 ?3L3� �d12�� w1M 2

C1 ?2C2 ?3C3� �� w12

C1

?22��& '

( ) w22 ?2 C2

?3,2

��& '( ) w3

2 .

� �

� �

�

?2 ?3

L1 ?2L2 ?3L3� �d12�� w1M 2

k* w12 J w2

2w3

2�& '( )��

k* J

� lj : j 1 � N� �� H 2 @� � H0

1 @� � +�

H2 H01 � � @� � L2 @� ��

lj u1 �� lj v1 �� 2 �d

t

t 1�

�t ��lim 0� j 1 � N� ��

u t� � u1 t� � u2 t� � u3 t� �� v t� � v1 t� � v2 t� �� v3 t� � � �

u t� � v t� �� 1 0� t ��

@ T 2 0 L�� 0 L��


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, , ,

(5.19)

where the unknown velocity vector and pressure are -periodic functions with respect to spatial variables, , and is the external force.

Let us introduce some definitions. Let be a space of trigonometric polynomi-als of the period with the values in such that and

. Let be the closure of in , let be the orthoprojector onto in , let and for all and from

. We remind (see, e.g., [11]) that is a positive operator with dis-crete spectrum and the bilinear operator is a continuous mapping from

into . In this case problem (5.19) can be rewritten in the form

, . (5.20)

It is well-known (see, e.g., [11]) that if and , then prob-lem (5.20) has a unique solution such that

, . (5.21)

One can prove (see [9] and [12]) that it possesses the property

(5.22)

for any . Here is the first eigenvalue of the operator in and

.

Lemma 5.1.

Let and let . Then

, (5.23)

, (5.24)

where is the -norm, is the -th component of the vector ,

, and is an absolute constant.

Proof.

Using the identity (see [12])

for and , it is easy to find that

.

Atu - uM� u E�� u Ep� � F x t�� x T 2 t 0�

Eu 0 ,� x T 2 , u x 0�� u0 x� � ,�

u x t�� u1 x t�� u2 x t�� p x t�� L - 0�F x t��

�

v x� � L R2 div v 0�v x� � xd

T2� 0� H � LLLL2 NH LLLL2 A N uM� uM�� B u v�� N u E�� v� u v

D A� � H HHHH2 � A

B u v�� D A� � D A� �� H

Atu -Au B u u�� NF t� �� u

t 0� u0 H�

u0 H NF t� � L� R+ H�� u t� �

u t� � C R+ H�� C t0 +� D A� �� t0 0�

1a�� Au �� 2 �d

t

t a�

�t ��lim F2

-2�� 1 1

a-71��& '

( )�

a 0� 71 2> L/� �2� A H

F NF t� �t ��lim�

u v� D A� � w u v��

B u u�� B v v�� Aw�� 2 w � Ew Au�

B u u�� B v v�� Aw�� c L Ewk Ew1 2�

wM 1 2�Au� � ��

. � L� wk

k w

k 1 2�� c

B u w�� Aw�� B w v�� Aw�� B w w�� Au�� 0�

u v� D A� � w u v��

B u u�� B v v�� Aw�� B w w�� Au��


Therefore, it is sufficient to estimate the norm . The incompressibili-ty condition gives us that

,

where is the derivative with respect to the variable . Consequently,

. (5.25)

This implies (5.23). Let us prove (5.24). For the sake of definiteness we let. We can also assume that . Then (5.25) gives us that

.

We use the inequalities (see, e.g., [11], [12])

and

,

where and are absolute constants (their explicit equations can be foundin [12]). These inequalities as well as a simply verifiable estimate

imply that

.

This proves (5.24) for .

Theorem 5.3.

1. A set of linearly independent continuousA set of linearly independent continuousA set of linearly independent continuousA set of linearly independent continuous

linear functionals on is an asymptotically determining setlinear functionals on is an asymptotically determining setlinear functionals on is an asymptotically determining setlinear functionals on is an asymptotically determining set

with respect to for problem with respect to for problem with respect to for problem with respect to for problem (5.20) in the class of solutions with property in the class of solutions with property in the class of solutions with property in the class of solutions with property

(5.21) if if if if

,,,, (5.26)

where is an absolute constant.where is an absolute constant.where is an absolute constant.where is an absolute constant.

2. Let be a set of linear functionals on Let be a set of linear functionals on Let be a set of linear functionals on Let be a set of linear functionals on

and letand letand letand let

,,,,

where is an absolute constant. Then every set is an asymptoticallywhere is an absolute constant. Then every set is an asymptoticallywhere is an absolute constant. Then every set is an asymptoticallywhere is an absolute constant. Then every set is an asymptotically

determining set with respect to for problem determining set with respect to for problem determining set with respect to for problem determining set with respect to for problem (5.20) in the class of solu- in the class of solu- in the class of solu- in the class of solu-

tions with property tions with property tions with property tions with property (5.21).... Here is the natural projection onto the -th Here is the natural projection onto the -th Here is the natural projection onto the -th Here is the natural projection onto the -th

component of the velocity vector, component of the velocity vector, component of the velocity vector, component of the velocity vector, ,,,, ....

B w w�� Ew 0�

w E�� w w1 A2 w2� w2 A2 w1� , w1 A1 w2 w2 A1 w1��

Ai xi

w E�� w2 2 w1 E�� w2

2w2 E�� w1

2�& '( )�

k 1� w �

w E�� w 2 w1 L4 Ew2 L4 w2 � Ew1�& '( )�

vL� a0 v

1 2�vM 1 2��

vL4 a1 v

1 2� Ev1 2��

a0 a1v �

L 2>/� � Ev��

w E�� w L>�� a1

2 a0�� Ew1 Ew 1 2� wM 1 2��

k 1�

� lj: j 1 2 � N� � ��

D A� � HHHH2 HHHH �HHHH1

+�

+�

D A� � H�� c1G c1-2 NF t� �

t ��lim& '

( ) 1��

c1� lj : j 1 � N� �� H2 @� �

+ 9�

+�

H2 @� � L2 @� �� c2-4 L 3� NF t� �

t ��lim& '

( ) 2��

c2 pk*�

HHHH1

pk

k

pk

u1 u2�� uk

� k 1 2��


C

h

a

p

t

e

r

5

Proof.

Let and be solutions to problem (5.20) possessing property (5.21).Then equations (5.20) and (5.23) imply that

for . As above, Theorem 2.1 gives us the estimate

, .

Therefore,

.

The inequality

enables us to obtain the estimate (see Exercise 2.12) andhence

for every . Therefore, we use dissipativity properties of solutions (see [8] and[9] as well as Chapter 2) to obtain that

,

where

.

Equation (5.22) for implies that the function possesses proper-ties (1.6) and (1.7), provided (5.26) holds. Therefore, we apply Lemma 1.1 to obtainthat as , provided

.

In order to prove the second part of the theorem, we use similar arguments. Forthe sake of definiteness let us consider the case only. It follows from (5.20)and (5.24) that

. (5.27)

The definition of completeness defect implies that

,

where .Therefore,

u t� � v t� �

12��

dtd

�� Ew2� - wM 2� 2 w � Ew Au� ��

w u v��

w C1 w� � +�

wM�� 1 w� � lj

w� �j

max�

w � a0 w1 2�

wM 1 2�� c1 1 w� � wM 1 2�� a0 +�

wM��

Ew2

w wM��

+�

D A� � D A1 2�� +�

1 2��

wM 2C,� 1 w� �2 1 ,�� +

�

1� Ew2��

, 0�

dtd

�� Ew t� � 2 � t� � Ew2�� C 1 w� �2 1 w� ��

� t� � - 1 ,�� +�

1� 2 a02 +

�- 1� Au t� � 2��

a -71� � 1�� t� �

Ew t� � 0� t ��

1 w �� 2 �d

t

t 1�

�t ��lim 0�

k 1�

12��

dtd

�� Ew2� - wM 2� c L Ew1

1 2� Ew� wM 1 2�Au� ��

Ew11 2�

C 1 w1� � + 9�

w1M 1 2�� C 1 w1� � + 9�

wM 1 2��

1 w1� � lj w1� �j

max�

D e t e r m i n i n g F u n c t i o n a l s i n t h e P r o b l e m o f N e r v e I m p u l s e T r a n s m i s s i o n 339

for any , where the constant depends on , and . Consequently,from (5.27) we obtain that

.

Therefore, we can choose and find that

,

where is an arbitrary number. Further arguments repeat those in the proofof the first assertion. Theorem 5.3 is proved.

It should be noted that assertion 1 of Theorem 5.3 and the results of Section 3 enableus to obtain estimates for the number of determining nodes and local volume avera-ges that are close to optimal (see the references in the survey [3]). At the same time,although assertion 2 uses only one component of the velocity vector, in general itmakes it necessary to consider a much greater number of determining functionals incomparison with assertion 1. It should also be noted that assertion 2 remains true ifinstead of we consider the projections of the velocity vector onto an arbitrarya priori chosen direction [3]. Furthermore, analogues of Theorems 1.3 and 4.4 can beproved for the Navier-Stokes system (5.19) (the corresponding variants of estimates(4.28) and (4.29) can be derived from the arguments in [2], [8], and [9]).


in the Problem of Nerve Impulsein the Problem of Nerve Impulsein the Problem of Nerve Impulsein the Problem of Nerve Impulse

TransmissionTransmissionTransmissionTransmission

We consider the following system of partial differential equations suggested byHodgkin and Huxley for the description of the mechanism of nerve impulse trans-mission:

, , , (6.1)

, , , . (6.2)

c L Ew11 2� Ew wM 1 2�

Au�

c L +9�

Ew wM Au� � � � C 1 w1� � Ew wM 1 2�Au� ��

C 1 w1� �� 2 < Lc2

-�� + 9��& '( ) Ew

2Au

2� � -2�� wM 2� �

�

� �

�

< 0� C < - �� L

dtd

�� Ew2 - wM 2� C 1 w1� �� 2 2 < Lc2

-�� +9��& '( ) Ew

2Au

2��

< , L c2 - 1� +9�

� � � ��

dtd

�� Ew2 - 2>

L��& '( )2

2 1 ,�� Lc2

-�� +9� Au2�� Ew

2� C 1 w1� �� 2�

, 0�

wk

Atu d0 Ax

2 u� g u v1 v2 v3� � �� 0� x 0 L�� t 0�

Atvj dj Ax2 vj� kj u� � vj hj u� �� 0� x 0 L�� t 0� j 1 2 3� ��


C

h

a

p

t

e

r

5

Here , , and

, (6.3)

where and . We also assume that and are thegiven continuously differentiable functions such that and ,

. In this model describes the electric potential in the nerve and isthe density of a chemical matter and can vary between and . Problem (6.1) and(6.2) has been studied by many authors (see, e.g., [9], [10], [13] and the referencestherein) for different boundary conditions. The results of numerical simulation givenin [13] show that the asymptotic behaviour of solutions to this problem can be quitecomplicated. In this chapter we focus on the existence and the structure of deter-mining functionals for problem (6.1) and (6.2). In particular, we prove that theasymptotic behaviour of densities is uniquely determined by sets of functionalsdefined on the electric potential only. Thus, the component of the state vector

is leading in some sense.We equip equations (6.1) and (6.2) with the initial data

, , , (6.4)

and with one of the following boundary conditions:

, , (6.5a)

, , (6.5b)

, , , (6.5c)

where . Thus, we have no boundary conditions for the function when the corresponding diffusion coefficient is equal to zero for some .

Let us now describe some properties of solutions to problem (6.1)–(6.5). Firstof all it should be noted (see, e.g., [10]) that the parallelepiped

is a positively invariant set for problem (6.1)–(6.5). This means that if the initial data belongs to for almost all ,

then

for and for all for which the solution to problem (6.1)–(6.5) exists.Let be the space consisting of vector-functions

, where , , . We equip it with the standard norm. Let

d0 0� dj

0�

g u v1 v2 v3� � �� 1 v13

v2 ,1 u�� 2 v34 ,2 u�� 3 ,3 u��

� j 0� ,1 ,3 0 ,2� � � kj u� � hj u� �kj u� � 0� 0 hj u� � 1� �

j 1 2 3� �� u vj

0 1

vj

u u

u v1� �� v2 v3� �

ut 0� u0 x� �� v

jt 0�

vj 0 x� �� j 1 2 3� ��

ux 0� u

x L� djv

jx 0�

djv

jx L�

0� � � � t 0�

Ax ux 0�

Ax ux L�

dj Ax vjx 0�

� dj Ax vjx L�

� 0� � � � t 0�

u x L t�� u x t�� dj vj x L t�� vj x t�� 0� � x R1 t 0�

j 1 2 3� �� vj

x t�� d

jj 1 2 3� ��

D U u v1 v2 v3� � �� : ,2 u ,1� � , 0 vj

1� � , j 1 2 3� �� " #$ %

R40�

U0 x� � u0 x� � v10 x� � v20 x� � v30 x� �� D x 0 L��

U t� � u x t�� v1 x t�� v2 x t�� v3 x t�� D

x 0 L�� t 0�H0 H� �4 L2 0 L�� 4

U x� � u v1 v2 v3� � �� u L2 0 L�� vj L2 0 L�� j 1 2 3� ��


.

Depending upon the boundary conditions (6.5 a, b, or c) we use the following nota-tions and , where

, or , or (6.6)

and

, or

, or , (6.7)

respectively. Hereinafter is the Sobolev space of the order on , and are subspaces in corresponding to the boundary conditions

(6.5a) and (6.5c). The norm in is defined by the equality

,

We use notations and for the norm and the inner product in . Further we assume that is the space of strongly continu-

ous functions on with the values in . The notation has a simi-lar meaning.

Let for all . Then for every vector problem(6.1)–(6.5) has a unique solution defined for all (see, e.g., [9], [10]).This solution lies in

for any segment and if , then

. (6.8)

Therefore, we can define the evolutionary semigroup in the space by the formula

,

where is a solution to problem (6.1)–(6.5) with the initial conditions

.

The dynamical system has been studied by many authors.In particular, it has been proved that it possesses a finite-dimensional global attrac-tor [9].

H0 D� � U x� � H0 : U x� � D for almost all x 0 L�� " #$ %

�

H1 V1� �4� H2 V2� �4�

V1 H01 0 L�� H1 0 L�� Hper

1 0 L��

V2 H2 0 L�� H01 0 L��

u x� � H 2 0 L�� : Axux 0 x� L� �

= 0� " #$ %

Hper2 0 L��

Hs 0 L�� s 0 L�� H0

1Hper

1 H1 0 L�� Hs 0 L��

us2 A

xs

u2

u2� A

xs

u x� � 2u x� � 2�

& '( ) xd

0

L

�� s 1 2 ��

. . .�� H L2 0 L�� C 0 T X��

0 T�� X Lp 0 T X��

dj

0� j 1 2 3� �� U0 H0 D� �U t� � H0 D� � t

C 0 T H0 D� �� L2 0 T H1��

0 T�� U0 H0 D� � H1

U t� � C 0 T H0 D� � H1 �� L2 0 T H2��

St H0 D� � H1

StU0 U t� � u x t�� v1 x t�� v2 x t�� v3 x t��

U t� �

U0 u0 x� � v10 x� � v20 x� � v30 x� ��

H0 D� � H1 St

� � �


C

h

a

p

t

e

r

5

If , then the corresponding evolutionary semigroup can bedefined in the space . In this case for any segment

we have

, (6.9)

if . This assertion can be easily obtained by using the general methodsof Chapter 2.

The following assertion is the main result of this section.

Theorem 6.1.

Let Let Let Let be a finite set of continuous linear functio-be a finite set of continuous linear functio-be a finite set of continuous linear functio-be a finite set of continuous linear functio-

nals on the space (see nals on the space (see nals on the space (see nals on the space (see (6.6) and and and and (6.7)). Assume that). Assume that). Assume that). Assume that

(6.10)

orororor

,,,, (6.11)

wherewherewherewhere

(6.12)

andandandand

(6.13)

with with with with ,,,, ,,,, ,,,, and and and and

,,,, ,,,,

....

Then is an asymptotically determining set with respect to the space Then is an asymptotically determining set with respect to the space Then is an asymptotically determining set with respect to the space Then is an asymptotically determining set with respect to the space

for problem for problem for problem for problem (6.1)–(6.5) in the sense that for any two solutions in the sense that for any two solutions in the sense that for any two solutions in the sense that for any two solutions

andandandand

satisfying either satisfying either satisfying either satisfying either (6.8) with with with with ,,,, or (6.9) with , or (6.9) with , or (6.9) with , or (6.9) with , ,,,, the con- the con- the con- the con-

ditionditionditiondition

d1 d2 d3 0� � �V1 V1 H1 0 L�� 3�� H0 D� � �

0 T��

St U0 U t� � C 0 T V1�� L2 0 T V2 H1 0 L�� 3��

U0 V1

� lj: j 1 � N� ��

Vs

s� 1 2��

+�

1� � +�

V1 H�� d0

d0 K1��

+�

2� � +�

V2 H�� d0

d02 K2

2��

K1 ,1 ,2�� 2j Aj Bj��

kj*

��

j 1�

3

*��

K2 2 �1 �2�� 2 5 ,1 ,2�� 22

jA

jB

j��

2 kj*

��& 'F G( )2

j 1�

3

*��& 'F G( )1 2�

��

21 3 �1� 22 �1� 23 4 �2�

Aj Aukj u� � : ,2 u ,1� �� " #$ %

max� Bj Au kj hj� � u� � : ,2 u ,1� �� " #$ %

max�

kj* k

ju� � : ,2 u ,1� �

� " #$ %

min�

� H0

U t� � u x t�� v1 x t�� v2 x t�� v3 x t��

U* t� � u* x t�� v1* x t�� v2

* x t�� v3* x t��

dj

0� dj

0� j 1 2 3� ��


forforforfor (6.14)

implies thatimplies thatimplies thatimplies that

.... (6.15)

Proof.

Assume that (6.10) is valid. Let

and

be solutions satisfying either (6.8) with , or (6.9) with , .It is clear that

where

.

Since , it is evident that

,

where . Let and , . It followsfrom (6.1) that

, , . (6.16)

If we multiply (6.16) by in , then it is easy to find that

. (6.17)

Equation (6.2) also implies that

. (6.18)

Thus, for any and , , we obtain that

lj

u �� lj

u* �� 2 �d

t

t 1�

�t ��lim 0� j 1 � N� ��

u t� � u* t� �� 2vj t� � vj

* t� �� 2

j 1�

3

*�� " #$ %

t ��lim 0�

U t� � u x t�� v1 x t�� v2 x t�� v3 x t��

U* t� � u* x t�� v1* x t�� v2

* x t�� v3* x t��

dj

0� dj

0� j 1 2 3� ��

G U U*�� g u v1 v2 v3� � �� g u* v1* v2

* v3*� � ��

�1 v13

v2 �2 v34 �3� �� u u*�� h U U*�� ,�

�

�

h U U*�� 1 v13 v2 v1

* 3 v2*�� ,1 u*�� 2 v3

4 v3* 4�� ,2 u*��

U U*� D

h U U*�� aj vj vj*�

j 1�

3

*�

aj ,1 ,2�� 2j�� w u u*�� j vj vj*�� j 1 2 3� ��

Atw d0 Ax

2 w� G U U*�� 0� x 0 L�� t 0�

w L2 0 L��

12��

dtd

�� w2� d0 Axw

2 �3 w2� � aj �j w�

j 1�

3

*�

12��

dtd

�� j2� dj Ax�j

2kj* �j

2� � Aj Bj�� j w��

0 + 1� � ?j 0� j 1 2 3� ��


C

h

a

p

t

e

r

5

(6.19)

Theorem 2.1 gives us that

for any . Therefore,

. (6.20)

Consequently, it follows from (6.19) that

(6.21)

for any , , and , where

.

We choose and obtain that

.

It is easy to see that if (6.10) is valid, then there exist and such that. Therefore, equation (6.21) gives us that

where . Thus, if (6.10) is valid, then equation (6.14) implies (6.15).

12��

dtd

�� w2 ?

j�

j2

j 1�

3

*�& 'F G( )

� d0 Ax

w2 �3 w

2 + kj* ?

j�

j2

j 1�

3

*� � �

1 +�� kj*?j �j

2aj ?j Aj Bj��

j 1�

3

* �j w��j 1�

3

*�

aj

?j

Aj

Bj

�� 2

4 1 +�� ?jk

j*

�� w2 .�

j 1�

3

*

� �

�

�

w2

C� 1� l

jw� � 2 1 1�� +

�

1� �� 2 Ax

w2

w2�& '

( )��j

max��

1 0�

Axw2 1

1 1�� +�

1� �� 2�� 1�& 'F G( )

w2� C

� 1� lj w� � 2

jmax��

12��

dtd

�� w2 ?

j�

j2

j 1�

3

*�& 'F G( )

� d0 J � + ? 1� � �� w2�

+ kj*?

j�

j2

j 1�

3

*

� �

� C� 1� l

jw� � 2

jmax��

0 + 1� � ?j 0� 1 0�

J � + ? 1� � �� 3d0��

1

1 1�� +�

1� �� 2�� 1�

aj

?j

Aj

Bj

�� 2

4 1 +�� ?jk

j*d0

��

j 1�

3

*��

?j

aj

Aj

Bj

�� 1��

J � + ? 1� � �� 3d0��

1

1 1�� +�

1� �� 2�� 1�

K1

1 +�� d0��

0 + 1� � 1 0�J � + ? 1� � �� 0�

w t� � 2 ?j�

jt� � 2

j 1�

3

*�

w02 ?

j�

j 02

j 1�

3

*�& 'F G( )

e J*� t� C� 1� e J* t �� l

jw �� 2

jmax � ,d

0

t

��

�

�

J* 0�


Let us now assume that (6.11) is valid. Since

,

then it follows from (6.16) that

(6.22)

Therefore, we can use equation (6.18) and obtain (cf. (6.19)) that

for any and . As above, we find that

for any . Therefore,

,

provided that

.

As in the first part of the proof, we can now conclude that if (6.11) is valid, then (6.14)implies the equations

and , . (6.23)

It follows from (6.17) that

.

Therefore, as above, we obtain equation (6.15) for the case (6.11). Theorem 6.1

is proved.

G U U*�� Ax2 w�� 3 A

xw

2� �1 �2�� w� h U U*�� & '( ) A

x2 w��

12��

dtd

�� Ax

w2�

d02�� A

x2 w

2 �3 Ax

w2� �

2d0�� 1 �2�� 2 w

2� � 2a

j2

d0�� j

2 .�j 1�

3

*��

�

�

12��

dtd

�� Axw2 ?j �j

2

j 1�

3

*�& 'F GF G( )

�d02�� Ax

2 w2 �3 Axw

2 + kj*?j �j

2

j 1�

3

*� � �

2d0�� 1 �2�� 2� 9

4��

aj2 Aj Bj�� 2

1 +�� 2 kj*� �2 d0

��

j 1�

3

*��& 'F GF G( )

w2�

�

�

0 + 1� � ?j

3 aj2 1 +�� k

j*d0� � 1��

Ax2 w

2 1

1 1�� +�

2� �� 2�� 1�& 'F G( )

w2� C

� 1� lj

w� � 2

jmax��

1 0�

12��

dtd

�� Ax

w2 ?

j�

j2

j 1�

3

*�& 'F GF G( )

� �3 Ax

w2 + k

j*?

j�

j2

j 1�

3

*� � C� 1� l

jw� � 2

jmax��

1

1 1�� +�

2� �� 2�� 1� 4

d02

�� 1 �2�� 2��9 a

j2 A

jB

j�� 2

2 1 +�� 2kj*� �2 d0

2��

j 1�

3

*� 0�

Ax

w t� �t ��lim 0� �

jt� �

t ��lim 0� j 1 2 3� ��

dtd

�� w2 �3 w

2� 3�3�� aj

2 �j t� � 2

j 1�

3

*�


C

h

a

p

t

e

r

5

As in the previous sections, modes, nodes and generalized local volume averages canbe choosen as determining functionals in Theorem 6.1.

Let be a basis in which consists of eigenvec-tors of the operator with one of the boundary conditions (6.5).Show that the set

is determining (in the sense of (6.14) and (6.15)) for problem (6.1)–(6.5) for large enough.

Show that in the case of the Neumann boundary conditions(6.5b) it is sufficient to choose the number in Exercise 6.1 suchthat

, . (6.24)

Obtain a similar estimate for the other boundary conditions (Hint:see Exercises 3.2–3.4).

Let

. (6.25)

Show that for every the estimate

(6.26)

holds for any (Hint: see Exercise 3.6).

Use estimate (6.26) instead of (6.20) in the proof of Theorem6.1 to show that the set of functionals (6.25) is determining for prob-lem (6.1)–(6.5), provided that .

Obtain the assertions similar to those given in Exercises 6.3and 6.4 for the following set of functionals

E x e r c i s e 6.1 ek

� � L2 0 L�� A

x2

� lj

u� � ej

x� �u x� � xd

0

L

�� , j 1 � N� �� " #$ %

�

N

E x e r c i s e 6.2

N

N L>��

Kj

d0�� j 1 or 2�

E x e r c i s e 6.3

� lj

: lj

u� � u xj

� �� , xj

j h� , hL

N�� , j 1 � N� ��

� " #$ %

�

w V1

Axw2 2

1 1�� h2�� w

2CN 1� lj w� � 2max��

1 0�

E x e r c i s e 6.4

N L 2 K1� � d0/��

E x e r c i s e 6.5

� lj

w� � 1h�� 7 �

h��& '

( ) u xj

�� d

0

h

�� ,

xj j h , hL

N�� , j 0 � N 1��

�

"

#

$

% ,

�


where the function possesses the properties

, .

It should be noted that in their work Hodgkin and Huxley used the following expres-sions (see [13]) for and :

, ,

where , ;

, ;

, .

Here . They also supposed that , , ,and . As calculations show, in this case and .Therefore (see Exercise 6.4), the nodes are determining for problem (6.1)–(6.5) when . Of course,similar estimates are valid for modes and generalized volume averages.

Thus, for the asymptotic dynamics of the system to be determined by a smallnumber of functionals, we should require the smallness of the parameter .However, using the results available (see [14]) on the analyticity of solutions toproblem (6.1)–(6.5) one can show (see Theorem 6.2 below) that the values of allcomponents of the state vector in two sufficiently close nodesuniquely determine the asymptotic dynamics of the system considered not depend-ing on the value of the parameter . Therewith some regularity conditions forthe coefficients of equations (6.1) and (6.2) are necessary.

Let us consider the periodic initial-boundary value problem (6.1)–(6.5c). As-sume that for all and the functions and are polynomials suchthat and for . In this case (see [14]) everysolution

possesses the following Gevrey regularity Gevrey regularity Gevrey regularity Gevrey regularity property: there exists suchthat

(6.27)

for some and for all . Here are the Fourier coefficients of thefunction :

7 x� � L� R1� �

7 x� � xd

��

�

� 1� supp 7 x� � 0 1�� 0

kj u� � hj u� �

kj

u� � �j

u� � 2j

u� �� hj

u� ��

ju� �

�j

u� � 2j

u� ��

�1 u� � e 0.1 u� 2.5�� 21 u� � 4 u� 18/� �exp�

�2 u� � 0.07e 0.05 u�� 22 u� � 1

1 0.1u� 3�� exp��

�3 u� � 0.1 e 0.1 u� 1�� 23 u� � 0.125 u 80/�� exp�

e z� � z e z 1�� /� ,1 115� ,2 12�� 1 120��2 36� K1 5.2 104�� K2 7.4 104��

xj

j h� , h l N/ ,�� j 0 1 2 � N� � � �� N 2.3 102 L d0/� ��

L d0/

U u v1 v2 v3� � ��

L d0/

dj 0� j kj u� � hj u� �kj u� � 0� 0 hj u� � 1� � u ,2 ,1��

U t� � u x t�� v1 x t�� v2 x t�� v3 x t��

t* 0�

Fl u t� �� 2Fl vj t� �� 2

j 1�

3

*�& 'F GF G( )

l ��

�

* e� l� C�

� 0� t t*� Fl

w� �w x� �


C

h

a

p

t

e

r

5

,

In particular, property (6.27) implies that every solution to problem (6.1)–(6.5c) be-comes a real analytic function for all large enough. This property enables us toprove the following assertion.

Theorem 6.2.

Let for all and let and be the polynomials pos-Let for all and let and be the polynomials pos-Let for all and let and be the polynomials pos-Let for all and let and be the polynomials pos-

sessing the propertiessessing the propertiessessing the propertiessessing the properties

,,,, forforforfor ....

Let and be two nodes such that andLet and be two nodes such that andLet and be two nodes such that andLet and be two nodes such that and

,,,, where is defined by formula where is defined by formula where is defined by formula where is defined by formula (6.12).... Then for every Then for every Then for every Then for every

two solutionstwo solutionstwo solutionstwo solutions

andandandand

to problem to problem to problem to problem (6.1)–(6.5 c) the condition the condition the condition the condition

implies their asymptotic closeness in the space implies their asymptotic closeness in the space implies their asymptotic closeness in the space implies their asymptotic closeness in the space ::::

.... (6.28)

Proof.

Let and let , . We introduce the notations:

, , and .

Let

.

As in the proof of Theorem 6.1, it is easy to find that

Fl

w� � 1L�� w x� � i

2> l

L�� x

� " #$ %

exp� xd

0

L

�� l 0 1O 2O ��

t

dj

0� j kj

u� � hj

u� �

kj

u� � 0� 0 hj

u� � 1� � u ,2 ,1��

x1 x2 0 x1 x2 L��x2 x1� 2 d0 K1/� K1

U t� � u x t�� v1 x t�� v2 x t�� v3 x t��

U* t� � u* x t�� v1* x t�� v2

* x t�� v3* x t��

u xl

t�� u* xl

t�� vj

xl

t�� vj* x

lt��

j 1�

3

*�� " #$ %

l 1 2��max

t ��lim 0�

H0

u t� � u* t� �� 2v

jt� � v

j* t� �� 2

j 1�

3

*�� " #$ %

t ��lim 0�

w u u*�� j vj vj*�� j 1 2 3� ��

M x : x1 x x2� �� M x2 x1�� w M2

w x� � 2xd

x1

x2

��

m t M�� w xl t�� j xl t�� j 1�

3

*�� " #$ %

l 1 2��max�

12��

dtd

�� w M2� d0 A

xw M

2 �3 w M2� � �


and

.


.

Therefore, for any and , , we have

where is defined by formula (6.12). Simple calculations give us that

for any . Consequently, if , then there exist and such that

,

where is a positive constant. As in the proof of Theorem 6.1, it follows that condi-tion as implies that

. (6.29)

Let us now prove (6.28). We do it by reductio ad absurdum. Assume that there existsa sequence such that

. (6.30)

Let and be sequences lying in the attractor of the dynamical systemgenerated by equations (6.1)–(6.5c) and such that

, . (6.31)

w xl

t�� Ax

w xl

t�� l 1 2��* a

j�

j M w M�j 1�

3

*��

12��

dtd

�� j M

2� kj* �

j M2� �

jx

lt�� A

x�

jx

lt��

l 1 2��* A

jB

j��

j M w M��

Axw x t�� Ax�j x t�� j 1�

3

*�� " #$ %

x 0 L�� max

t 0�sup ��

0 + 1� � ?j

aj

Aj

Bj

�� 1�� j 1 2 3� ��

12��

dtd

�� w t� � M2 ?

j�

j M2

j 1�

3

*�& 'F GF G( )

� d0 Ax

w M2 �3 w M

2 + kj*?

j�

j M2

j 1�

3

*� � �

K1� 1 +�� 1�w M

2� � m t M�� ,�

�

K1

w M2

w x� � 2xd

x1

x2

� 1 1�� M 2

2�� A

xw x� � 2

xd

x1

x2

�� C1 M w x1� � 2� ��

1 0� M 2 2 d0 K1/� 0 + 1� �1 0�

dtd

�� w M2 ?j �j M

2

j 1�

3

*�& 'F GF G( )

J w M2 ?j �j M

2

j 1�

3

*�& 'F GF G( )�� C m t M��

Jm t M�� 0� t ��

U t� � U * t� �� Mt ��lim w M

2 ?j �j M2

j 1�

3

*�& 'F G( )

t ��lim 0�

tn

+��

U tn

� � U* tn

� ��n ��

lim 0�

Vn

� � Vn*� �

U tn

� � Vn

� 0� U* tn

� � Vn*� 0�


C

h

a

p

t

e

r

5

Using the compactness of the attractor we obtain that there exist a sequence and elements such that and . Since

,

it follows from (6.29) and (6.31) that

.

Therefore, for . However, the Gevrey regularity property im-plies that elements of the attractor are real analytic functions. The theorem on theuniqueness of such functions gives us that for . Hence,

as . Therefore, equation (6.31) implies that

.

This contradicts assumption (6.30). Theorem 6.2 is proved.

It should be noted that the connection between the Gevrey regularity and the exis-tence of two determining nodes was established in the paper [15] for the first time.The results similar to Theorem 6.2 can also be obtained for other equations (see thereferences in [3]). However, the requirements of the spatial unidimensionalityof the problem and the Gevrey regularity of its solutions are crucial.


for Second Order in Time Equationsfor Second Order in Time Equationsfor Second Order in Time Equationsfor Second Order in Time Equations

In a separable Hilbert space we consider the problem

, , , (7.1)

where the dot over stands for the derivative with respect to , is a positiveoperator with discrete spectrum, is a constant, and is a continuousmapping from into with the property

(7.2)

for some and for all such that . Assumethat for any , , and problem (7.1) is uniquely solvable inthe class of functions

(7.3)

and defines a process in the space with the evolu-tionary operator given by the formula

, (7.4)

nk

� �V V*� Vnk

V� Vnk

* V*�

Vn

Vn*� M U t

n� � U* t

n� �� M U t

n� � V

n� U* t

n� � V

n*��

V V*� M Vn

kV

nk

*�Mk ��

lim 0� �

V x� � V* x� �� x M

V x� � V* x� � x 0 L�� Vn

kVn

k

*� 0� k ��

U tn

k� � U* t

nk

� ��k ��lim 0�

H

u�� u� A u� � B u t�� ut s� u0� u�

t s� u1�

u t A

� 0� B u t�� D A?� � R� H

B u1 t�� B u2 t�� M K� � A? u1 u2��

0 ? 1 2/�� uj D A1 2�� A1 2� uj K�s R u0 D A1 2�� u1 H

C s +�� D A1 2�� C1 s +�� H� �� S t �� D A1 2�� H��

S t s�� u0 u1�� u t� � u� t� ��

D e t e r m i n i n g F u n c t i o n a l s f o r S e c o n d O r d e r i n T i m e E q u a t i o n s 351

where is a solution to problem (7.1) in the class (7.3). Assume that the process is pointwise dissipative, i.e. there exists such that

, (7.5)

for all initial data . The nonlinear wave equation (see the bookby A. V. Babin and M. I. Vishik [8])

is an example of problem (7.1) which possesses all the properties listed above. Here is a bounded domain in and the function possesses the pro-

perties:

,

,

where is the first eigenvalue of the operator with the Dirichlet boundaryconditions on , and are constants, for and

is arbitrary for .

Theorem 7.1.

Let be a set of continuous linear functionalsLet be a set of continuous linear functionalsLet be a set of continuous linear functionalsLet be a set of continuous linear functionals

on . Assume thaton . Assume thaton . Assume thaton . Assume that

,,,, (7.6)

where is the radius of dissipativity (see where is the radius of dissipativity (see where is the radius of dissipativity (see where is the radius of dissipativity (see (7.5)), and ), and ), and ), and

are the constants from are the constants from are the constants from are the constants from (7.2),,,, and is the first eigenvalue of the operator and is the first eigenvalue of the operator and is the first eigenvalue of the operator and is the first eigenvalue of the operator

.... Then is an asymptotically determining set for problem Then is an asymptotically determining set for problem Then is an asymptotically determining set for problem Then is an asymptotically determining set for problem (7.1) in the in the in the in the

sense that for a pair of solutions and from the class sense that for a pair of solutions and from the class sense that for a pair of solutions and from the class sense that for a pair of solutions and from the class (7.3) thethethethe

conditionconditionconditioncondition

forforforfor (7.7)


.... (7.8)

u t� �� S t �� R 0�

S t s�� y0 �R� t s t0 y0� ��

y0 u0 u1��

A2u

t2A�� uA

tA�� uM�� f u� � , x @ , t 0 ,��

u A@ 0 , ut 0� u0 x� � , uA

tA��t 0�

u1 x� � ,� � ��!!"!!$

@ Rd f u� � C1 R� �

71 +�� u2 C1� f v� � vd

0

u

� C2 u f u� � C312�� 71 +�� u2� ��

f 9 u� � C4 1 u2��

71 M�A@ C

j0� + 0� 2 2 d 2�� 1�� d 3�

2 d 2�

� lj : j 1 � N� �� D A1 2��

+ +�

D A1 2�� H��

M R� � 4 4711� �2�� 3 2�

��

11 2?��

�

R M K� � ? 0 1 2/ ��71

A �

u1 t� � u2 t� �

lj

u1 �� u2 �� d

t

t 1�

�t ��lim 0� j 1 � N� ��

A1 2� u1 t� � u2 t� �� 2u� 1 t� � u� 2 t� �� 2�

� " #$ %

t ��lim 0�


C

h

a

p

t

e

r

5

Proof.

We rewrite problem (7.1) in the form

, , (7.9)

where

, , .

Lemma 7.1.

There exists an exponential operator in the space

and

, (7.10)

where is the first eigenvalue of the operator .

Proof.

Let . Then it is evident that ,where is a solution to the problem

, , , (7.11)

(see Section 3.7 for the solvability of this problem and the properties of solu-tions). Let us consider the functional

, ,

on the space . It is clear that

(7.12)

Moreover, for a solution to problem (7.11) from the class (7.3) with we have that

. (7.13)

Therefore, it follows from (7.12) and (7.13) that

Hence, for and we obtain that

ydtd

�� y� � y t�� yt s� y0�

y u u�� y u� ; �u� A u�� y t�� 0 B u t��

t�� exp � �D A1 2�� H��

t�� exp y�

2 1�2

71��

71 � t

471 2 �2��

� " #$ %

y�

exp��

71 A

y0 u0 ; u1� �� y t� � e t� y0 u t� � u� t� �� u t� �

u�� u� A u� � 0� ut 0� u0� u�

t 0� u1�

V u� � 12�� u

2A1 2� u

2�& '( ) - u u�

�2��u��& '

( )�� 0 - ��

� D A1 2�� H��

12�� 1 -

�� u�2 1

2�� A1 2� u

2� V u� �

12�� -

2��& '( ) u�

2 12�� 71

1� - �� A1 2� u2 .�

� �

�

u t� � s 0�

dtd

��V u t� �� -�� u�2� - A1 2� u

2��

dtd

��V u t� �� 2V u t� ��

� -� 12�� -

2��& '( ) 2�& '

( ) u�2� - 1

2�� 71

1� - ��& '( ) 2�& '

( ) A1 2� u2 .�

�

�

�

- 1 2/� � �� 2 2 711� � 2�� 1� ��


,

.

This implies estimate (7.10). Lemma 7.1 is proved.


.

Therefore, with the help of Lemma 7.1 for the difference of two solutions , , we obtain the estimate

, (7.14)

where and the constants and have the form

, .

By virtue of the dissipativity (7.5) we can assume that for all .Therefore, equations (7.14) and (7.2) imply that

.

The interpolation inequality (see Exercise 2.1.12)

, ,

Theorem 2.1, and the result of Exercise 2.12 give us that

,

where . Therefore,

where . If we introduce a new unknownfunction in this inequality, then we obtain the equation

dtd

��V u t� �� 2V u t� �� 0�

14�� u�

2A1 2� u

2�& '( ) V u� � 3

4�� 71

1� � 2�& '( ) u�

2A1 2� u

2�& '( )� �

y t� � e t s�� y s� � e t �� y �� d

s

t

��

yj

t� � �u

jt� � u� j t� �� j 1 2��

y t� ��

De 2 t s�� y s� ��

D e 2 t �� B u1 �� B u2 �� d

s

t

��

y t� � y1 t� � y2 t� �� D 2

D 2 1 711� �2�� 2

�71

471 2 � 2��

yj t� ��

R� t s s0� �

y t� ��

De 2 t s�� y s� ��

DM R� � e 2 t �� A? u1 �� u2 �� d

s

t

��

A?u u1 2?�

A1 2� u2?� 0 ? 1

2��

A?u C�

lj u� �j

max +�

1 2?�A1 2� u��

+�

+�

D A1 2�� H��

y t� ��

De 2 t s�� y s� ��

DM R� � +�

1 2?�e 2 t �� y ��

��d

s

t

��

C � D R� �� e 2 t �� N�

u �� ,d

s

t

��

� �

�

�

N�

u� � lj

u� � : j 1 2 � N� � �� max�� t� � e2 t y t� �

��


C

h

a

p

t

e

r

5

,

where . We apply Gronwall’s lemma to obtain

.

After integration by parts we get

.

If equation (7.6) holds, then . Therewith it is evident that for we have the estimate

Therefore, using the dissipativity property we obtain that

for and . If we fix and tend the parameter to infinity, thenwith the help of (7.7) we find that

for any . This implies (7.8). Theorem 7.1 is proved.

Unfortunately, because of the fact that condition (7.2) is assumed to hold only for, Theorem 7.1 cannot be applied to the problem on nonlinear plate

oscillations considered in Chapter 4. However, the arguments in the proof of Theo-rem 7.1 can be slightly modified and the theorem can still be proved for this caseusing the properties of solutions to linear nonautonomous problems (see Section 4.2).However, instead of a modification we suggest another approach (see also [3]) whichhelps us to prove the assertions on the existence of sets of determining functionalsfor second order in time equations. As an example, let us consider a problem of plateoscillations .

� t� � D� s� � � � �� d

s

t

� C e2 �N�

u �� d

s

t

��

� DM R� � +�

1 2?��

� t� � D� s� � e� t s�� C e� t e �� e2; N�

u ;� �� ;d

s

�

�� ! !" #! !$ %

�d

s

t

��

y t� ��

D y s� ��

2 �� t s�� exp C e 2 �� t �� N�

u �� d

s

t

��

J 2 �� 0��0 a t s��

y t� ��

D y s� ��

e J t s�� C e J t �� N�

u �� d

t a�

t

�C e J t �� N

�u �� .d

s

t a�

�

� �

�

�

y t� ��

DR e J t s�� C N�

u �� d

t a�

t

� C R �� e J a��

t s� 0 a t s�� a t

y t� ��

t ��lim C R �� e J a��

a 0�

0 ? 1 2/��


Thus, in a separable Hilbert space we consider the equation

We assume that is an operator with discrete spectrum and the function liesin and possesses the properties:

a) , (7.17)

where , , and is the first eigenvalue of the opera-tor ;

b) there exist numbers such that

, (7.18)

with some constant . We also require the existence of and such that

, . (7.19)

These assumptions enable us to state (see Sections 4.3 and 4.5) that if

, , , , (7.20)

then problem (7.15) and (7.16) is uniquely solvable in the class of functions

. (7.21)

Therewith there exists such that

, (7.22)

for any solution to problem (7.15) and (7.16).

Theorem 7.2.

Assume that conditions Assume that conditions Assume that conditions Assume that conditions (7.17)–(7.20) hold. Let hold. Let hold. Let hold. Let

be a set of continuous linear functionals on be a set of continuous linear functionals on be a set of continuous linear functionals on be a set of continuous linear functionals on .... Then there exists Then there exists Then there exists Then there exists

depending both on and the parameter of equation depending both on and the parameter of equation depending both on and the parameter of equation depending both on and the parameter of equation (7.15) such that the such that the such that the such that the

condition implies that is an asymptotically deter-condition implies that is an asymptotically deter-condition implies that is an asymptotically deter-condition implies that is an asymptotically deter-

mining set of functionals with respect to for problem mining set of functionals with respect to for problem mining set of functionals with respect to for problem mining set of functionals with respect to for problem (7.15) and and and and

(7.16) in the class of solutions in the class of solutions in the class of solutions in the class of solutions ,,,, i.e. for two solutions and i.e. for two solutions and i.e. for two solutions and i.e. for two solutions and

from the conditionfrom the conditionfrom the conditionfrom the condition

,,,, (7.23)


H

u�� u� A2u M A1 2� u2

& '( ) Au Lu� � � � p t� � ,�

ut 0� u0 , u�

t 0� u1 .� ��!"!$

(7.15)

(7.16)

A M z� �C1 R+� �

z� � M ;� � ;d

0

z

� a z� b��

0 a 71�� b R 71A

aj 0�

zM z� � a1 z� �� a2 z1 �� a3�� z 0�

� 0�0 ? 1�� C 0�

Lu C A?u� u D A?� �

u0 D A� � u1 H p t� � L� R+ H�� 0�

� C R+ D A� �� C1 R+ H��

R 0�

Au t� � 2u� t� � 2� R2� t t0 u0 u1��

u t� � �

� lj: j 1 2 � N� � ��

D A� � +0 0�R

+ +�

D A� � H�� +0� �

D A� � H�� u1 t� � u2 t� �

�

lj

u1 �� u2 �� 2 �d

t

t 1�

�t ��lim 0� l

j�


C

h

a

p

t

e

r

5

.... (7.24)

Proof.

Let and be solutions to problem (7.15) and (7.16) lying in . Dueto equation (7.22) we can assume that these solutions possess the property

, , . (7.25)

Let us consider the function as a solution to equation

, (7.26)

where

.

It follows from (7.19) and (7.25) that

. (7.27)

Let us consider the functional

(7.28)

on the space , where

and the positive parameters and will be chosen below. It is clear that for we have

,

where . Moreover,

.

Therefore, the value can be chosen such that

(7.29)

for all , where and are positive numbers depending on . Let usnow estimate the value . Due to (7.26) we have that

u�1 t� � u� 2 t� �� 2A u1 t� � u2 t� �� 2�

� " #$ %

t ��lim 0�

u1 t� � u2 t� � �

Auj

t� � 2u� j t� � 2� R2� t 0� j 1 2��

u t� � u1 t� � u2 t� ��

u�� u� A2u M A1 2� u1 t� � 2& '( ) Au� � � F u1 t� � u2 t� ��

F u1 u2�� M A1 2� u22

& '( ) M A1 2� u1

2& '( )� Au2 L u1 u2��

F u1 t� � u2 t� �� CR

A1 2� u A?u�& '( )�

V u u� t�� 12�� E u u� t�� - u u��

�2�� u

2�� " #$ %

��

� D A� � H��

E u u� t�� u�2

Au2

M A1 2� u1 t� � 2& '( ) A1 2� u

2 6 u2� � ��

6 -u u��

E u u� t�� u�2

Au2

mR

A1 2� u2 6 u

2� � ��

mR

M z� � : 0 z 711� R2� �� min�

12�� u�

2� u u�� 2�� u�

2� 12�� u�

2 � u2��

6

�1 Au2

u�2�& '

( ) V u t�� 2 Au2

u�2�& '

( )� �

0 - �� 1 �2 R

d td/� �V u t� � u� t� � t��

12��

dtd

�� E u t� � u� t� � t�� u� t� � 2� 6 u t� � u� t� ��

M 9 A1 2� u1 t� � 2& '( ) Au1 t� � u�1 t� �� A1 2� u t� � 2

F u1 t� � u2 t� �� u� t� �� .

� �

� �

�


With the help of (7.25) and (7.27) we obtain that

.

Using (7.26) and (7.27) it is also easy to find that

where

.

We choose and use the estimate of the form

, , ,

to obtain that

Therefore, using the estimate

and equation (7.29) we obtain the inequality

,

provided . Here is a positive constant. As above, thiseasily implies (7.24), provided (7.23) holds. Theorem 7.2 is proved.

Show that the method used in the proof of Theorem 7.2 alsoenables us to obtain the assertion of Theorem 7.1 for problem (7.1).

Using the results of Section 4.2 related to the linear variant ofequation (7.15), prove that the method of the proof of Theorem 7.1can also be applied in the proof of Theorem 7.2.

Thus, the methods presented in the proofs of Theorems 7.1 and 7.2 are close to eachother. The same methods with slight modifications can also be used in the studyof problems like (7.1) with additional retarded terms (see [3]).

12��

dtd

�� E u u� t�� 2�� u� t� � 2� CR A1 2� u

2A?u

2�& '( )��

dtd

�� u u�� 2�� u u��

� " #$ %

u�2

u u�� u��

u�2

Au2� M

RA1 2� u

2C

RA1 2� u

2A?u

2�& '( ) u ,� �

�

�

MR

M z� � : 0 z 711�R2� �

� " #$ %

max�

- � 4/�

A2u Au2

u1 2�� + Au C+ u�� 0 2 1� � + 0�

dtd

��V u u� t�� dtd

�� 12�� E u u� t�� - u u�

�2��u��& '

( )�� " #$ %

�8�� Au

2u� 2�& '

( )� CR

u t� � 2 .�

�

�

�

u t� � 2C�

lj u� � 2

jmax 2 +

�D A� � H�� Au

2��

dtd

��V u t� � u� t� � t�� JV u t� � u� t� � t�� C lj u t� �� 2

jmax�

+�

D A� � H�� +0��

16�� C

R1�� J

E x e r c i s e 7.1

E x e r c i s e 7.2


C

h

a

p

t

e

r

5

Using the estimates for the difference of two solutions to equ-ation (7.15) proved in Lemmata 4.6.1 and 4.6.2, find an analogueof Theorems 1.3 and 4.4 for the problem (7.15) and (7.16).

§ 8 On Boundary Determining Functionals§ 8 On Boundary Determining Functionals§ 8 On Boundary Determining Functionals§ 8 On Boundary Determining Functionals

The fact (see Sections 5–7 as well as paper [3]) that in some cases determining func-tionals can be defined on some auxiliary space admits in our opinion an interestinggeneralization which leads to the concept of boundary determining functionals.We now clarify this by giving the following simple example.

In a smooth bounded domain we consider a parabolic equation withthe nonlinear boundary condition

(8.1)

Assume that is a positive parameter, and are continuously differenti-able functions on such that

, , (8.2)

where and are constants. Let

. (8.3)

Here is a set of functions on that are twice continu-ously differentiable with respect to and continuously differentiable with respectto . The notation has a similar meaning, the bar denotes the closureof a set.

Let and be two solutions to problem (8.1) lying in the class (we do not discuss the existence of such solutions here and refer the reader to

the book [7]). We consider the difference . Then (8.1) evidentlyimplies that

E x e r c i s e 7.3

@ Rd0

uAtA�� - uM f u� � , x @ , t 0 ,��

uAnA�� A@

h u� � A@� 0 , ut 0� u0 x� � .� �

�!!"!!$

- f z� � h z� �R1

f 9 z� � �� h 9 z� � 2�� 0� 2 0�

� C2 1� @ R+�� C1 0� @ R+��

C2 1� @ R+�� u x t�� @ R+�x

t C1 0� @ R+��

u1 x t�� u2 x t��

u t� � u1 t� � u2 t� ��

12��

dtd

�� u t� �L2 @� �2 - Eu t� �

L2 @� �2

f u1 t� �� f u2 t� �� u t� �� L2 @� �

� �

- u t� � h u1 t� �� h u2 t� �� .d

A@��

�

�

O n B o u n d a r y D e t e r m i n i n g F u n c t i o n a l s 359

Using (8.2) we obtain that

. (8.4)

One can show that there exist constants and depending on the domain only and such that

, (8.5)

. (8.6)

Here is the Sobolev space of the order on the boundary of the domain. Equations (8.4)–(8.6) enable us to obtain the following assertion.

Theorem 8.1.

Let be a set of continuous linear functionalsLet be a set of continuous linear functionalsLet be a set of continuous linear functionalsLet be a set of continuous linear functionals

on the space on the space on the space on the space .... Assume that and Assume that and Assume that and Assume that and

,,,, (8.7)

where the constants , and are defined in equations where the constants , and are defined in equations where the constants , and are defined in equations where the constants , and are defined in equations (8.1),,,,

(8.2), , , , (8.5), and , and , and , and (8.6). Then is an asymptotically determining set with. Then is an asymptotically determining set with. Then is an asymptotically determining set with. Then is an asymptotically determining set with

respect to for problem respect to for problem respect to for problem respect to for problem (8.1) in the class of classical solutions in the class of classical solutions in the class of classical solutions in the class of classical solutions ....

Proof.

Let , where are solutions to problem (8.1).Theorem 2.1 implies that

(8.8)

for any . Equations (8.5) and (8.6) imply that

.

Therefore, equation (8.4) gives us that

Using estimate (8.5) once again we get

(8.9)

12��

dtd

�� u t� �L2 @� �2 - Eu t� �

L2 @� �2 � u t� �

L2 @� �2�� -2 u t� �

L2 A@� �2��

c1 c2 @

uL2 @� �2 c1 u

L2 A@� �2 Eu

L2 @� �2�& '

( )�

uH1 2� A@� �2

c2 uL2 @� �2 Eu

L2 @� �2�& '

( )�

Hs A@� � s

@

� lj: j 1 � N� ��

H1 2� A@� � � c1 -�

+�

+�

H1 2� A@� � L2 A@� �� - �c1�

- 1 c1�� c2 1 2��

1 2�+0�

- � 2 c1� � � c2�

L2 @� � �

u t� � u1 t� � u2 t� �� uj

t� � �

uL2 A@� �2 C

� ,� lj u� � 2 1 ,�� +�

2u

H1 2� @� �2�

jmax�

, 0�

uL2 A@� �2

C� ,� l

ju� � 2

jmax 1 c1�� c2 1 ,�� +

�

2u

L2 A@� �2 Eu

L2 @� �2�& '

( )��

12��

dtd

�� uL2 @� �2 - u

L2 A @� �2 Eu

L2 @� �2� � u t� �

L2 @� �2��

- 1 2�� u t� �L2 A@� �

- 1 c1�� c2 1 ,�� 1 2�� +�

2u

L2 A@� �2 Eu

L2 @� �2�& '

( )�

C lj u� � 2 .j

max

�

�

�

�

12��

dtd

�� u2 -

c1�� 1 1 c1�� c2 1 ,�� 1 2�� +

�

2� �c1-��

� " #$ %

u2� C l

ju� � 2

jmax� ,


C

h

a

p

t

e

r

5

provided that

. (8.10)

It is evident that (8.10) with some follows from (8.7). Therefore, inequality(8.9) enables us to complete the proof of the theorem.

Thus, the analogue of Theorem 3.1 for smooth surfaces enables us to state that prob-lem (6.1) has finite determining sets of boundary local surface averages.

An assertion similar to Theorem 8.1 can also be obtained (see [3]) for a nonlinearwave equation of the form

, , ,

, , , .

Here is a smooth open subset on the boundary of , and are bound-ed continuously differentiable functions, and and are positive parameters.

1 1 c1�� c2 1 ,�� 1 2�� +�

2� �c1-�� 0�

, 0�

A2u

At2�� uAtA�� uM f u� �� x @ Rd0 t 0�

uAnA��

� uAtA

��

�� 8 u �& '

( )�� u A@ \� 0� ut 0� u0 x� �� uA

tA��

t 0�u1 x� ��

� @ f u� � 8 u� ��


1. FOIAS C., PRODI G. Sur le comportement global des solutions non stationnaires

des equations de Navier-Stokes en dimension deux // Rend. Sem. Mat. Univ. Padova. –1967. –Vol. 39. –P. 1–34.

2. LADYZHENSKAYA O. A. A dynamical system generated by Navier-Stokes

equations // J. of Soviet Mathematics. –1975. –Vol. 3. –P. 458–479.3. CHUESHOV I. D. Theory of functionals that uniquely determine the asymptotic

dynamics of infinite-dimentional dissipative systems // Russian Math. Surveys. –1998. –Vol. 53. –#4. –P. 731–776.

4. LIONS J.-L., MAGENES E. Non-homogeneous boundary value problems and ap-

plications. –New York: Springer, 1972. –Vol.1.5. MAZ’YA V. G. Sobolev spaces. –Leningrad: Leningrad Univ. Press, 1985

(in Russian).6. TRIEBEL H. Interpolation theory, function spaces, differential operators.

–Amsterdam: North-Holland, 1978.7. LADYZHENSKAYA O. A., SOLONNIKOV V. A., URALTCEVA N. N. Linear and quasiline-

ar equations of parabolic type. –Providence: Amer. Math. Soc., 1968.8. BABIN A. V., VISHIK M. I. Attractors of evolution equations. –Amsterdam:

North-Holland, 1992.9. TEMAM R. Infinite-dimensional dynamical systems in Mechanics and

Physics. –New York: Springer, 1988.10. SMOLLER J. Shock waves and reaction-diffusion equations. –New York:

Springer, 1983.11. TEMAM R. Navier-Stokes equations, theory and numerical analysis. 3rd edn.

–Amsterdam: North-Holland, 1984.12. FOIAS C., MANLEY O. P., TEMAM R., TREVE Y. M. Asymptotic analysis of the

Navier-Stokes equations // Physica D. –1983. –Vol. 9. –P. 157–188.13. HASSARD B. D., KAZARINOFF N. D., WAN Y.-H. Theory and application of Hopf

bifurcation. –Cambridge: Cambridge Univ. Press, 1981.14. PROMISLOW K. Time analyticity and Gevrey regularity for solutions of a class

of dissipative partial differential equations // Nonlinear Anal., TMA. –1991. –Vol. 16. –P. 959–980.

15. KUKAVICA I. On the number of determining nodes for the Ginzburg-Landau

equation // Nonlinearity. –1992. –Vol. 5. –P. 997–1006.

C h a p t e r 6

Homoclinic ChaosHomoclinic ChaosHomoclinic ChaosHomoclinic Chaos

in Infinite-Dimensional Systemsin Infinite-Dimensional Systemsin Infinite-Dimensional Systemsin Infinite-Dimensional Systems

C o n t e n t s

. . . . § 1 Bernoulli Shift as a Model of Chaos . . . . . . . . . . . . . . . . . . . . 365

. . . . § 2 Exponential Dichotomy and Difference Equations . . . . . . . 369

. . . . § 3 Hyperbolicity of Invariant Sets for Differentiable Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 377

. . . . § 4 Anosov’s Lemma on -trajectories . . . . . . . . . . . . . . . . . . . . 381

. . . . § 5 Birkhoff-Smale Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

. . . . § 6 Possibility of Chaos in the Problemof Nonlinear Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . 396

. . . . § 7 On the Existence of Transversal Homoclinic Trajectories . . 402

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

�

In this chapter we consider some questions on the asymptotic behaviour of a dis-crete dynamical system. We remind (see Chapter 1) that a discrete dynamical sys-tem is defined as a pair consisting of a metric space and a continuousmapping of into itself. Most assertions on the existence and properties of attrac-tors given in Chapter 1 remain true for these systems. It should be noted that the fol-lowing examples of discrete dynamical systems are the most interesting from thepoint of view of applications: a) systems generated by monodromy operators (periodmappings) of evolutionary equations, with coefficients being periodic in time;b) systems generated by difference schemes of the type

, in a Banach space (see Examples 1.5 and 1.6 of Chap-ter 1).

The main goal of this chapter is to give a strict mathematical description of oneof the mechanisms of a complicated (irregular, chaotic) behaviour of trajectories.We deal with the phenomenon of the so-called homoclinic chaos. This phenomenonis well-known and is described by the famous Smale theorem (see, e.g., [1–3]) for fi-nite-dimensional systems. This theorem is of general nature and can be proved forinfinite-dimensional systems. Its proof given in Section 5 is based on an infinite-di-mensional variant of Anosov’s lemma on -trajectories (see Section 4). The conside-rations of this Chapter are based on the paper [4] devoted to the finite-dimensionalcase as well as on the results concerning exponential dichotomies of infinite-dimen-sional systems given in Chapter 7 of the book [5]. We follow the arguments given in [6]while proving Anosov’s lemma.

§ 1 Bernoulli Shift as a Model of Chaos§ 1 Bernoulli Shift as a Model of Chaos§ 1 Bernoulli Shift as a Model of Chaos§ 1 Bernoulli Shift as a Model of Chaos

Mathematical simulation of complicated dynamical processes which take place in realsystems requires that the notion of a state of chaos be formalized. One of the possibleapproaches to the introduction of this notion relies on a selection of a class of expli-citly solvable models with complicated (in some sense) behaviour of trajectories.Then we can associate every model of the class with a definite type of chaotic beha-viour and use these models as standard ones comparing their dynamical structurewith a qualitative behaviour of the dynamical system considered. A discrete dynami-cal system known as the Bernoulli shift is one of these explicitly solvable models.

Let and let

,

i.e. is a set of two-sided infinite sequences the elements of which are the inte-gers . Let us equip the set with a metric

X S�� X

X

� 1� un 1� u

n��

F un

� �� n 0 1 2 � � �� X

�

m 2

�m

x x 1� x0 x1 � � � �� : xj

1 2 m� � �� , j Z��

�

�m

1 2 m� � � �m

366 H o m o c l i n i c C h a o s i n I n f i n i t e - D i m e n s i o n a l S y s t e m s

6

C

h

a

p

t

e

r

. (1.1)

Here and are elements of . Other methodsof introduction of a metric in are given in Example 1.1.7 and Exercise 1.1.5.

Show that the function satisfies all the axioms ofa metric.

Let and be elements of the set .Assume that for and for some integer . Prove that

.

Assume that equation holds for ,where is a natural number. Show that for all (Hint: if ).

Let and let

. (1.2)

Prove that for any the relation

holds, where is an integer with the property

.

Show that the space with metric (1.1) is a compact met-ric space.

In the space we define a mapping which shifts every sequence one symbolleft, i.e.

, , .

Evidently, is invertible and the relations

,

hold for all . Therefore, the mapping is a homeomorphism.The discrete dynamical system is called the Bernoulli shift the Bernoulli shift the Bernoulli shift the Bernoulli shift of the

space of sequences of symbols. Let us study the dynamical properties of the sys-tem .

Prove that has fixed points exactly. What struc-ture do they have?

d x y�� 2 i�x

iy

i�

1 xi yi��

i ��

�

��

x xi : i Z�� y yi : i Z�� m

�m

E x e r c i s e 1.1 d x y��

E x e r c i s e 1.2 x xi

� � y yi

� � �m

xi

yi

� i N� N

d x y�� 2 N� 1��

E x e r c i s e 1.3 d x y�� 2 N�� x y� �m�N xi yi� i N 1��d x y�� 2 i� 1� xi yi�

E x e r c i s e 1.4 x �m

�

�N x� � y �m : xi yi� for i N��

0 � 1� �

�N �� 2� x� � y : d x y�� N �� 1� x� �� N ��

N �� 1 ��ln2ln

�� N �� 1��


�m

S

Sx� �i

xi 1�� i Z� x x

i� �

m��

S

d S x S y�� 2 d x y�� d S 1� x S 1� y�� 2 d x y�� x y� �

m� S

�m

S�� m

�m

S��


S�� m

B e r n o u l l i S h i f t a s a M o d e l o f C h a o s 367

We call an arbitrary ordered collection with a segment (of the length ). Each element can be considered as an or-dered infinite family of finite segments while the elements of the set can be con-structed from segments. In particular, using the segment we canconstruct a periodic element by the formula

, , . (1.3)

Let be a segment of the length and let be an element defined by (1.3). Prove that is a periodic

point of the period of the dynamical system , i.e..

Prove that for any natural there exists a periodic pointof the minimal period equal to .

Prove that the set of all periodic points is dense in , i.e. forevery and there exists a periodic point with theproperty (Hint: use the result of Exercise 1.4).

Prove that the set of nonperiodic points is not countable.

Let and be fi-xed points of the system . Let be an elementof such that for and for , where

and are natural numbers. Prove that

, . (1.4)

Assume that an element possesses property (1.4) with and .If , then the set

is called a heteroclinic trajectory heteroclinic trajectory heteroclinic trajectory heteroclinic trajectory that connects the fixed points and .If , then is called a homoclinic trajectory homoclinic trajectory homoclinic trajectory homoclinic trajectory of the point . Theelements of a heteroclinic (homoclinic, respectively) trajectory are called hetero-clinic (homoclinic, respectively) points.

Prove that for any pair of fixed points there exists an infinitenumber of heteroclinic trajectories connecting them whereas thecorresponding set of heteroclinic points is dense in .

Let

and

a �1 �N

� �� j

1 m� �� N x �

m�

�m

a �1 �N

� �� a �

m�

aNk j� �

j� j 1 N� �� k Z�

E x e r c i s e 1.7 a �1 �N� �� N

a �m� a

N �m S�� SN a a�

E x e r c i s e 1.8 N

N


x �m

� � 0� a

d x a��


E x e r c i s e 1.11 a � � � � � � �� b � � � ��

mS�� C c

i� �

�m

ci

�� i M1�� ci

� i M2M1 M2

Sncn ��!

lim a� Sncn �!

lim b�

c �m

� c a� c b�a b�

"a b� Snc : n Z��

a b

a b� "a "a a�� a


�m


"1 Sna : n Z�� Sna : n 0 1 N1 1��

"2 Snb : n Z�� Snb : n 0 1 N2 1��


6

C

h

a

p

t

e

r

be cycles (periodic trajectories). Prove that there exists a hetero-clinic trajectory that connects the cycles and , i.e. such that

,

and

, .

For every there exists only a finite number of segments of the length . There-fore, the set of all segments is countable, i.e. we can assume that

, therewith the length of the segment is not less than the lengthof . Let us construct an element from taking for

and sequentially putting all the segments to the right of the zeroth posi-tion. As a result, we obtain an element of the form

, . (1.5)

Prove that a positive semitrajectory with having the form (1.5) is dense in , i.e. for every and

there exists such that .

Prove that the semitrajectory constructed in Exercise 1.14returns to an -vicinity of every point infinite numberof times (Hint: see Exercises 1.4 and 1.9).

Construct a negative semitrajectory which is dense in .

Thus, summing up the results of the exercises given above, we obtain the followingassertion.

Theorem 1.1.

The dynamical system of the Bernoulli shift of sequencesThe dynamical system of the Bernoulli shift of sequencesThe dynamical system of the Bernoulli shift of sequencesThe dynamical system of the Bernoulli shift of sequences

of symbols possesses the properties:of symbols possesses the properties:of symbols possesses the properties:of symbols possesses the properties:

1) there exists a finite number of fixed points;there exists a finite number of fixed points;there exists a finite number of fixed points;there exists a finite number of fixed points;

2) there exist periodic orbits of any minimal period and the set of thesethere exist periodic orbits of any minimal period and the set of thesethere exist periodic orbits of any minimal period and the set of thesethere exist periodic orbits of any minimal period and the set of these

orbits is dense in the phase space ;orbits is dense in the phase space ;orbits is dense in the phase space ;orbits is dense in the phase space ;

3) the set of nonperiodic points is uncountable;the set of nonperiodic points is uncountable;the set of nonperiodic points is uncountable;the set of nonperiodic points is uncountable;

4) heteroclinic and homoclinic points are dense in the phase space;heteroclinic and homoclinic points are dense in the phase space;heteroclinic and homoclinic points are dense in the phase space;heteroclinic and homoclinic points are dense in the phase space;

5) there exist everywhere dense trajectories.there exist everywhere dense trajectories.there exist everywhere dense trajectories.there exist everywhere dense trajectories.

All these properties clearly imply the extraordinarity and complexity of the dyna-mics in the system . They also give a motivation for the following definitions.

"1 2� Snc : n Z�� "1"2

Snc "1�� dist d Snc x�� x "1�

inf 0!# n ��!

Snc "2�� dist d Snc x�� x "2�

inf 0!� n +�!

N N

� � ak :��k 1 2 � �� ak 1�

ak b bi : i Z�� m bi 1�i 0� a

k

b 1 1 1 a1 a2 a3 � � � � � � �� aj ��

E x e r c i s e 1.14 "+ Snb , n 0� �b �

mx �

m�

� 0� n n x �� d x Snb��

E x e r c i s e 1.15 "+� x �m�

E x e r c i s e 1.16 "– Snc : n 0��

m

�m

S�� m

�m

�m S��

E x p o n e n t i a l D i c h o t o m y a n d D i f f e r e n c e E q u a t i o n s 369

Let be a discrete dynamical system. The dynamics of the system is called chaotic chaotic chaotic chaotic if there exists a natural number such that the mapping istopologically conjugate to the Bernoulli shift for some , i.e. there exists a homeo-morphism such that for all . We also saythat chaotic dynamics is observed in the system if there exist a number and a set in invariant with respect to such that the restrictionof to is topologically conjugate to the Bernoulli shift.

It turns out that if a dynamical system has a fixed point and a correspon-ding homoclinic trajectory, then chaotic dynamics can be observed in this systemunder some additional conditions (this assertion is the core of the Smale theorem).Therefore, we often speak about homoclinic chaos in this situation. It should also benoted that the approach presented here is just one of the possible methods used todescribe chaotic behaviour (for example, other approaches can be found in [1] aswell as in book [7], the latter contains a survey of methods used to study the dynam-ics of complicated systems and processes).

§ 2 Exponential Dichotomy§ 2 Exponential Dichotomy§ 2 Exponential Dichotomy§ 2 Exponential Dichotomy

and Difference Equationsand Difference Equationsand Difference Equationsand Difference Equations

This is an auxiliary section. Nonautonomous linear difference equations of the form

, , (2.1)

in a Banach space are considered here. We assume that is a family of linearbounded operators in , is a sequence of vectors from . Some results bothon the dichotomy (splitting) of solutions to homogeneous equation (2.1)and on the existence and properties of bounded solutions to nonhomogeneous equa-tion are given here. We mostly follow the arguments given in book [5] as well asin paper [4] devoted to the finite-dimensional case.

Thus, let be a sequence of linear bounded operators in a Banachspace . Let us consider a homogeneous difference equation

, , (2.2)

where is an interval in , i.e. a set of integers of the form

,

where and are given numbers, we allow the cases and .Evidently, any solution to difference equation (2.2) possesses the pro-perty

, , ,

X f�� X f�� k f k

m

h : X �m

! h f k x� �� S h x� �� x X�X f�� k

Y X f k f kY Y�� f k Y

X f��

xn 1� An xn hn�� n Z�

X An

� X h

nX

hn

0#� �

An

: n Z�� X

xn 1� A

nx

n� n J�

J Z

J n Z : m1 n m2� ��

m1 m2 m1 �� m2 +��x

n: n J��

xm

$ m n�� xn

� m n m n� J�


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where for and . The mapping is called an evolutionary operator evolutionary operator evolutionary operator evolutionary operator of problem (2.2).

Prove that for all we have

.

Let be a family of projectors (i.e. ) in such that . Show that

, , ,

i.e. the evolutionary operator maps into .

Prove that solutions to nonhomogeneous differenceequation (2.1) possess the property

, .

Let us give the following definition. A family of linear bounded operators is said to possess an exponential dichotomy exponential dichotomy exponential dichotomy exponential dichotomy over an interval with constants

and if there exists a family of projectors such that

a) , ;

b) , , ;

c) for the evolutionary operator is a one-to-one mappingof the subspace onto and the following estimateholds:

, , .

If these conditions are fulfilled, then it is also said that difference equation (2.2) ad-mits an exponential dichotomy over . It should be noted that the cases and

are the most interesting for further considerations, where isthe set of all nonnegative (nonpositive) integers.

The simplest case when difference equation (2.2) admits an exponential dicho-tomy is described in the following example.

E x a m p l e 2.1 (autonomous case)

Assume that equation (2.2) is autonomous, i.e. for all , and the spec-trum does not intersect the unit circumference . Linearoperators possessing this property are often called hyperbolic (with respect tothe fixed point ). It is well-known (see, e.g., [8]) that in this case thereexists a projector with the properties:

$ m n�� Am 1� A

n% %� m n� $ m m�� I�

$ m n��

E x e r c i s e 2.1 m n k $ m k�� $ m n�� $ n k��

E x e r c i s e 2.2 Pn

: n J�� Pn2 P

n� X

Pn 1� A

nA

nPn

�

Pm$ m n�� $ m n�� P

n� m n m n� J�

$ m n�� Pn

X Pm

X

E x e r c i s e 2.3 xn�

xm

$ m n�� xn

$ m k 1�� hk

k n�

m 1�

�� m n�

An

� J

K 0� 0 q 1� � Pn

: n J��

Pn 1� A

nA

nP

n� n n 1�� J�

$ m n�� Pn

K qm n�� m n m n� J�

n m $ n m�� 1 P

m�� X 1 P

n�� X

$ n m�� 1� 1 Pn�� K qn m�� m n� m n� J�

J J Z�J Z &� Z+ Z–� �

An A# n

' A� � z C : z =1��

x 0�P


a) , i.e. the subspaces and are invariant with re-spect to ;

b) the spectrum of the restriction of the operator to liesstrictly inside of the unit disc;

c) the spectrum of the restriction of to the subspace lies outside the unit disc.

Let be a linear bounded operator in a Banach space andlet be its spectral radius. Show that forany there exists a constant such that

,

(Hint: use the formula the proof of which can befound in [9], for example).

Applying the result of Exercise 2.4 to the restriction of the operator to , we ob-tain that there exist and such that

, . (2.3)

It is also evident that the restriction of the operator to is invertible andthe spectrum of the inverse operator lies inside the unit disc. Therefore,

, , (2.4)

where the constants and can be chosen the same as in (2.3). Theevolutionary operator of the difference equation has theform , . Therefore, the equality and estimates(2.3) and (2.4) imply that the equation admits an exponential dicho-tomy over , provided the spectrum of the operator does not intersect the unitcircumference.

Assume that for the operator there exists a projector such that and estimates (2.3) and (2.4) hold with .Show that the spectrum of the operator does not intersect theunit circumference, i.e. is hyperbolic.

Thus, the hyperbolicity of the linear operator is equivalent to the exponential di-chotomy over of the difference equation with the projectors in-dependent of . Therefore, the dichotomy property of difference equation (2.2)should be considered as an extension of the notion of hyperbolicity to the nonauto-nomous case. The meaning of this notion is explained in the following two exercises.

Let be a hyperbolic operator. Show that the space can bedecomposed into a direct sum of stable and unstable sub-spaces, i.e. therewith

AP P A� PX 1 P�� XA

' APX

� � A PX

' A 1 P�� X� � A

1 P�� X

E x e r c i s e 2.4 C X

( z : z ' C� �� max#q (� M

q1

Cn Mq

qn� n 0 1 2 � � ��

( Cn 1 n�

n �!lim�

A PX

K 0� 0 q 1� �

AnP K qn� n 0A 1 P�� X

A n� 1 P�� K qn� n 0K 0� 0 q 1� �$ m n�� x

n 1� Axn

�$ m n�� Am n�� m n AP PA�

xn 1� Ax

n�

Z A

E x e r c i s e 2.5 A P

AP PA� 0 q 1� �A

A

A

Z xn 1� Ax

n� P

n

n

E x e r c i s e 2.6 A X

Xs Xu

X Xs Xu��


6

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, , ,

, , ,

with some constants and .

Let be a plane and let be an operator defined bythe formula

, .

Show that the operator is hyperbolic. Evaluate and display gra-phically stable and unstable subspaces on the plane. Displaygraphically the trajectory of some point that liesneither in , nor in .

The next assertion (its proof can be found in the book [5]) plays an important role inthe study of existence conditions of exponential dichotomy of a family of operators

.

Theorem 2.1.

Let be a sequence of linear bounded operators in a Ba-Let be a sequence of linear bounded operators in a Ba-Let be a sequence of linear bounded operators in a Ba-Let be a sequence of linear bounded operators in a Ba-

nach space nach space nach space nach space .... Then the foll Then the foll Then the foll Then the folloooowing assertions are equivalent:wing assertions are equivalent:wing assertions are equivalent:wing assertions are equivalent:

(i) the sequence possesses an exponential dichotomy overthe sequence possesses an exponential dichotomy overthe sequence possesses an exponential dichotomy overthe sequence possesses an exponential dichotomy over

,,,,

(ii) for any bounded sequence from there exists a uniquefor any bounded sequence from there exists a uniquefor any bounded sequence from there exists a uniquefor any bounded sequence from there exists a unique

bounded solution to the nonhomogeneous differencebounded solution to the nonhomogeneous differencebounded solution to the nonhomogeneous differencebounded solution to the nonhomogeneous difference


,,,, .... (2.5)

In the case when the sequence possesses an exponential dichotomy, solutionsto difference equation (2.5) can be constructed using the Green function the Green function the Green function the Green function :

Prove that .

Prove that for any bounded sequence from a solution to equation (2.5) has the form

, .

Anx K qn x� x Xs� n 0

Anx K 1� q n� x x Xu� n 0

K 0� 0 q 1� �

E x e r c i s e 2.7 X R2� A

A x1 x2�� 2 x1 x2 x1 x2�)�� x x1 x2)� � R2��

A

Xs Xu

Anx : n Z�� x

Xs Xu

An

: n Z��

An : n Z�� X

An : n Z�� Z

hn : n Z�� X

xn : n Z��

xn 1� A

nx

nh

n�� n Z�

An

�

G n m�� $ n m�� P

m, n m ,

$ m n�� 1�� 1 Pm�� , n m .��*�*�

�

E x e r c i s e 2.8 G n m�� K q n m��

E x e r c i s e 2.9 hn

: n Z�� X

xn

G n m 1�� hm

m Z�� n Z�


Moreover, the following estimate is valid:

.

The properties of the Green function enable us to prove the following assertionon the uniqueness of the family of projectors .

Lemma 2.1.

Let a sequence possess an exponential dichotomy over . Then

the projectors are uniquely defined.

Proof.

Assume that there exist two collections of projectors and forwhich the sequence possesses an exponential dichotomy. Let and be Green functions constructed with the help of these collec-tions. Then Theorem 2.1 enables us to state (see Exercise 2.9) that

for all and for any bounded sequence . Assuming that for and for , we find that

, , , .

This equality with gives us that . Thus, the lemma is proved.

In particular, Theorem 2.1 implies that in order to prove the existence of an expo-nential dichotomy it is sufficient to make sure that equation (2.5) is uniquely solv-able for any bounded right-hand side. It is convenient to consider this differenceequation in the space of sequences of elements of for which the norm

(2.6)

is finite. Assume that the condition

(2.7)

is valid. Then for any the sequence lies in .Consequently, equation

, (2.8)

defines a linear bounded operator acting in the space . Therewith as-sertion (ii) of Theorem 2.1 is equivalent to the assertion on the invertibility of theoperator given by equation (2.8).

xn

nsup K

1 q�1 q�� h

nnsup�

Pn�

An

� ZP

n: n Z��

Pn

� Qn

� A

n� G

Pn m��

GQ

n m��

GP n m 1�� hm

m Z�� GQ n m 1�� hm

m Z��

n Z� hn

� X� hm

0�m k 1�� h

mh� m k 1��

GP n k�� h GQ n k�� h� h X� n k� Z� n k

n k� Pn h Qnh�

lX

�l� Z X�� # xxxx x

n: n Z�� X

xxxxl�

xn

� l�

xn

: n Z��

sup� �

An

: n Z��

sup ��

xxxx xn

� lX

�� yn

xn

An 1� x

n 1�� lX

�

L xxxx� �n xn

An 1� x

n 1�� xxxx xn

� lX

��

lX

�l� Z X��

L


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The assertion given below provides a sufficient condition of invertibility of theoperator . Due to Theorem 2.1 this condition guarantees the existence of an expo-nential dichotomy for the corresponding difference equation. This assertion will beused in Section 4 in the proof of Anosov’s lemma. It is a slightly weakened variantof a lemma proved in [6].

Theorem 2.2.

Assume that a sequence of operators satisfies conditionAssume that a sequence of operators satisfies conditionAssume that a sequence of operators satisfies conditionAssume that a sequence of operators satisfies condition

(2.7).... Let there exist a family of projectors such that Let there exist a family of projectors such that Let there exist a family of projectors such that Let there exist a family of projectors such that

,,,, ,,,, (2.9)

,,,, ,,,, (2.10)

for all . We also assume that the operator is invertiblefor all . We also assume that the operator is invertiblefor all . We also assume that the operator is invertiblefor all . We also assume that the operator is invertible

as a mapping from into and the estimatesas a mapping from into and the estimatesas a mapping from into and the estimatesas a mapping from into and the estimates

,,,, (2.11)

are valid for every . Ifare valid for every . Ifare valid for every . Ifare valid for every . If

,,,, ,,,, (2.12)

then the operator acting in according to formula then the operator acting in according to formula then the operator acting in according to formula then the operator acting in according to formula (2.8) is invertible is invertible is invertible is invertible

and and and and ....

Proof.

Let us first prove the injectivity of the mapping . Assume that there exists anonzero element such that , i.e. for all .Let us prove that the sequence possesses the property

(2.13)

for all . Indeed, let there exist such that

. (2.14)

It is evident that this equation is only possible when . Let us con-sider the value

(2.15)

It is clear that

.

Since

,

L

An : n Z�� Qn : n Z��

Qn

K� 1 Qn

� K�

Qn 1� An 1 Qn�� +� 1 Qn 1�� An Qn +�

n Z� 1 Qn 1�� A

n

1 Qn

�� X 1 Qn 1�� X

An

Qn

,� 1 Qn 1�� A

n� � 1� 1 Q

n 1�� ,�

n Z�

K, 18�� + 1

8��

L lX

�

L 1� 2 K 1��

L

xxxx xn

� lX

�� Lxxxx 0� xn

An 1� x

n 1�� n Z�x

n�

1 Qn�� xn Qn xn�

n Z� m Z�

1 Qm

�� xm

Qm

xm

�

1 Qm

�� xm

0�

Nm 1� 1 Qm 1�� xm 1� Qm 1� xm 1��#

1 Qm 1�� Am xm Qm 1� Am xm .�

�

�

1 Qn 1�� A

nx

n1 Q

n 1�� An

1 Qn

�� xn

1 Qn 1�� A

nQ

nx

n�

1 Qn 1�� A

n� � 1� 1 Q

n 1�� An

1 Qn

�� 1 Qn

��


it follows from (2.11) that

for every and for all . Therefore, we use estimates (2.10) to find that

. (2.16)

Then it is evident that

Therefore, estimates (2.9)–(2.11) imply that

, . (2.17)

Thus, equations (2.15)–(2.17) lead us to the estimate

.


.

Therefore,

.

Hence, if conditions (2.12) hold, then

. (2.18)

When proving (2.18) we use the fact that

.

Thus, equation (2.18) follows from (2.14), i.e. implies . Hence,

for all .

Moreover, (2.18) gives us that

, .

Therefore, as . This contradicts the assumption .Thus, for all estimate (2.13) is valid. In particular it leads us to the inequality

. (2.19)


for all . We use conditions (2.12) to find that

, . (2.20)

1 Qn

�� x , 1 Qn 1�� A

n1 Q

n�� x�

x X� n Z�

1 Qn 1�� A

nx

n, 1� 1 Q

n�� x

nx

n+�

Qn 1� A

nx

nQ

n 1� An

Qn

xn

Qn 1� A

n1 Q

n�� x

n�

Qn 1� AnQn% Qn 1� An 1 Qn�� - ./ 0 x

n .

� �

�

Qn 1� A

nx

nK, +�� x

n� n Z�

Nm 1� , 1� 1 Q

m�� x

m2+ K,�� x

m�

xm

Qm

xm

1 Qm

�� xm

� 2 1 Qm

�� xm

��

Nm 1� , 1� 2K,� 4+�� 1 Q

m�� x

m�

1 Qm 1�� xm 1� Qm 1� xm 1�� 7 1 Qm�� xm 0� �

, 1� 8K 8 Qn 8

Nm

0� Nm 1� 0�

1 Qn

�� xn

Qn

xn

� n m

K xn% 1 Qn�� xn 7n m� 1 Qm�� xm n m

xn +�! n +�! xxxx xn� lX��

n Z�

xn

1 Qn

�� xn

Qn

xn

� 2 Qn

xn

� �

Qn 1� xn 1� Qn 1� An xn 2 K, +�� Qn xn��

n Z�

Qn 1� xn 1�12�� Qn xn� n Z�


6

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If , then inequality (2.19) gives us that there exists such that. Therefore, it follows from (2.20) that

for all . We tend to obtain that which is impossibledue to (2.9) and the boundedness of the sequence . Therefore, there does notexist a nonzero such that . Thus, the mapping is injective.

Let us now prove the surjectivity of . Let us consider an operator in thespace acting according to the formula

, ,

where the operator acts from into and is inverse to . It follows from (2.9) and (2.11) that

, . (2.21)

It is evident that

Since

,

we have that

Consequently,

Therefore, inequalities (2.10), (2.11), and (2.12) give us that

,

i.e. . That means that the operator is invertible and

. (2.22)

Let be an arbitrary element of . Then it is evident that the element is a solution to equation . Moreover, it follows from (2.21) and

(2.22) that

.

Hence, is surjective and . Theorem 2.2 is proved.

xxxx xn

� 0�� m Z�Q

mx

m0�

Qn

xn

2m n� Qm

xm

0�

n m� n ��! Qn

xn

+�!x

n�

xxxx lX�� L xxxx 0� L

L R

lX�

R yyyy� �n Qn yn Bn 1 Qn 1�� yn 1�� yyyy yn� lX��

Bn 1 Qn 1�� An� � 1�� 1 Qn 1�� X 1 Qn�� X1 Qn 1�� An 1 Q

n�� X

Ryyyyl�

K ,�� yyyyl�

� yyyy lX��

L R yyyy� �n yn� 1 Qn�� yn Bn 1 Qn 1�� yn 1��

An 1� Qn 1� yn 1��

�

An 1� Bn 1� 1 Qn�� yn .�

�

1 Qn�� An 1� Bn 1� 1 Qn�� 1 Qn��

L R yyyy� �n yn� Bn 1 Qn 1�� yn 1�� An 1� Qn 1� yn 1��

Qn

An 1� 1 Q

n 1�� Bn 1� 1 Q

n�� y

n .

�

�

�

L R yyyy� �n yn� Bn 1 Qn 1�� yn 1�% An 1� Qn 1� yn 1�%

Qn An 1� 1 Qn 1�� Bn 1� 1 Qn�� yn .% %

� �

�

�

L R y yy yy yy y�l�

, 2 +�� yyyyl�

12�� yyyy

l��

LR 1� 1 2�� LR

LR� � 1� 1 LR 1�� 1� 2� �

hhhh hn

� � lX� yyyy �

R LR� � 1� hhhh� Ly hy hy hy h�

yyyyl�

2 K ,�� hhhhl�

�

L L 1� 2 K 1��

H y p e r b o l i c i t y o f I n v a r i a n t S e t s f o r D i f f e r e n t i a b l e M a p p i n g s 377

§ 3 Hyperbolicity of Invariant Sets§ 3 Hyperbolicity of Invariant Sets§ 3 Hyperbolicity of Invariant Sets§ 3 Hyperbolicity of Invariant Sets

for Differentiable Mappingsfor Differentiable Mappingsfor Differentiable Mappingsfor Differentiable Mappings

Let us remind the definition of the differentiable mapping. Let and be Banachspaces and let be an open set in . The mapping from into is called(Frechét) differentiable differentiable differentiable differentiable at the point if there exists a linear bounded ope-rator from into such that

.

If the mapping is differentiable at every point , then the mapping acts from into the Banach space of all linear bounded oper-

ators from into . If is continuous, then the mapping issaid to be continuously differentiable continuously differentiable continuously differentiable continuously differentiable (or -mapping) on . The notion ofthe derivative of any order can also be introduced by means of induction. For example,

is the Frechét derivative of the mapping .

Let and be continuously differentiable mappings from into and from into , respectively. Moreover, let

and be open sets such that . Prove that is a -mapping on and obtain a chain rule for the

differentiation of a composed function

, .

Let be a continuously differentiable mapping from into and let be the -th degree of the mapping , i.e.

, , . Prove that is a -map-ping on and

. (3.1)

Now we give the definition of a hyperbolic set. Assume that is a continuously dif-ferentiable mapping from a Banach space into itself and is a subset in whichis invariant with respect to . The set is called hyperbolic hyperbolic hyperbolic hyperbolic (withrespect to ) if there exists a collection of projectors such that

a) continuously depends on with respect to the operatornorm;

b) for every

; (3.2)

c) the mappings are invertible for every as linear operatorsfrom into ;

X Y

� X f � Y

x ��D f x� � X Y

1v��

v 0!lim f x v�� f x� �� D f x� � v� 0�

f x �� D f :x D f x� �! � � X Y��

X Y D f : � � X Y�� ! f

C1 �

D2 f x� � D f : � � X Y�� !

E x e r c i s e 3.1 g f

� X� Y � Y� Z

� � g �� f 1 g� � x� � �f g x� �� C1 �

D f 1g� � x� � D f g x� �� Dg x� �� x ��

E x e r c i s e 3.2 f X

X f n n f f n x� � �f f n 1� x� �� n 1 f 1 x� � f x� �# f n C1

X

D f n� � x� � D f f n 1� x� �� D f f n 2� x� �� D f x� �% % %�

f

X 2 X

f f 2� � 2�� 2f P x� � : x 2��

P x� � x 2�

x 2�D f x� � P x� �% P f x� �� D f x� �%�

Df x� � x 2�1 P x� �� X I P f x� �� X


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d) for every the following equations hold:

, , (3.3)

, , (3.4)

with the constants and independent of . Here is the -th degree of the mapping ( for

and ).

It should be noted that properties (b) and (c) as well as formula (3.1) enable usto state that maps into and is an invertibleoperator. Therefore, the value in the left-hand side of inequality (3.4) exists.

Let , where is a fixed point of a -mapping ,i.e. . Then for the set to be hyperbolic it is necessaryand sufficient that the spectrum of the linear operator doesnot intersect the unit circumference (Hint: see Example 2.1).

Let be an invariant hyperbolic set of a -mapping and let be a complete trajectory (in ) for , i.e. is a sequence of points from such that for all . Let us consider a difference equation ob-tained as a result of linearization of the mapping along :

, . (3.5)

Prove that the evolutionary operator of differenceequation (3.5) has the form

, , .

Prove that difference equation (3.5) admits an exponential di-chotomy over with (i) the constants and given by equations(3.3) and (3.4) and (ii) the projectors involved in thedefinition of the hyperbolicity.

It should be noted that property (a) of uniform continuity implies that the projectors are similar to one another, provided the values of are close enough.

The proof of this fact is based on the following assertion.

Lemma 3.1.

Let and be projectors in a Banach space . Assume that

, , (3.6)

for some constant . Then the operator

(3.7)

x 2�

D f n� � x� � P x� � K qn� n 0

D f n� � x� �� 1� 1 P f n x� �� K qn� n 0

K 0� 0 q 1� � x 2�f n n f f n x� � f f n 1� x� �� n 1 f 1 x� � f x� �#

D f n� � x� � 1 P x� �� X 1 P f n x� ��

E x e r c i s e 3.3 2 x0� � x0 C1 f

f x0� � x0� 2D f x0� �

2 C1 f " xn : n Z�� 2 f " xn� � 2

f xn� � xn 1�� n Z�f "

un 1� D f x

n� �u

n� n Z�

E x e r c i s e 3.4 $ m n��

$ m n�� D f m n�� xn

� �� m n� m n� Z�

E x e r c i s e 3.5

Z K q

Pn P xn� ��

P x� � x

P Q X

P K� 1 P� K� P Q� 12K��

K 1

J P Q 1 P�� 1 Q��


possesses the property and is invertible, therewith

. (3.8)

Proof.

Since , we have

.


.

Hence, the operator can be defined as the following absolutely convergentseries

.

This implies estimate (3.8). The permutability property is evident.Lemma 3.1 is proved.

Let be a connected compact set and let bea family of projectors for which condition (a) of the hyperbolicity de-finition holds. Then all operators are similar to one another, i.e.for any there exists an invertible operator suchthat .

The following assertion contains a description of a situation when the hyperbolicityof the invariant set is equivalent to the existence of an exponential dichotomy for dif-ference equation (3.5) (cf. Exercise 3.5).

Theorem 3.1.

Let be a continuously differentiable mapping of the space intoLet be a continuously differentiable mapping of the space intoLet be a continuously differentiable mapping of the space intoLet be a continuously differentiable mapping of the space into

itself. itself. itself. itself. Let Let Let Let be a hyperbolic fixed point of ( ) and let be a hyperbolic fixed point of ( ) and let be a hyperbolic fixed point of ( ) and let be a hyperbolic fixed point of ( ) and let

be a homoclinic trajectory (not equal to ) of the mapping , i.e.be a homoclinic trajectory (not equal to ) of the mapping , i.e.be a homoclinic trajectory (not equal to ) of the mapping , i.e.be a homoclinic trajectory (not equal to ) of the mapping , i.e.

,,,, ,,,, ,,,, .... (3.9)

Then the set is hyperbolic if and only if the diffe-Then the set is hyperbolic if and only if the diffe-Then the set is hyperbolic if and only if the diffe-Then the set is hyperbolic if and only if the diffe-

rence equationrence equationrence equationrence equation

,,,, ,,,, (3.10)

possesses an exponential dichotomy over possesses an exponential dichotomy over possesses an exponential dichotomy over possesses an exponential dichotomy over ....

Proof.

If is hyperbolic, then (see Exercise 3.5) equation (3.10) possesses an expo-nential dichotomy over . Let us prove the converse assertion. Assume that equa-tion (3.10) possesses an exponential dichotomy over with projectors

PJ J Q�

J 1� 1 2 K P Q�%�� 1��

P2 1 P�� 2� 1�

J 1� J P2� 1 P�� 2� P Q P�� 1 P�� P Q��

J 1� 2 K P Q� 1��

J 1�

J 1� 1 J�� n

n 0�

�

��

PJ J Q�

E x e r c i s e 3.6 2 P x� � : x 2��

P x� �x y� 2� J Jx y��

P x� � J P y� � J 1��

f x� � X

x0 f f x0� � x0� yn: n Z�� x0 f

f yn� � yn 1�� n Z� yn x0! n �&!

2 x0� yn

: n Z�� 3�

un 1� D f yn� �un� n Z�

Z

2Z

Z Pn

: n Z��


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and constants and . Let us denote the spectral projector of the operator corresponding to the part of the spectrum inside the unit disc by . Without lossof generality we can assume that

, ,

, .

Thus, for every the projector is defined: , .The structure of the evolutionary operator of difference equation (3.10) (see Exer-cise 3.4) enables us to verify properties (b)–(d) of the definition of a hyperbolic set.In order to prove property (a) it is sufficient to verify that

as . (3.11)

Since is a compact set, then

. (3.12)

Let us consider the following difference equations

, , (3.13)

and

, , (3.14)

where is an integer. It is evident that equation (3.14) admits an exponential di-chotomy over with constants and and projectors . Let and be the Green functions (see Section 2) of difference equations(3.13) and (3.14). We consider the sequence

, .

Since (see Exercise 2.8)

, , (3.15)

we have that the sequence is bounded. Moreover, it is easy to prove (see Exer-cise 2.9) that is a solution to the difference equation

.

It follows from (3.12) and (3.15) that the sequence is bounded. Therefore,(see Exercise 2.9),

.

If we take in this formula, then from the definition of the Green function weobtain that

.

K q D f x0� �P

D f x0� �� n P K qn� n 0

D f x0� �� n� 1 P�� K q n�� n 0

x 2� P x� � P x0� � P� P yk� � Pk�

Pk P� 0! k �&!

2

M D f x� � : x 2�� sup ��

vn 1� D f x0� � v

n� n Z�

wn 1�k� �

D f yn k�� w

nk� �� n Z�

k

Z K q Pn

k� � Pn k�� G n m��

G k� � n m��

xn

G n 0�� z G k� � n 0�� z�� z X�

G n 0�� K q n� G k� � n 0�� K q n�

xn

� x

n�

xn 1� D f x0� � x

n� h

nD f x0� � D f y

n k�� G k� � n 0�� z#�

hn

�

G n 0�� z G k� � n 0�� z� xn

# G n m 1�� hm

m Z��

n 0�

P Pk

�� z G 0 m 1�� D f x0� � D f ym k�� G k� � m 0�� z

m Z��



.

Consequently,

where is an arbitrary natural number. Upon simple calculations we find that

for every . It follows that

, .

We assume that to obtain that

.

This implies equation (3.11). Therefore, Theorem 3.1 is proved.

It should be noted that in the case when the set from The-orem 3.1 is hyperbolic the elements of the homoclinic trajectory are called transversal homoclinic points transversal homoclinic points transversal homoclinic points transversal homoclinic points. The point is that in some cases (see,e.g., [4]) it can be proved that the hyperbolicity of is equivalent to the transversa-lity property at every point of the stable and unstable mani-folds of a fixed point (roughly speaking, transversality means that the surfaces

and intersect at the point at a nonzero angle). In this case thetrajectory is often called a transversal homoclinic trajectory transversal homoclinic trajectory transversal homoclinic trajectory transversal homoclinic trajectory.

§§§§ 4 Anosov’s Lemma on -trajectories4 Anosov’s Lemma on -trajectories4 Anosov’s Lemma on -trajectories4 Anosov’s Lemma on -trajectories

Let be a -mapping of a Banach space into itself. A sequence in is called a -pseudotrajectory -pseudotrajectory -pseudotrajectory -pseudotrajectory (or -pseudoorbit) of the mapping if forall the equation

P Pk

�� z K 2 D f x0� � D f ym k�� q

m m 1��z% %

m Z��

P Pk� K 2 D f x0� � D f ym k�� q2 m 1�%m Z��

K 2 D f x0� � D f ym k�� q2 m 1�

m N��% 2 M q2 m 1�

m N��

m N�max

� ��

,

� �

�

N

P Pk

� 2 K 2

q 1 q2�� D f x0� � D f y

m k�� 2 M q 2 N 2��m N�max

- ./ 0�

N 1

P Pk

�k �&!

lim 4 K 2 M

1 q2�� q2 N 1�� N 1 2 � ��

N +�!

P Pk�k �&!

lim 0�

2 x0� yn : n Z�� 3�yn " yn: n Z��

2yn Ws x0� � W u x0� �

x0W s x0� � W u x0� � yn

"

�

f C1 X yn

: n Z�� X + + f

n Z�y

n 1� f yn

� �� +�

A n o s o v ’ s L e m m a o n - t r a j e c t o r i e s�


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is valid. A sequence is called an -trajectory -trajectory -trajectory -trajectory of the mapping cor-responding to a -pseudotrajectory if

(a) for any ;

(b) for all .

It should be noted that condition (a) means that is an orbit (complete tra-jectory) of the mapping . Moreover, if a pair of -mappings and is given,then the notion of the -trajectory of the mapping corresponding to a -pseu-doorbit of the mapping can be introduced.

The following assertion is the main result of this section.

Theorem 4.1.

Let be a -mapping of a Banach space into itself and let beLet be a -mapping of a Banach space into itself and let beLet be a -mapping of a Banach space into itself and let beLet be a -mapping of a Banach space into itself and let be

a hyperbolica hyperbolica hyperbolica hyperbolic invariant set. Assume that there exists a -vicini- invariant set. Assume that there exists a -vicini- invariant set. Assume that there exists a -vicini- invariant set. Assume that there exists a -vicini-

ty of the set such that and are bounded and uniformlyty of the set such that and are bounded and uniformlyty of the set such that and are bounded and uniformlyty of the set such that and are bounded and uniformly

continuous on the closure of the set continuous on the closure of the set continuous on the closure of the set continuous on the closure of the set .... Then there exists posses- Then there exists posses- Then there exists posses- Then there exists posses-

sing the property that for every there exists suchsing the property that for every there exists suchsing the property that for every there exists suchsing the property that for every there exists such

that any -pseudoorbit lying in has a unique -trajectorythat any -pseudoorbit lying in has a unique -trajectorythat any -pseudoorbit lying in has a unique -trajectorythat any -pseudoorbit lying in has a unique -trajectory

corresponding to . corresponding to . corresponding to . corresponding to .

As the following theorem shows, the property of the mapping to have an -trajec-tory is rough, i.e. this property also remains true for mappings that are close to .

Theorem 4.2.

Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem Assume that the hypotheses of Theorem 4.1 hold for the mapping . hold for the mapping . hold for the mapping . hold for the mapping .

LetLetLetLet be a set of continuously differentiable mappings of the spacebe a set of continuously differentiable mappings of the spacebe a set of continuously differentiable mappings of the spacebe a set of continuously differentiable mappings of the space

into itself such that the following estimates hold on the closure of the into itself such that the following estimates hold on the closure of the into itself such that the following estimates hold on the closure of the into itself such that the following estimates hold on the closure of the

-vicinity of the set :-vicinity of the set :-vicinity of the set :-vicinity of the set :

,,,, .... (4.1)

Then can be chosen to possess the property that for every Then can be chosen to possess the property that for every Then can be chosen to possess the property that for every Then can be chosen to possess the property that for every

there exist and such that for any -pseudotra-there exist and such that for any -pseudotra-there exist and such that for any -pseudotra-there exist and such that for any -pseudotra-

jectory (lying in ) of the mapping and for any jectory (lying in ) of the mapping and for any jectory (lying in ) of the mapping and for any jectory (lying in ) of the mapping and for any

there exists a unique trajectory of the mapping with the pro-there exists a unique trajectory of the mapping with the pro-there exists a unique trajectory of the mapping with the pro-there exists a unique trajectory of the mapping with the pro-

pertypertypertyperty

for all for all for all for all ....

It is clear that Theorem 4.1 is a corollary of Theorem 4.2 the proof of which is basedon the lemmata below.

xn

: n Z�� f

+ yn

: n Z�� f xn� � xn 1�� n Z�

xn

yn

� �� n Z�xn�

f C1 f g

� g +f

f C1 X 2f 2� � 2�� 4

� 2 f x� � D f x� �� 0 0�

0 � �0�� + �� + 0��+ yn : n Z�� 2 �

xn :� n Z� yn�

f �f

f

�5 f� � g

X �

4 � 2

f x� � g x� �� 5� D f x� � D g x� �� 5�

�0 0� � 0 �0 ��+ �� + 0�� 5 5 �� 0�� +

yn : n Z�� 2 f g �5 f� ��xn : n Z�� g

yn

xn

� �� n Z�


Lemma 4.1.

Let be an open set in a Banach space and let be a con-

tinuously differentiable mapping. Assume that for some point

there exist an operator and a number such that

(4.2)

for all with the property . Assume that for some

the inequality

(4.3)

is valid with . Then for any -mapping such that

(4.4)

and

(4.5)

for , the equation has a unique solution with

the property .

Proof.

Let and let

.

For we have that

Since

,

it follows from (4.2) that

(4.6)

for all and from . Now we rewrite the equation in the form

.

� X � : � X!y ��

D� y� �� 1� �0 0�

D� x� � D� y� �� 2 D� y� �� 1�- ./ 0

1��

x x y� �0� � 0 �0��

� y� � q � 2 D� y� �� 1�- ./ 0

1�%�

0 q 1� � C1 � : � X!

� x� � � x� �� 1 q�� 2 D� y� �� 1�- ./ 0

1��

D� x� � D� x� �� 12�� 2 D� y� �� 1�- ./ 0

1��

x y� �0� � x� � 0� x

x y� ��

6 2 D� y� �� 1�� 1��

5 x� � � x� � � y� �� D� y� � x y��

x1 x2� B�0z : z y� �0�� #�

5 x1� � 5 x2� �� x1� � � x2� �� D� y� � x1 x2��

D� x1 7 x2 x1�� D� y� �� x1 x2�� 7 .d

0

1

8

� �

�

5 x1� � 5 x2� �� D� x1 7 x2 x1�� D� y� �� 7 x1 x2�d

0

1

8�

5 x1� � 5 x2� �� 6 x1 x2��

x1 x2 B�0� x� � 0�

x T x� � y D� y� �� 1�� x� � � x� �� y� � 5 x� �� #�



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Let us show that the mapping has a unique fixed point in the ball . It is evident that

for any . Since , we obtain from (4.6) that

.

Therefore, estimates (4.3) and (4.4) imply that

for ,

i.e. maps the ball into itself. This mapping is contractive in . Indeed,

,

where


.

This equation and inequality (4.6) imply the estimate

.

Therefore, the mapping has a unique fixed point in the ball . The lemma is proved.

Let the hypotheses of Theorems 4.1 and 4.2 hold. We assume that in (4.1).Then for any element the following estimates hold:

, , , (4.7)

where is a constant. In particular, these estimates are valid for the mapping .

Lemma 4.2.

Let be a -pseudotrajectory of the mapping lying in .

Then for any the sequence is a -pseu-

dotrajectory of the mapping . Here has the form

, , , (4.8)

and is a constant from (4.7).

T B� x :��x y� ��

T x� � y� D� y� �� 1�� x� � � x� �� y� � 5 x� �� - .

/ 0�

x B�� 5 y� � 0�

5 x� � 5 x� � 5 y� �� 6 x y� 6��

T x� � y� �� x B��

T B� B�

T x1� � T x2� �� 126�� x1 x2�� 5 x1� � 5 x2� ��- .

/ 0�

� x1 x2�� x1� � � x2� �� x1� �� x2� ��#

D� x1 7 x1 x2�� D� x1 7 x1 x2�� 7 x1 x2�� .d

0

1

8

�

�

� x1 x2�� 12��6 x1 x2��

T x1� � T x2� �� 34�� x1 x2��

T B� x :��x y� ��

5 1�g �5 f� ��

g x� � M� Dg x� � M� x ��M 0� f

yn : n Z�� + f 2k 1 zn yn k : n Z�#� + Mk 1�%

f k Mk

Mk

1 M M k� � �� k 1 M0 1�

M


Proof.

Let us use induction to prove that

, . (4.9)

Since is a -pseudotrajectory, then it is evident that for inequality(4.9) is valid. Assume that equation (4.9) is valid for some and prove esti-mate (4.9) for :

.

With the help of (4.7) we obtain that

.


Lemma 4.3.

Let be a -pseudoorbit of the mapping lying in . Let be

a trajectory of the mapping such that

, , (4.10)

for some . If

, (4.11)

then

, (4.12)

where has the form (4.8).

Proof.

We first note that

Therefore, it is evident that

. (4.13)

Here we use the estimate

which follows from (4.7) and holds when the segment connecting the points and lies in . Condition (4.11) guarantees the fulfillment of this propertyat each stage of reasoning. If we repeat the arguments from the proof of (4.13),then it is easy to complete the proof of (4.12) using induction as in Lemma 4.2.Lemma 4.3 is proved.

yn k i� f i y

n k� �� + M

i 1�%� 1 i k� �

yn

� + i 1�i 1

i 1�

ynk i 1� � f i 1� y

nk� �� y

nk i 1� � f ynk i�� f y

nk i�� f f i ynk

� ��

ynk i 1� � f i 1� y

nk� �� + M y

nk i� f i ynk

� �� + M Mi 1�+%�� + M

i%�

yn

� + f 2 xn

� g �5 f� ��

ynk

xnk

� �� n Z�

k 1

� + 5�� max Mk

4�

yn

xn

� � + 5�� max Mk

%�

Mk

ynk 1� x

nk 1�� ynk 1� g x

nk� ��

ynk 1� f y

nk� �� f y

nk� � g y

nk� �� g y

nk� � g x

nk� �� .� �

�

�

�

ynk 1� xnk 1�� + 5 M ynk xnk�� + 5�� max 1 M��

g y� � g x� �� Dg y 7 x y�� 7 x y�%d

0

1

8 M x y��

x

y �



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Lemma 4.4.

Let and . Assume that

. (4.14)

Then the estimates

, , (4.15)

, (4.16)

(4.17)

are valid for and for every mapping .

Proof.

As above, let us use induction. If , it is evident that equations (4.15)–(4.17) hold. The transition from to in (4.15) is evident. Let us considerestimate (4.16):

. (4.18)

Condition (4.14) and the induction assumption give us that lies in the ballwith the centre at the point lying in . Therefore, it follows from(4.18) that

.

The transition from to in (4.17) can be made in a similar way. Lemma 4.4is proved.

Lemma 4.5.

There exists such that the equations

(4.19)

and

(4.20)

are valid in the -vicinity of the set for any function

. Here as .

The proof follows from the definition of the class of functions and estimates(4.7) and (4.17).

y 2� x y� ��

� 5�� max 1 M k 1�� 4�

f j y� � f j x� �� M j x y�� D f j x� � M j�

f j y� � g j x� �� 5�� 1 M j� �� max�

f j x� � g j x� �� 5 1 M j 1��

j 1 2 k� � �� g �5 f� ��

j 1�j j 1�

f j 1� y� � g j 1� x� �� f f j y� �� f g j x� �� - ./ 0 f g j x� �� g g j x� ��

gj x� �f j y� � 2� �

f j 1� y� � g j 1� x� �� M f j y� � g j x� �� 5� � 5�� max 1 M j 1��

j j 1�

49 4�

f k x� � g k x� �� : x � 9��

sup (k5� ��

D f k� � x� � D g k� � x� �� : x � 9��

sup (k 5� ��

49 � 9 2g �5 f� �� (

k5� � 0! 5 0!

�5 f� �


Let us also introduce the values

(4.21)

and

. (4.22)

The requirement of the uniform continuity of the derivative (see the hypo-theses of Theorem 4.1) and the projectors (see the hyperbolicity definition)enables us to state that

, as . (4.23)

Let be a -pseudotrajectory of the mapping lying in . Then due toLemma 4.2 the sequence is a -pseudotrajectory of themapping . Let us consider the mappings and in the space

(for the definition see Section 2) given by the equalities

, (4.24)

, (4.25)

where is an element from . Thus, the construction of -trajec-tories of the mapping and corresponding to the sequence is reduced tosolving of the equations

and

in the ball . Let us show that for large enough Lemma 4.1 can beapplied to the mappings and . Let us start with the mapping .

Lemma 4.6.

The function is a -smooth mapping in with the properties

, (4.26)

, . (4.27)

Proof.

Estimate (4.26) follows from the fact that is a -pseudotrajec-tory. Then it is evident that

, (4.28)

where and lie in . Therefore, simple cal-culations and equation (4.21) give us (4.27).

:k;� � D f k y� � D f k x� �� : y 2 , x y� ;��

� ��

sup�

: ;� � P y� � P x� �� , x y 2 , x y� ;��

sup�

Df x� �P x� �

:k;� � 0! : ;� � 0! ; 0!

yn� + f 2y

ny

nk: n Z�� +M

k 1�f k � xxxx� � � xxxx� �

lX

�l� Z X�� #

� xxxx� �� n yn

xn

f k yn 1� x

n 1��

� xxxx� �� n yn

xn

g k yn 1� x

n 1��

xxxx xn

: n Z�� lX

� �f k g k y

n�

� xxxx� � 0� � xxxx� � 0�

xxxx : xxxxlX� �� k

� � �

� C1 lX

�

� 0� � Mk 1�+�

D� xxxx� � D� 0� �� :k �� xxxxlX� ��

yn

� +Mk 1�

D� xxxx� � hhhh� �n hn

D f k yn 1� x

n 1�� hn 1��

xxxx xn : n Z�� hhhh hn : n Z�� lX�



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In order to deduce relations (4.2) and (4.3) from inequalities (4.26) and (4.27) for, we use Theorem 2.2. Consider the operator . It is clear that

, .

Let us show that equations (2.9)–(2.11) are valid for and and then estimate the corresponding constants. Property (2.9) follows

from the hyperbolicity definition. Equations (3.3) and (4.15) imply that

and .

Further, the permutability property (3.2) gives us that

Hence (see (4.22)),

.

Similarly, we find that

.

The operator

is invertible if (see Lemma 3.1)

.

Moreover,

,

provided . Due to the hyperbolicity of the set , the operator is an invertible mapping from into . Therefore,

since

,

the operator is invertible as a mapping from into. Moreover, by virtue of (3.4) we have that

.

Thus, under the conditions

, , , (4.29)

Theorem 2.2 implies that the operator is invertible and .Let us fix some . If

yyyy 0� L D� 0� ��

L hhhh� �n hn

D f k yn 1�� h

n 1�� hhhh hn

� �

An

D f k yn 1�� Q

n�

P yn 1��

An

Qn

K q k� An

M k�

Qn 1� An 1 Qn�� Qn 1� An AnQn�� 1 Qn��

Qn 1� P f k y

n 1�� An

1 Qn

�� .

� �

�

Qn 1� A

n1 Q

n�� : M

k 1�+� � An

1 Qn

�% % : Mk 1�+� �M k K%� �

1 Qn 1�� A

nQ

n: M

k 1�+� �M k K%�

Jn

Qn 1� P f k y

n 1�� 1 Qn 1�� 1 P f k y

n 1��

Qn 1� P f k y

n 1�� : Mk 1�+� � 1

2K��

Jn1� 1 2 K: Mk 1�+� �� 1� 2��

2 K: Mk 1�+� � 1 2�� 2An 1 Qn�� X 1 P f k yn 1�� X

1 Qn 1�� A

n1 Q

n�� J

nA

n1 Q

n��

1 Qn 1�� An 1 Qn�� 1 Qn�� X1 Qn 1�� X

1 Qn 1�� An� � 1� 1 Qn 1�� An 1 Qn�� 1�Jn

1� 1 Qn 1�� % 2 K 2 q k��

4 K: Mk 1�+� � 1� 8KM k: M

k 1�+� � 1� 16 K 3q k 1�

L D� 0� �� L 1� 2K 1�� 0�


, , (4.30)

then by Lemma 4.6 relations (4.2) and (4.3) hold with , , and . If

, (4.31)

then equations (4.4) and (4.5) also hold with , , and . Hence,under conditions (4.29)–(4.31) there exists a unique solution to equation possessing the property . This means that for any -pseudoorbit

(lying in ) of the mapping there exists a unique trajectory of the mapping such that

, ,

provided conditions (4.29)–(4.31) hold. Therefore, under the additional condition

and due to Lemma 4.3 we get

, .

These properties are sufficient for the completion of the proof of Theorem 4.2.Let us fix such that . We choose ( is defined

in Lemma 4.5) such that

for all .

Let us fix an arbitrary and take . Now we choose and such that the following conditions hold:

, ,

, , .

It is clear that under such a choice of and any -pseudoorbit (from ) of themapping has a unique -trajectory of the mapping . Thus, Theorem 4.2 is proved.

Let the hypotheses of Theorem 4.1 hold. Show that thereexist and such that for any two trajectories and of a dynamical system the conditions

, ,

imply that , . In other words, any two trajectories ofthe system that are close to a hyperbolic invariant set cannotremain arbitrarily close to each other all the time.

Mk 1�+ 1

2�� 4K 2�� 1� �%� :

k�� 4K 2�� 1��

y 0� � �� q 1 2��

(k5� � 1

2�� 1�� min 4K 2�� 1��

yyyy 0� � �� q 1 2�� xxxx� � 0�

xxxxl�

�� + yn :�n Z� 2 f zn : n Z��

g �5 f� ��

znk

ynk

� �� n Z�

� + 5� �� Mk

4�

yn

zn

� � + 5� �� Mk

� n Z�

k 16 K 3q k 1� �0 49 4� � 49

:k�� 4K 2�� 1��

�0

2Mk

��

� 0 �0�� 2 Mk� � 1��+ �� +� 5 5 ��

4 K: Mk 1�+� � 1� 8KMk: Mk 1�+� � 1�

Mk 1�+ 1

4�� 2K 1�� 1�� (

k5� � 1

4�� 1�� min 2K 1�� 1�� 2 + 5�� M

k��

+ 5 + 2f � g

E x e r c i s e 4.1

4 0� + 0� xn

: n Z�� y

n: n Z�� X f��

xn2�� dist 4� y

n2�� dist 4� x

ny

n�

nsup +�

xn yn# n Z�X f��



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Show that Theorem 4.1 admits the following strengthening:if the hypotheses of Theorem 4.1 hold, then there exists suchthat for every there exists with the propertythat for any -pseudoorbit such that thereexists a unique -trajectory.

Prove the analogue of the assertion of Exercise 4.2 for Theo-rem 4.2.

Let be a periodic orbit of the mapping ,i.e. for all and for some . Assumethat the hypotheses of Theorem 4.2 hold. Then for smallenough every mapping possesses a periodic trajectoryof the period .

§ 5 Birkhoff-Smale Theorem§ 5 Birkhoff-Smale Theorem§ 5 Birkhoff-Smale Theorem§ 5 Birkhoff-Smale Theorem

One of the most interesting corollaries of Anosov’s lemma is the Birkhoff-Smale the-orem that provides conditions under which the chaotic dynamics is observed ina discrete dynamical system . We remind (see Section 1) that by definitionthe possibility of chaotic dynamics means that there exists an invariant set in thespace such that the restriction of some degree of the mapping on is to-pologically equivalent to the Bernoulli shift in the space of two-sided infinitesequences of symbols.

Theorem 5.1.

Let be a continuously differentiable mapping of a Banach space Let be a continuously differentiable mapping of a Banach space Let be a continuously differentiable mapping of a Banach space Let be a continuously differentiable mapping of a Banach space

into itself. Let be a hyperbolic fixed point of and let into itself. Let be a hyperbolic fixed point of and let into itself. Let be a hyperbolic fixed point of and let into itself. Let be a hyperbolic fixed point of and let

be a homoclinic trajectory of the mapping that does not coincide withbe a homoclinic trajectory of the mapping that does not coincide withbe a homoclinic trajectory of the mapping that does not coincide withbe a homoclinic trajectory of the mapping that does not coincide with

, i.e., i.e., i.e., i.e.

;;;; ,,,, ,,,, ;;;; , ....

Assume that the trajectory is transversal, i.e. the setAssume that the trajectory is transversal, i.e. the setAssume that the trajectory is transversal, i.e. the setAssume that the trajectory is transversal, i.e. the set

is hyperbolic with respect to and there exists a vicinity of the set is hyperbolic with respect to and there exists a vicinity of the set is hyperbolic with respect to and there exists a vicinity of the set is hyperbolic with respect to and there exists a vicinity of the set

such that and are bounded and uniformly continuous on thesuch that and are bounded and uniformly continuous on thesuch that and are bounded and uniformly continuous on thesuch that and are bounded and uniformly continuous on the

closure . By we denote a set of continuously differentiable map-closure . By we denote a set of continuously differentiable map-closure . By we denote a set of continuously differentiable map-closure . By we denote a set of continuously differentiable map-

pings of the space into itself such thatpings of the space into itself such thatpings of the space into itself such thatpings of the space into itself such that

,,,, ,,,, ....

E x e r c i s e 4.2

�0 0�� 0 �0�� + + �� + y

n� y

n2�� dist +�

�

E x e r c i s e 4.3

E x e r c i s e 4.4 2 xn

: n Z�� f

f k xn

� � xn k�# x

n� n Z� k 1

5 0�g �5 f� ��

k

X f�� Y

X f k f Y

S �m

m

f X

x0 X� f yn

: n Z�� f

x0

f x0� � x0� f yn� � yn 1�� yn x0� n Z� yn x0! n �&!

yn

: n Z��

2 x0� yn

: n Z�� 3�

f � 2f x� � Df x� �

� �5 f� �g X

f x� � g x� �� 5� Df x� � D g x� �� 5� x ��

B i r k h o f f - S m a l e T h e o r e m 391

Then there exists such that for any mapping and for anyThen there exists such that for any mapping and for anyThen there exists such that for any mapping and for anyThen there exists such that for any mapping and for any

there exist a natural number and a continuous mapping of the there exist a natural number and a continuous mapping of the there exist a natural number and a continuous mapping of the there exist a natural number and a continuous mapping of the

space into a compact subset in such thatspace into a compact subset in such thatspace into a compact subset in such thatspace into a compact subset in such that

a) is strictly invariant with respect to , i.e. ; is strictly invariant with respect to , i.e. ; is strictly invariant with respect to , i.e. ; is strictly invariant with respect to , i.e. ;

b) if and are elementsif and are elementsif and are elementsif and are elements

of such that for some , then ;of such that for some , then ;of such that for some , then ;of such that for some , then ;

c) the restriction of on is topologically conjugate to the Bernoullithe restriction of on is topologically conjugate to the Bernoullithe restriction of on is topologically conjugate to the Bernoullithe restriction of on is topologically conjugate to the Bernoulli

shift in , i.e.shift in , i.e.shift in , i.e.shift in , i.e.

,,,, ....

Moreover, if in addition we assume that for the mapping there existsMoreover, if in addition we assume that for the mapping there existsMoreover, if in addition we assume that for the mapping there existsMoreover, if in addition we assume that for the mapping there exists

such that for any two trajectories and such that for any two trajectories and such that for any two trajectories and such that for any two trajectories and

(of the mapping ) lying in the -vicinity of the set the condition(of the mapping ) lying in the -vicinity of the set the condition(of the mapping ) lying in the -vicinity of the set the condition(of the mapping ) lying in the -vicinity of the set the condition

for some implies that for all for some implies that for all for some implies that for all for some implies that for all ,,,, then the map- then the map- then the map- then the map-

ping is a homeomorphism.ping is a homeomorphism.ping is a homeomorphism.ping is a homeomorphism.

The proof of this theorem is based on Anosov’s lemma and mostly follows the stan-dard scheme (see, e.g., [4]) used in the finite-dimensional case. The only difficultyarising in the infinite-dimensional case is the proof of the continuity of the mapping

. It can be overcome with the help of the lemma presented below which is bor-rowed from the thesis by Jürgen Kalkbrenner (Augsburg, 1994) in fact.

It should also be noted that the condition under which is a homeomorphismholds if the mapping does not “glue” the points in some vicinity of the set , i.e.the equality implies .

Lemma 5.1.

Let the hypotheses of Theorem 5.1 hold. Let us introduce the notation

,

where and . Let be a segment (lying

in ) of a pseudoorbit of the mapping :

, , . (5.1)

Assume that and are segments of or-

bits of the mapping :

, , , (5.2)

such that

, . (5.3)

Then there exist , and such that conditions (5.1)–(5.3) imply the inequality

. (5.4)

5 0� g �5 f� ��m 2 l <

�m

Y < �m

� �# X

Y < �m

� �� gl gl Y� � Y�a a 1� a0 a1 � � �� a 9 a 9 1� a 90 a 91 � � �� m ai a 9i� i 0 < a� � < a 9� ��

g l Y

S �m

g l < a� �� < S a� �� a �m�

g

�0 0� xn

: n Z�� xn

: n Z�� g �0 2

xn0xn0

� n0 Z� xn

xn

� n Z�<

<

<g 2

g x� � g x� �� x x�

J= J= k0 >�� # k Z : k k0� >=��

k0 Z� > =� N� z zn

: n J=�� 2 + - f

zn

2� zn 1� f z

n� �� +� n n 1�� J=�

x xn

: n J=�� x xn

: n J=�� g �5 f� ��

g xn

� � xn 1�� g x

n� � x

n 1�� n n 1�� J=�

zn

xn

� �� zn

xn

� ��

+ 5 �� 0� > N�

xk0

xk0� 21 =� ��


6

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Proof.


, (5.5)

where

.

Since the set is hyperbolic with respect to , there exists a family of projec-tors for which equations (3.2)–(3.4) are valid. Therefore,

It means that

Consequently, equation (3.4) implies that

.

Let us estimate the value . It can be rewritten in the form

.

It follows from (5.3) that . Hence, using (4.20)and (4.21), for small enough we obtain that

, (5.6)

where as and as . Therefore,

, (5.7)

provided . Further, we substitute the value for in (5.5) to obtain that

.

Therefore, using (3.2) we find that

xk0 >� x

k0 >�� D f >� � zk0

� � xk0

xk0

�� Rk0

��

Rk0

g> xk0

� � g> xk0

� �� D f >� � zk0

� � xk0

xk0

��

2 f

P x� � : x 2��

1 P f > zk0

� �� xk0 >� x

k0 >��

D f >� � zk0� � 1 P zk0

� �� xk0xk0

�� 1 P f > zk0� �� Rk0

.�

�

�

1 P zk0� �� xk0

xk0��

D f >� � zk0� �� 1� 1 P f > zk0

� �� xk0 >� xk0 >�� Rk0�� .

�

�

1 P zk0

� �� xk0

xk0

�� K q> xk0 >� x

k0 >�� Rk0

��

Rk0

Rk0

D g>� � 7 xk0

1 7�� xk0

�� D f > zk0

� �� 7d

0

1

8 xk0

xk0

�� %�

7xk01 7�� xk0

zk0��

� 0�

Rk0

(> 5� � :> �� xk0

xk0

��

(> 5� � 0! 5 0! :> �� 0! � 0!

1 P zk0

� �� xk0

xk0

�� K q> xk0 >� x

k0 >�� xk0

xk0

��

(> 5� � :> �� 1� k0 >� k0

xk0

xk0

� D f > zk0 >�� x

k0 >� xk0 >�� R

k0 >��

P f > zk0 >�� xk0xk0

��

D f > zk0 >�� P z

k0 >�� xk0 >� x

k0 >�� P f > zk0 >�� R

k0 >� .�

�

�


Hence, equations (3.3) and (5.6) with instead of give us that

(5.8)

Since

,

it follows from (4.15) and (5.1) that

for small enough. Therefore,

,

where as (cf. (4.22)). Consequently, estimate (5.8) implies that

(5.9)

It is evident that estimates (5.7) and (5.9) enable us to choose the parameters, and such that

.

Using this inequality with instead of we obtain that

for all . Therefore,

.

If we continue to argue like that, then we find that

.

Since , this and estimate (5.3) imply (5.4).Lemma 5.1 is proved.


Let be distinct integers. Let us choose and fix the parame-ters and the integer such that (i) Theorem 4.2 and Lemma 5.1can be applied to the hyperbolic set and (ii)

k0 >� k0

P f > zk0 >�� xk0xk0

��

K q > (> 5� � :> �� xk0 >� xk0 >�� .

�

�

zk0f > zk0 >�� f j zk0 j�� f j f zk0 j� 1��

� ��

j 0�

> 1�

��

zk0f > zk0 m�� M j

j 0�

> 1�

�+ + > 1 M>�� %� �

+

P zk0

� � P f > zk0 >�� : + > 1 M>�� %� � : + >�� #�

: 7� � 0! 7 0!

P zk0� � xk0

xk0��

K q> (> 5� � :> �� K 1� : + >�� %� � �� xk0 >� x

k0 >�� .

�

�

> 5 �� +

xk0xk0

� 12�� xk xk� : k J1 k0 >�� max�

k k0

xk

xk

� 12�� x

nx

n� : n J2 k0 >�� max�

k J1 k0 >��

xk0

xk0

� 14�� x

nx

n� : n J2 k0 >�� max�

xk0

xk0

� 2 =� xn

xn

� : n J= k0 >�� max�

xn

xn

� xn

zn

� xn

zn

��

p1 p2 pm 1��

� 5 +� � 0� > 0�2 x0� y

n: n Z�� 3�


6

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. (5.10)

Assume that is such that

for and (5.11)

for all . Let us consider the segments of the orbits of the map-ping of the form

, ,

.

The length of every such segment is . Let . Let usconsider a sequence of elements made up of the segments by the formula

. (5.12)

It is clear that and by virtue of (5.11) is a -pseudoorbit of the mapping. Therefore, due to Theorem 4.2 there exists a unique trajectory

of the mapping such that

, (5.13)

where and is the -th element of thesegment , , . Let us define the mapping from into by the formula

, (5.14)

where is the zeroth element of the trajectory . Since the trajectory possessing the property (5.13) is uniquely defined, equation (5.14) defines a map-ping from into .

If we substitute for in (5.13) and use the equations

,

we obtain that

for all and . Therefore, the equality with being the Bernoulli shift in leads us to the equation

.

Consequently, the uniqueness property of the -trajectory in Theorem 4.2 gives usthe equation

, .

� 12�� y

pi

yp

j� y

pi

x0�� : i j� = 1 m 1 i�� j��

min�

N

yn

x0� +2�� n N p

i1� �� n p

iN��

i 1 2 m 1�� Ci

f

Ci

yp

iN� y

pi y

pi

N�� i 1 2 m 1��

Cm

x0 x0 x0� � ��

2 N 1� a a 1� a0 a1� � �m��"a Ci

"a Ca 1�Ca0

Ca1� ��

"a 2� "a +f wn wn a� �# :�n Z� g

wn N i j� �� zi j a� ��

n N i j� �� N i 1�� 2 N 1�� j� �� zi j

a� � j

Caii Z� j 1 2 2 N 1�� < �

m

X

< a� � w0�

w0 wn

� wn

�

�m

X

i 1� i

wn N i 1� j� �� wn N i j� �� 2 N 1� � g2 N 1� wn N i j� �� - ./ 0� �

g2 N 1� wn N i j� �� z

i 1� j� a� ��

i Z� j 1 2 2 N 1�� zi 1� j� a� � zi j Sa� ��S �m

g2 N 1� wn N i j� �� zi 1� j� Sa� ��

�

wn

Sa� � g2 N 1� wn

a� �� n Z�


This implies that

, , (5.15)

i.e. property (c) is valid for . It follows from (5.15) that

.

Therefore, the set is strictly invariant with respect to . Thus, as-sertion (a) is proved.

Let us prove the continuity of the mapping . Assume that the sequence of ele-ments of tends to

as . This means (see Exercise 1.4) that for any there exists such that

for , . (5.16)

Assume that and are -pseudoorbits in constructed ac-cording to (5.12) for the symbols and , respectively. Equation (5.16) impliesthat for . Let and be -trajectories corres-ponding to and , respectively. Lemma 5.1 gives us that

, (5.17)

provided , i.e. for any equations (5.17) is valid for .This means that

as . Thus, the mapping is continuous and is a compact strict-ly invariant set with respect to .

Let us now prove nontriviality property (b) of the mapping . Let be such that for some . Let and be -trajectories corres-ponding to the symbols and , respectively. Then

Therefore, it follows both from (5.13) and the definition of the elements that

,

where and . We apply (5.10) to obtain that

. (5.18)

Therefore, if , then

.

Hence, . This completes the proof of assertion (b).

< Sa� � g2 N 1� < a� �� a �m

�

l 2 N 1��

g2 N 1� < S 1� a� �� < a� ��Y < �m� �� g2N 1�

<a s� � a 1�

s� �a0

s� �a1

s� � � � � �� m a a 1� a0 a1 � � � �� m� s +�! M N�

s0 s0 M� ��

ais� �

ai� i M� s s0

" s� � zk

s� �� " zk� � + 2a s� � a

zks� �

zk� k M 2 N 1�� wn� wns� ��

" " s� �

w0s� �

w0� 21 =� ��

M 2N 1�� => = N� s s0 = M��

< a s� �� < a� �� w0s� �

w0� 0!#

s +�! < Y < �m

� ��g2 N 1�

< a a 9� �m

�a

ia 9

i� i 0 w

n� w 9

n� �

a a 9

wn N i N 1�� w 9

n N i N 1�� zi N 1�� a� � z

i N 1�� a 9� ��

wn N i N 1�� z

i N 1�� a� �� w 9n N i N 1�� z

i N 1�� a 9� �� .��

�

zi j

a� �

w 2N 1�� i w 92N 1�� i� yqi

yq 9i

� 2 ��

qi

pa

i� q 9

ip

a 9i

�

w 2N 1�� iw 92N 1�� i

� 0�

i 0

g 2N 1�� i w0� � g 2N 1�� i w 90� ��

< a� � w0 w 90 < a 9� �#�#


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If the trajectories of the mapping cannot be “glued” (see the hypotheses ofTheorem 5.1), then for some equation (5.18) gives us that , i.e.

if . Thus, the mapping is injective in this case. Since is a compact metric space, then the injectivity and continuity of imply that isa homeomorphism from onto . Theorem 5.1 is proved.

It should be noted that equations (5.13) and (5.14) imply that the set lies in the -vicinity of the hyperbolic set . Therewith, the values and in-volved in the statement of the theorem depend on and one can state that for anyvicinity of the set there exist and such that the conclusions of Theo-rem 5.1 are valid and . It is also clear that the set is notuniquely determined.

Assume that in Theorem 5.1. Prove that the mapping can be constructed such that , where

is a homoclinic orbit of the mapping .

Prove the Birkhoff theorem: if the hypotheses of Theorem 5.1hold, then for any small enough there exist and such that for every mapping there exist periodic trajec-tories of the mapping of any minimal period in the -vicinityof the set .

Use Theorem 1.1 to describe all the possible types of beha-viour of the trajectories of the mapping on a set

.

In conclusion, it should be noted that different infinite-dimensional versions of Ano-sov’s lemma and the Birkhoff-Smale theorem have been considered by many authors(see, e.g., [6], [10], [11], [12], and the references therein).

§ 6 Possibility of Chaos in the Problem§ 6 Possibility of Chaos in the Problem§ 6 Possibility of Chaos in the Problem§ 6 Possibility of Chaos in the Problem

of Nonlinear Oscillations of a Plateof Nonlinear Oscillations of a Plateof Nonlinear Oscillations of a Plateof Nonlinear Oscillations of a Plate

In this section the Birkhoff-Smale theorem is applied to prove the existence of chaoticregimes in the problem of nonlinear plate oscillations subjected to a periodic load.The results presented here are close to the assertions proved in [13]. However, themethods used differ from those in [13].

g

i Z� w0 w 90�< a� � < a 9� �� a a 9� < �

m

< <�

m< �

m� �

Y < �m

� �� 2 5 l

�� 2 5 l

< �m

� � �� Y < �m

� ��

E x e r c i s e 5.1 g f� << �m� � x0� ynl : n 0� 3?

yn : n Z�� f

E x e r c i s e 5.2

� 0� 5 0� l N�g �5 f� ��

g l �2

E x e r c i s e 5.3

g

W g k < �m

� �� k 1�

l

3�

P o s s i b i l i t y o f C h a o s i n t h e P r o b l e m o f N o n l i n e a r O s c i l l a t i o n s o f a P l a t e 397

Let us remind the statement of the problem. We consider its abstract versionas in Chapter 4. Let be a separable Hilbert space and let be a positive operatorwith discrete spectrum in , i.e. there exists an orthonormalized basis in such that

, , .

The following problem is considered:

Here , and are positive parameters, is an element of the space , is a linear operator in subordinate to , i.e.

, (6.3)

where is a constant. The problem of the form (6.1) and (6.2) was studied in Chap-ter 4 in details (nonlinearity of a more general type was considered there). The re-sults of Section 4.3 imply that problem (6.1) and (6.2) is uniquely solvable in theclass of functions

. (6.4)

Moreover, one can prove (cf. Exercise 4.3.9) that Cauchy problem (6.1) and (6.2)is uniquely solvable on the whole time axis, i.e. in the class

.

This fact as well as the continuous dependence of solutions on the initial conditions(see (4.3.20)) enables us to state that the monodromy operator acting in

according to the formula

(6.5)

is a homeomorphism of the space (see Exercise 4.3.11). Here is a solutionto problem (6.1) and (6.2)

The aim of this section is to prove the fact that under some conditions on and chaotic dynamics is observed in the discrete dynamical system for some

set of parameters , and .

Lemma 6.1.

The mapping defined by equality (6.5) is a diffeomorphism of the

space .

Proof.

We use the method applied to prove Lemma 4.7.3. Let be a solutionto problem (6.1) and (6.2) with the initial conditions and let

H A

H ek

� H

Aek

,k

ek

� 0 ,1 ,2�� ,n

n �!lim ��

(6.1)

(6.2)

u�� "u� A2u ; A1 2� u 2 6�� Au L u� � � � h : t ,cos�

ut 0� u0� F1� D A� �� , u�

t 0� u1 H .��*�*�

" ; 6� � : h H L

H A

Lu K Au�

K

�+ C R+ D A� �� C1 R+ H�� @�

� C R D A� �� C1 R H�� @�

G � �D A� � HA�

G u0 u1)� � u2B:��- .

/ 0 u�2B:��- .

/ 0�- ./ 0�

� u t� �

L

h � G�� " ; 6� � :

G

� D A� � HA�

u1 t� �y u0 u1��


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be a solution to it with the initial condions .Let us consider a linearization of problem (6.1) and (6.2) along the solution :

As in the proof of Theorem 4.2.1, it is easy to find that problem (6.6) and (6.7) isuniquely solvable in the class of functions (6.4). Let .It is evident that is a weak solution to problem

where

A simple calculation shows that

,

where

,

.

We assume that and . In this case (see Section 4.3) the esti-mates

,

,

, (6.10)

are valid on any segment . Here is a constant. Therefore,

, , ,

where and are constants depending on and T. Hence,

, .

u2 t� � y z� u0 z0� u1 z1�� #u1 t� �

(6.6)

(6.7)

w�� t� � "w� t� � A2w ; A1 2� u1 t� � 2 6�� Aw� � � �

2� ; A1 2� u1 t� � A1 2� w t� �� Au1 t� � L w = 0 ,�

wt 0� z0� w�

t 0�� z1 .��**�**�

v t� � u2 t� � u1 t� �� w t� ��v t� �

(6.8)

(6.9)

v�� "v� t� � A2v t� �� F t� �� F u1 t� � u2 t� � w t� �� ,#

vt 0� 0 , v�

t 0� 0 ,� ��*�*�

F u1 u2 w� �� ; A1 2� u22 6�

- ./ 0� Au2 ; A1 2� u1

2 6�- ./ 0 Au1� +

; A1 2� u12 6�

- ./ 0 Aw 2; Au1 w�� Au1 L u2 u1� w�� .��

�

F u1 u2 w� �� F1 u1 u2 w� �� F2 u1 u2 w� �� F1 F2�#

F1 ; A1 2� u22 6�

- ./ 0� Av Lv� 2; Au1 v�� A u1 w��

F2 ; A1 2� u1 u2�� 2� A u1 w�� 2; Au1 w�� Aw��

y�

R� z�

1�

Auj t� � CR T��

A u1 t� � u2 t� �� CR T� z�

�

Aw t� � CR T� z�

�

0 T�� CR T�

F1 C1 Av� F2 C2 z�

2� t 0 T��

C1 C2 R

F t� � 2C1 Av

2C2 z

�

4�� t 0 T��


Therefore, the energy equation

(6.11)

for problem (6.8) and (6.9) leads us to the estimate

, .

Using Gronwall’s lemma we find that

, ,

where the constant depends on and . This estimate implies that the map-ping

(6.12)

is a Frechét derivative of the mapping defined by equality (6.5). Here is a solution to problem (6.6) and (6.7). It follows from (6.10) and (6.6) that is a continuous linear mapping of into itself. Using (6.10) it is also easy to seethat continuously depends on with respect tothe operator norm. Lemma 6.1 is proved.

Further we will also need the following assertion.

Lemma 6.2.

Let be the monodromy operator of problem (6.1) and (6.2) with

and , . Assume that for equation (6.3) is va-

lid and , . Moreover, assume that

, , .

Then the estimates

(6.13)

and

(6.14)

are valid. Here is a ball of the radius in ,

is an arbitrary number while the constant depends on and but

does not depend on the parameters , and provided they

vary in bounded sets.

12�� v� t� � 2

Av t� � 2�- ./ 0 " v� �� 2 �d

0

t

8� F �� v� �� d

0

t

8�

v� t� � 2Av t� � 2� C1 Av �� 2

C2 z4�- .

/ 0 �d

0

t

8� t 0 T��

v� t� � 2Av t� � 2� C z

4� t 0 T��

C T R

G 9 : z z0 z1)� �# w2B:��- .

/ 0 w�2B:��- .

/ 0)- ./ 0!

G w t� �G 9

�

G 9 G 9 u0 u1�� y u0 u1)� ��

Gj

L Lj

� h hj

� j 1 2�� L Lj

�h

jH� j 1 2��

LjA 1� (� h

j(� j 1 2��

G1 y� � G2 y� ��y B

R�

sup C L1 L2�� A 1� h1 h2��- ./ 0�

G 91 y� � G 92 y� ��y B

R�

sup C L1 L2�� A 1� h1 h2��- ./ 0�

BR

R � D A� � HA� R 0�C R (

: " ;� � 0 6


6

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Proof.

Let be a solution to problem (6.1) and (6.2) with and ,. It is evident (see Section 4.3) that

, . (6.15)

Therefore, it is easy to find that the difference satisfiesthe equation

where the function can be estimated as follows:

.

As in the proof of Lemma 6.1, we now use energy equality (6.11) and Gronwall’slemma to obtain the estimate

. (6.16)

This implies inequality (6.13). Estimate (6.14) can be obtained in a similar way.In its proof equations (6.12), (6.15), and (6.16) are used. We suggest the readerto carry out the corresponding reasonings himself/herself. Lemma 6.2 is proved.

Let us now prove that there exist an operator and a vector such that the corres-ponding mapping possesses a hyperbolic homoclinic trajectory. To do that, we usethe following well-known result (see, e.g., [1], [13], as well as Section 7) related to theDuffing equation.

Theorem 6.1.

Let Let Let Let be a monodromy operator corresponding to the Duffing be a monodromy operator corresponding to the Duffing be a monodromy operator corresponding to the Duffing be a monodromy operator corresponding to the Duffing


,,,, (6.17)

i.e. the mapping of the plane i.e. the mapping of the plane i.e. the mapping of the plane i.e. the mapping of the plane into itself acting according to the formula into itself acting according to the formula into itself acting according to the formula into itself acting according to the formula

,,,, (6.18)

where where where where is a solution to equation is a solution to equation is a solution to equation is a solution to equation (6.17) such that such that such that such that and and and and

.... All the parameters contained in All the parameters contained in All the parameters contained in All the parameters contained in (6.17) are assumed to be positive. are assumed to be positive. are assumed to be positive. are assumed to be positive.

Let us also assume thatLet us also assume thatLet us also assume thatLet us also assume that

.... (6.19)

uj t� � L Lj� h hj�j 1 2��

Auj

t� � C C R ( T� �� #� t 0 T��

u t� � u1 t� � u2 t� ��

u�� "u A2u� � F t u1 u2� �� ,�

ut 0� 0 , u�

t 0� 0 ,� ��*�*�

F t u1 u2� ��

F t u1 u2� �� C1 Au C2 L1 L2�� A 1� h1 h2��- ./ 0��

u� t� � 2Au t� � 2� C L1 L2�� A 1� 2

h1 h2� 2�- ./ 0�

L h

G

g : R2 R2!

x�� x� �x3� � f : tcos% x�+��

R2

g x0 x1�� x2B:��- .

/ 0 x� 2B:��- .

/ 0)- ./ 0�

x t� � x 0� � x0� x� 0� � �x1�

f fcr

+2 3 2�

3: 2��

B:2 ��- ./ 0cosh% %#�


Then there exists Then there exists Then there exists Then there exists such that for every such that for every such that for every such that for every the mapping the mapping the mapping the mapping pos-pos-pos-pos-

sesses a fixed point sesses a fixed point sesses a fixed point sesses a fixed point and a homoclinic trajectory and a homoclinic trajectory and a homoclinic trajectory and a homoclinic trajectory to it, to it, to it, to it, ,,,,

therewith the set therewith the set therewith the set therewith the set is hyperbolic. is hyperbolic. is hyperbolic. is hyperbolic.

Let be the orthoprojector onto the one-dimensional subspace generated by theeigenvector in . We consider problem (6.1) and (6.2) with in-stead of and , where is a positive number. Then it is evident thatevery solution to problem (6.1) and (6.2) with the initial conditions and

has the form

,

where is a solution to the Duffing equation

(6.20)

with the initial conditions and . In particular, this means thatthe two-dimensional subspace of the space is strict-ly invariant with respect to the corresponding monodromy operator while the re-striction of to coincides with the monodromy operator corresponding to theDuffing equation (6.20). Therefore, if is small enough and the conditions

,

hold, then the mapping possesses a hyperbolic invariant set

consisting of the fixed point and its homoclinic trajectory

, where .

Thus, if and for some the condition holds, then there exists anopen set in the space of parameters such that for every the mo-nodromy operator corresponding to problem (6.1) and (6.2) with instead of and possesses a hyperbolic set consisting of a fixed point anda homoclinic trajectory. This fact as well as Lemmata 6.1 and 6.2 enables us to applythe Birkhoff-Smale theorem and prove the following assertion.

Theorem 6.2.

Let Let Let Let and let the condition and let the condition and let the condition and let the condition hold for some hold for some hold for some hold for some .... Then there Then there Then there Then there

exist exist exist exist and an open set and an open set and an open set and an open set in the metric space in the metric space in the metric space in the metric space such that if such that if such that if such that if

,,,, ,,,,

then some degree then some degree then some degree then some degree of the monodromy operator of the monodromy operator of the monodromy operator of the monodromy operator of problem of problem of problem of problem (6.1) andandandand

(6.2) possesses a compact strictly invariant set possesses a compact strictly invariant set possesses a compact strictly invariant set possesses a compact strictly invariant set in the space in the space in the space in the space

in which the mapping in which the mapping in which the mapping in which the mapping is topologically conjugate to the Bernoulli shift of is topologically conjugate to the Bernoulli shift of is topologically conjugate to the Bernoulli shift of is topologically conjugate to the Bernoulli shift of

�0 0� � 0 �0�� g

z yn

: n Z�� yn

z�z� y

n: n Z�� 3

Pk

ek

H Lk

L 1 Pk

�� L h h

ke

k%� h

k

u0 c0 ek

�u1 c1 e

k�

u t� � x t� � ek

�

x t� �

x�� "x� ,k 6 ,k�� x� ;,k2

x3� � hk : tcos�

x 0� � c0� x� 0� � c1��

kLin e

k0)� � 0 e

k)� ��

Gk

Gk

�k

hk

0 ,k 6� �h

k

"��2,

k6 ,

k��

3: 2;�� 6

,k

�� 1�- ./ 01 2� B:

2 ,k6 ,

k��

��cosh�

Gk

2k

z ek

� yn

ek

: n Z�%� 3�

z0 ek z1 ek)� �

yn

ek

: n Z�� yn

yn0 y

n1)� � R2��

: 0� k 0 ,k

6� �� " h

k�� " h

k��

Gk

Lk

L 1 Pk

�� L h h

ke

k�

: 0� 0 ,k

6� � k

> 0� � R+ HA

Lek,

k1� >� " h��

Gl G

Y GlY = Y� � �

Gl


6

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sequences of sequences of sequences of sequences of symbols, i.e. there exists a homeomorphism symbols, i.e. there exists a homeomorphism symbols, i.e. there exists a homeomorphism symbols, i.e. there exists a homeomorphism

such thatsuch thatsuch thatsuch that

,,,, ....

Prove that if the hypotheses of Theorem 6.2 hold, then equa-tion (6.1) possesses an infinite number of periodic solutions with pe-riods multiple to .

Apply Theorem 6.2 to the Berger approximation of the prob-lem of nonlinear plate oscillations:

§ 7 On the Existence of § 7 On the Existence of § 7 On the Existence of § 7 On the Existence of TransversalTransversalTransversalTransversal

Homoclinic TrajectoriesHomoclinic TrajectoriesHomoclinic TrajectoriesHomoclinic Trajectories

Undoubtedly, Theorem 6.1 on the existence of a transversal (hyperbolic) homoclinictrajectory of the monodromy operator for the periodic perturbation of the Duffingequation is the main fact which makes it possible to apply the Birkhoff-Smaletheorem and to prove the possibility of chaotic dynamics in the problem of plateoscillations. In this connection, the question as to what kind of generic conditionguarantees the existence of a transversal homoclinic orbit of monodromy operatorsgenerated by ordinary differential equations gains importance. Extensive literatureis devoted to this question (see, e.g., [1], [2] and the references therein). There areseveral approaches to this problem. All of them enable us to construct systems withtransversal homoclinic trajectories as small perturbations of “simple” systems withhomoclinic (not transversal!) orbits. In some cases the corresponding conditions onperturbations can be formulated in terms of the Melnikov function.

This section is devoted to the exposition and discussion of the results obtainedby K. Palmer [14]. These results help us to describe some classes of systems of ordi-nary differential equations which generate dynamical systems with transversal ho-

m < : �m

Y!

Gl < a� �� < Sa� �� a �m

�

E x e r c i s e 6.1

:

E x e r c i s e 6.2

C2u

t2C�� " uC

tC�� 42u ; Du x t�� 2xd

E8 6�

- .F G/ 0

u +4��

+ ( uCx1C�� = h x� � : t , x x1 x2�� E R2�� , t 0 ,�cos

u EC 4u EC 0 , ut 0� u0 x� � , uC

tC��t 0�

u1 x� � .� � � ��****�****�

O n t h e E x i s t e n c e o f T r a n s v e r s a l H o m o c l i n i c T r a j e c t o r i e s 403

moclinic orbits. Such differential equations are obtained as periodic perturbationsof autonomous equations with homoclinic trajectories.

In the space let us consider a system of equations

, , (7.1)

where is a twice continuously differentiable mapping. Assume that theCauchy problem for equation (7.1) is uniquely solvable for any initial condition

. Let us also assume that there exist a fixed point anda trajectory homoclinic to , i.e. a solution to equation (7.1) such that

as . Exercises 7.1 and 7.2 given below give us the examples of thecases when these conditions hold. We remind that every second order equation

can be rewritten as a system of the form (7.1) if we take and.

Consider the Duffing equation

, .

Prove that the curve is an orbit of the corres-ponding system (7.1) homoclinic to . Here .

Assume that for a function there exist a num-ber and a pair of points such that

; , ;

; ; .

Then system (7.1) corresponding to possesses an or-bit homoclinic to that passes through the point .

Unfortunately, as the cycle of Exercises 7.3–7.5 shows, the homoclinic orbit of au-tonomous equation (7.1) cannot be used directly to construct a discrete dynamicalsystem with a transversal homoclinic trajectory.

For every define the mapping by the for-mula , where is a solution to equation(7.1) with the initial condition . Show that is a diffeomorphismin with a fixed point and a family of homoclinic orbits

, where , .

Prove that the derivative of the mapping constructed inExercise 7.3 can be evaluated using the formula ,where is a solution to problem

, .

Here is a solution to equation (7.1) with the initialcondition .

Rn

x� t� � g x t� �� x t� � Rn�

g : Rn Rn!

x 0� � x0� z0 g z0� �=0� �z t� � z0� z0

z t� � z0! t �&!x��

V x� �� 0� x1 x�x2 x��

E x e r c i s e 7.1

x�� x� �x3� 0� � � 0�

z t� � 5 t� � 5� t� �� R2��0 5 t� � 2 �� tsech�

E x e r c i s e 7.2 U x� � C3 R� ��E a b�

U a� � U b� � E� � U x� � E� x a b��

U 9 a� � 0� U 99 a� � 0� U 9 b� � 0�x�� U 9 x� �� 0�

a 0�� b 0��

E x e r c i s e 7.3 � 0� f : Rn Rn!f x0� � x � x0)� �� x t x0)� �

x0 f

Rn z0yn

s : n Z�� yn z s n �� s 0 ��

E x e r c i s e 7.4 f 9 f

f 9 x0� �w y � w�� y t w��

y� t� � g 9 x t� �� y t� �� y 0� � w0�

x t� � x t x0)� ��x0


6

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Let be the mapping constructed in Exercise 7.3 and let be a homoclinic orbit of equation (7.1). Show that

is a bounded solution to the difference equa-tion , where (Hint: the function

satisfies the equation ).

Thus, due to Theorems 2.1 and 3.1 the result of Exercise 7.5 implies that the set

cannot be hyperbolic with respect to the mapping defined by the formula, where is a solution to equation (7.1) with the initial

condition . Nevertheless we can indicate some quite simple conditions on theclass of perturbations periodic with respect to under which the mo-nodromy operator of the problem

, , (7.2)

possesses a transversal (hyperbolic) homoclinic trajectory for small enough.

Further we will use the notion of exponential dichotomy for ordinary differentialequations (see [15], [16] as well as [5] and the references therein)

Let be a continuous and bounded matrix function on the real axis.We consider the problem

, , , (7.3)

in the space . It is easy to see that it is solvable for every initial condition.Therefore, we can define the evolutionary operator , , by the for-mula

, ,

where is a solution to problem (7.3).

Prove that

,

for all and the following matrix equations hold:

, . (7.4)

Prove the inequality

, .

E x e r c i s e 7.5 f

z t� � : t Z�� w

n= z� n�� : n Z�� w

n 1� f 9 yn

� �wn

� yn

z n�� w t� � z� t� �� w� g 9 z t� �� w�

2 z0� z n�� : n Z�� 3�

f

f x0� � x � x0)� �� x t x0)� �x0

h t x >� �� t

x� t� � g x t� �� >h t x t� � >� �� x t� � Rn�

>

A t� � n nA

x� t� � A t� � x t� �� t R� xt s� x0�

X Rn�$ t s�� t s� R�

$ t s�� x0 x t� � x t s x0)�� #� t s� R�

x t� �

E x e r c i s e 7.6

$ t s�� $ t �� $ � s�� $ t t�� I�

t s �� R�dtd

��$ t s�� A t� � $ t s�� dsd

��$ t s�� $ t s�� A s� ��

E x e r c i s e 7.7

$ t s�� A �� d

s

t

8� �* *� �* *� �

exp� t s


Let be some interval of the real axis. We say that equation (7.3) admits an expo- expo- expo- expo-

nential dichotomy nential dichotomy nential dichotomy nential dichotomy over the interval if there exist constants anda family of projectors continuously depending on and such that

, ; (7.5)

, , (7.6)

, , (7.7)

for .

Let be a constant matrix. Prove that equation (7.3)admits an exponential dichotomy over if and only if the eigenva-lues on do not lie on the imaginary axis.

The assertion contained in Exercise 7.8 as well as the following theorem on theroughness enables us to construct examples of equations possessing an exponentialdichotomy.

Theorem 7.1.

Assume that problemAssume that problemAssume that problemAssume that problem (7.3) possesses an exponential dichotomy over possesses an exponential dichotomy over possesses an exponential dichotomy over possesses an exponential dichotomy over

an interval . Then there exists such that equationan interval . Then there exists such that equationan interval . Then there exists such that equationan interval . Then there exists such that equation

(7.8)

possesses an exponential dichotomy over , provided for possesses an exponential dichotomy over , provided for possesses an exponential dichotomy over , provided for possesses an exponential dichotomy over , provided for ....

Moreover, the dimensions of the corresponding projectors for Moreover, the dimensions of the corresponding projectors for Moreover, the dimensions of the corresponding projectors for Moreover, the dimensions of the corresponding projectors for (7.3) and and and and

(7.8) are the same. are the same. are the same. are the same.

The proof of this theorem can be found in [15] or [16], for example.

The exercises given below contain some simple facts on systems possessing an expo-nential dichotomy. We will use them in our further considerations.

Prove that equations (7.5)–(7.7) imply the estimates

, ,

, ,

for any .

Assume that equation (7.3) admits an exponential dichotomyover . Prove that , where

(7.9)

(Hint: , ).

K �� 0�P t� � : t �� t

P t� � $ t s�� $ t s�� P s� �� t s

$ t s�� P s� � K e � t s�� t s

$ t s�� I P s� �� K e � s t�� t s�t s� �

E x e r c i s e 7.8 A t� � A#R

A

� 0�

x� t� � A t� � B t� �� x t� ��

B t� � �� t �

E x e r c i s e 7.9

$ t s�� 7 K 1� e� t s�� 1 P s� �� 7 t s

$ t s�� 7 K 1� e� s t�� P s� � 7 t s�

7 X Rn#�


R+ 0 +�� P 0� �X V+�

V+ 7 X Rn#� : $ t 0�� 7t 0�sup ��

� ��

#

$ t 0�� 7 K 1� e� t 1 P 0� �� 7 t 0


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If equation (7.3) possesses an exponential dichotomy over, then , where

(7.10)

(Hint: , ).

Assume that equation (7.3) possesses an exponential dicho-tomy over (over , respectively). Show that any solution toproblem (7.3) bounded on (on , respectively) decreasesat exponential velocity as (as , respectively).

Assume that equation (7.3) possesses an exponential dicho-tomy over the half-interval , where is a real number.Prove that equation (7.3) possesses an exponential dichotomy overany semiaxis of the form .(Hint: ).

Prove the analogue of the assertion of Exercise 7.13 for thesemiaxis .

Prove that for problem (7.3) to possess an exponential dicho-tomy over it is necessary and sufficient that equation (7.3) pos-sesses an exponential dichotomy both over and and has nonontrivial solutions bounded on the whole axis .

Prove that the spaces and (see (7.9) and (7.10)) pos-sess the properties

, ,

provided problem (7.3) possesses an exponential dichotomy over .

Consider the following equation adjoint to (7.3):

, (7.11)

where is the transposed matrix. Prove that the evolutionaryoperator of problem (7.11) has the form .

Assume that problem (7.3) possesses an exponential dichoto-my over an interval . Then equation (7.11) possesses exponentialdichotomy over with the same constants and projectors

.

Assume that problem (7.3) possesses an exponential dichoto-my both over and . Let , where aredefined by equalities (7.9) and (7.10). Show that the dimensions ofthe spaces of solutions to problems (7.3) and (7.11) bounded on thewhole axis are finite and coincide.


R– � 0 �� I P 0� �� X V–�

V– 7 X Rn#� : $ t 0�� 7t 0�sup ��

� ��

�

$ t 0�� 7 K 1� e� t P 0� � 7 t 0�


R+ R–R+ R–

t +�! t ��!


a +�� a

b +��P t� � $ t a�� P a� � $ a t��


� a ��


RR+ R–R

E x e r c i s e 7.16 V+ V–

V+ V–@ 0� � V+ V–� X Rn#�

R


y� t� � A* t� �� y t� ��A* t� �H t s�� H t s�� $ s t�� *�


K �� 0�Q t� � I P t� �*��


R+ R– V+dim V–dim� n� V&


Assume that problem (7.3) possesses an exponential dichoto-my over . Then for any and the difference equation

possesses an exponential dichotomy over (for the definition see Section 2).

Let us now return to problem (7.1). Assume that is a hyperbolic fixed point for(7.1), i.e. the matrix does not have any eigenvalues on the imaginary axis. Let

be a trajectory homoclinic to . Using Theorem 7.1 on the roughness and theresults of Exercises 7.13 and 7.14 we can prove that the equation

(7.12)

possesses an exponential dichotomy over both semiaxes and . Moreover, thedimensions of the corresponding projectors are the same and coincide with the di-mension of the spectral subspace of the matrix corresponding to the spec-trum in the left semiplane. Therewith it is easy to prove that ,where have form (7.9) and (7.10). The result of Exercise 7.15 implies that equa-tion (7.12) cannot possess an exponential dichotomy over ( is a solu-tion to (7.12) bounded on ) while Exercise 7.19 gives that the number of linearlyindependent bounded (on ) solutions to (7.12) and to the adjoint equation

(7.13)

is the same. These facts enable us to formulate Palmer’s theorem (see [14]) as fol-lows.

Theorem 7.2.

Assume that Assume that Assume that Assume that is a twice continuously differentiable function from is a twice continuously differentiable function from is a twice continuously differentiable function from is a twice continuously differentiable function from

into into into into and equation and equation and equation and equation

possesses a fixed hyperbolic point possesses a fixed hyperbolic point possesses a fixed hyperbolic point possesses a fixed hyperbolic point and a trajectory and a trajectory and a trajectory and a trajectory homo- homo- homo- homo-

clinic to clinic to clinic to clinic to . We also assume that . We also assume that . We also assume that . We also assume that is a unique (up to a scalar fac- is a unique (up to a scalar fac- is a unique (up to a scalar fac- is a unique (up to a scalar fac-

tor) solution to equationtor) solution to equationtor) solution to equationtor) solution to equation

(7.14)

bounded on . Let bounded on . Let bounded on . Let bounded on . Let be a continuously differentiable vector func- be a continuously differentiable vector func- be a continuously differentiable vector func- be a continuously differentiable vector func-

tion tion tion tion -periodic with respect to -periodic with respect to -periodic with respect to -periodic with respect to and defined for and defined for and defined for and defined for ,,,, ,,,,

,,,, .... If If If If

,,,, ,,,, (7.15)

where where where where is a bounded (unique up to a constant factor) solution to is a bounded (unique up to a constant factor) solution to is a bounded (unique up to a constant factor) solution to is a bounded (unique up to a constant factor) solution to

the equation adjoint to the equation adjoint to the equation adjoint to the equation adjoint to (7.14), then there exist , then there exist , then there exist , then there exist and and and and such that for such that for such that for such that for

the perturbed equation the perturbed equation the perturbed equation the perturbed equation

(7.16)


R s R� � 0�x

n$ s �n� s�� x

n 1��Z

z0g 9 z0� �

z t� � z0

y� g 9 z t� �� y�

R+ R–

g 9 z0� �V+dim V–dim� n�

V&R y t� � z� t� ��

RR

y� g 9 z t� �� *y�

g x� �Rn Rn

x� g x� ��z0 z t� � : t R��

z0 y t� � z� t� ��

y� g 9 z t� �� y�R h t x >� ��

T t t R� x z t� �� 40�> '0� > R�

I t� � h t z t� � 0� �� Rn td

��

�

8 0� I t� � ht

t z t� � 0� �� Rn

td

��

�

8 0�

I t� �4 '

0 > '� �

x� g x� � >h t x >� ��


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possesses the following properties:possesses the following properties:possesses the following properties:possesses the following properties:

(a) there exists a unique -periodic solution such thatthere exists a unique -periodic solution such thatthere exists a unique -periodic solution such thatthere exists a unique -periodic solution such that

,,,, ,,,,

andandandand

,,,, ;;;;

(b) there exists a solution bounded on and such thatthere exists a solution bounded on and such thatthere exists a solution bounded on and such thatthere exists a solution bounded on and such that

,,,, ,,,,

,,,,

andandandand

;;;;

(c) the linearized equationthe linearized equationthe linearized equationthe linearized equation

,,,, (7.17)

where is equal to either or , possesses an ex-where is equal to either or , possesses an ex-where is equal to either or , possesses an ex-where is equal to either or , possesses an ex-

ponential dichotomy over .ponential dichotomy over .ponential dichotomy over .ponential dichotomy over .

This theorem immediately implies (see Exercise 7.20 and Theorem 3.1) thatunder conditions (7.15) the monodromy operator for problem (7.16) has a hyperbo-lic fixed point in a vicinity of the orbit and a transversal trajectory ho-moclinic to it.

We will not prove Theorem 7.2 here. Its proof can be found in paper [14]. We onlyoutline the scheme of reasoning which enables us to construct a homoclinic trajecto-ry . Here we pay the main attention to the role of conditions (7.15). If wechange the variable in equation (7.16), then we obtain the equation

.

We use this equation to construct a mapping from into acting according to the formula

. (7.18)

We remind that is the space of times continuously differentiable bound-ed functions from into with bounded derivatives with respect to up to the -th order, inclusive.

Thus, the existence of bounded solutions to problem (7.16) is equivalent to thesolvability of the equation . It is clear that . Therefore, inorder to construct solutions to equation we should apply an appropri-ate version of the theorem on implicit functions. Its standard statement requiresthat the operator be invertible. However, it is easy to check that theoperator has the form

T 70 t >��

z0 70 t >�� 4� t R�

z0 70 t >�� t R�sup 0! > 0!

7 t >�� R

7 t >�� z t� �� 4� t R�

7 t >�� z t� ��t R�sup 0 >� ��

7 t >�� 70 t >�� t �&!

lim 0�

y� g 9 5 t >�� >h9x t 5 t >�� >� ��

y�

5 t >�� 7 t >�� 70 t >�� R

z t� � : t R��

7 t :�� x z t� � J��

J� g z t� � J�� g z t� �� >h t z t� � J� >� ��

� Cb1 R Rn�� RA

Cb0 R Rn��

J >�� J >�� ! J� g z t� � J�� g z t� �� >h t z t� � J� >� ��

Cbk R X�� k

R X t k

� J >�� 0� � 0 0�� 0�� J >�� 0�

L DJ� 0 0�� L


.

Therefore, it possesses a nonzero kernel . Hence, we should use the modi-fied (nonstandard) theorem on implicit functions (see Theorem 4.1 in [14]). Roughlyspeaking, we should make one more change and consider the equation

. (7.19)

If this equation is solvable and the solution depends on smoothly, then sa-tisfies the equation

, (7.20)

where is the derivative of with respect to the parameter . This equation canbe obtained by differentiation of the identity with respect to .Due to the smoothness properties of the mapping , it follows from (7.20) the solva-bility of the problem

, (7.21)

which is equivalent to the differential equation

(7.22)

in the class of bounded solutions. It is easy to prove that the first condition in (7.15)is necessary for the solvability of (7.22) (it is also sufficient, as it is shown in [14]).

Further, the necessary condition of the dichotomicity of (7.17) for on is the condition of the absence of nonzero solutions to equation

(7.17) bounded on . However, this equation can be rewritten in the form

, (7.23)

where is determined with (7.19). If we assume that equation (7.23) hasnonzero solutions, then we differentiate equation (7.23) with respect to at zero,as above, to obtain that

, (7.24)

where is a solution to equation (7.21), , , and isa solution to equation (7.23). Equation (7.17) transforms into (7.14) when .Therefore, the condition of uniqueness of bounded solutions to (7.14) gives us that

, therewith we can assume that . Hence, equation (7.24) trans-forms into an equation for of the form

, (7.25)

where

L y� � t� � y� t� � g 9 z t� �� y t� ��

L z� 0��

J >w�

� >w >�� 0�w > w

DJ� >w >�� w >w>�� D>� >w >�� 0�

w> w >� >w >� � >�� 0� >

�

DJ� 0 0�� w0 D>� 0 0�� 0�

w� 0 g 9 z t� �� w0 h t z t� � 0� ��

5 t >�� 7 t >�� R

R

DJ � >w >�� y >� � 0�

w w >� ��>

DJJ � 0 0�� w0 DJ> � 0 0�� y0 DJ � 0 0�� y1� 0�

w0 y0 y 0� �� y1 D>y 0� �� y >� �> 0�

y0 c0 z� t� �� c0 1�y1

y�1 g 9 z t� �� y1� a t� ��

a t� � DJJ � 0 0�� w0 DJ> � 0 0�� z�- ./ 0 t� ��

Dx x

g z t� �� w0 t� � Dx

h t z t� � 0� �� z� t� � .

� �

�


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Here is a solution to (7.22). A simple calculation shows that equation (7.25)can be rewritten in the form

. (7.26)

The second condition in (7.15) means that equation (7.26) cannot have solutionsbounded on the whole axis. It follows that equation (7.23) has no nonzero solutions,i.e. equation (7.17) is dichotomous for .

Thus, the first condition in (7.15) guarantees the existence of a homoclinic tra-jectory while the second one guarantees the exponential dichotomicity ofthe linearization of the equation along this trajectory.

As to the existence and properties of the periodic solution , this situa-tion is much easier since the point is hyperbolic. The standard theorem on im-plicit functions works here.

It should be noted that condition (7.15) can be modified a little. If we consider a“shifted” homoclinic trajectory for instead of in Theo-rem 7.2, then the first condition in (7.15) can be rewritten in the form

.

If we change the variable , then we obtain that

. (7.27)

It is evident that

.

Therefore, the second condition in (7.15) leads us to the requirement .Thus, if the function has a simple root ( , ), then theassertions of Theorem 7.2 hold if we substitute the value for in (b).Performing the corresponding shift in the function , we obtain the asser-tions of the theorem in the original form. Thus, condition (7.15) is equivalent to therequirement

, for some , (7.28)

where has form (7.17).

In conclusion we apply Theorem 7.2 to prove Theorem 6.1. The unperturbed Duffingequation can be rewritten in the form

(7.29)

w0 t� �

dtd

�� y1 t� � w� 0 t� �� g 9 z t� �� y1 t� � w� 0 t� �� ht t z t� � 0� ��

5 t >�� 7 t >��

7 t >��

70 t >�� z0

zs

t� � z t s�� s R� z t� �

s� �4 I t s�� h t z t s�� 0� �� - ./ 0

Rntd

��

�

8# 0�

t t s�!

s� �4 I t� � h t s� z t� � 0� �� - ./ 0

Rntd

��

�

8�

49 s� � I t s�� ht

t z t s�� 0� �� - ./ 0

Rntd

��

�

8�

49 s� � 0�s� �4 s0 s0� �4 0� 49 s0� � 0�

z t s0�� z t� �7 t >��

s0� �4 0� 49 s0� � 0� s0 R�

s� �4

x�1 x2 ,�

x�2 x1 �x13 .��

��


The equation linearized along the homoclinic orbit (see Exer-cise 7.1) has the form

(7.30)

Let us show that system (7.30) has no solutions which are bounded on the axis andnot proportional to . Indeed, if is another bounded solution,then due to the fact that as , the Wronskian

possesses the properties

and .

This implies that and therefore is proportional to .Evidently, the equation adjoint to (7.30) has the form

(7.31)

Since we have that

,

a solution to (7.31) bounded on has the form . Let us nowconsider the corresponding function . Since in this case

, we have

,

where

.

Calculations (try to do them yourself) give us that

.

Therefore, equation has simple roots under condition (6.19). Thus, the as-sertion of Theorem 6.1 follows from Theorem 7.2.

It should be noted that in this case the function coincides with the famousMelnikov function arising in the geometric approach to the study of the transversali-ty (see, e.g., [1], [2] and the references therein). Therewith conditions (7.28) trans-form into the standard requirements on the Melnikov function which guarantee theappearance of homoclinic chaos.

z t� � 5 t� � 5� t� ��

y�1 y2 ,�

y�2 y1 3�5 t� �2 y1 .��

z� t� � w t� � v t� � v� t� �� 5� t� � 5�� t� �� 0! t �! W t� � �

v t� � 5�� t� � v� t� � 5� t� ��

dtd

��W t� � 0� W t� �t �!lim 0�

W t� � 0# v t� � 5� t� �

y�1 y2� 3�5 t� �2 y2� ,�

y�2 y1 .��

y��2 y2� 3�5 t� �2 y2� 0�

R I t� � 5�� t� � 5� t� �)�� s� �4 h t x >� ��

0 f : tcos x2+�)� ��

s� �4 5� t� � f : t s�� cos 5� t� �+�� td

��

�

8�

5 t� �2 �� tsech

8 �� e t e t��

- ./ 0

1��

s� �4 2 f: 2��

:ssinB:2 ��- ./ 0cosh

�� 43�� +

3 2��%�

s� �4 0�

s� �4


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Addition to the English translation:

The monographs by Piljugin [1*] and by Palmer [2*] have appeared after publicationof the Russian version of the book. Both monographs contain an extensive bibliogra-phy and are closely related to the subject of Chapter 6.


1. GUCKENHEIMER J., HOLMES P. Nonlinear oscillations, dynamical systems and

bifurcations of vector fields. –New York: Springer, 1983.2. WIGGINS S. Global bifurcations and chaos: analytical methods. –New York:

Springer, 1988.3. NITECKI Z. Differentiable dynamics. –Cambrige: M.I.T. Press, 1971.4. PALMER K. J. Exponential dichotomies, the shadowing lemma and transversal

homoclinic points // Dynamics Reported. –1988. –Vol. 1. –P. 265–306.5. HENRY D. Geometric theory of semilinear parabolic equations. Lect. Notes

in Math. 840. –Berlin: Springer, 1981.6. CHOW S.-N., LIN X.-B., PALMER K. J. A shadowing lemma with applications

to semilinear parabolic equations // SIAM J. Math. Anal. –1989. –Vol. 20. –P. 547–557.

7. NICOLIS G., PRIGOGINE I. Exploring complexity. An introduction. –New York: W. H. Freeman and Company, 1989.

8. KATO T. Perturbation theory for linear operators. –Berlin; Heidelberg; New York.: Springer, 1966.

9. RUDIN W. Functional analysis. 2nd ed. –New York: McGraw Hill, 1991.10. LERMAN L. M ., SHILNIKOV L. P. Homoclinic structures in infinite-dimentional

systems // Siberian Math. J. –1988. –Vol. 29. –#3. –P. 408–417.11. HALE J. K., LIN X.-B. Symbolic dynamics and nonlinear semiflows // Ann. Mat.

Pura ed Appl. –1986. –Vol. 144. –P. 229–259.12. LANI-WAYDA B. Hyperbolic sets, shadowing and persistance for noninvertible

mappings in Banach spaces. –Essex: Longman, 1995.13. HOLMES P., MARSDEN J. A partial differential equation with infinitely many pe-

riodic orbits: chaotic oscillations of a forced beam // Arch. Rat. Mech Anal. –1981. –Vol. 76. –P. 135–165.

14. PALMER K. J. Exponential dichotomies and transversal homoclinic points //J. Diff. Eqs. –1984. –Vol. 55. –P. 225–256.

15. HARTMAN P. Ordinary differential equations. –New York; London; Sydney: John Wiley, 1964.

16. DALECKY Y. L., KREIN M. G. Stability of solutions of differential equations

in Banach space. –Providence: Amer. Math. Soc., 1974.

1*. PILJUGIN S. JU. Shadowing in dynamical systems. Lect. Notes in Math. 1706.–Berlin; Heidelberg; New York: Springer, 1999.

2*. PALMER K. Shadowing in dynamical systems. Theory and Applications.–Dordrecht; Boston; London: Kluwer, 2000.

IndexIndexIndexIndex

�-pseudotrajectory 381�-trajectory 382

AAAA

Attractorfractal exponential 61, 103global 20minimal 21regular 39weak 21, 32

BBBB

Belousov-Zhabotinsky equations 334Berger equation 217Bernoulli shift 14, 365Burgers equation 96

CCCC

Cahn-Hilliard equation 97Cantor set 52Chaotic dynamics 369Chirikov mapping 26Completeness defect 296Cycle 17

DDDD

Determining functionals 286, 292boundary 358mixed 332

Determining nodes 313, 332Diameter of a set 53Dichotomy

exponential 370, 405Dimension of a set

fractal 52Hausdorff 54

Dimension of dynamical system 11Duffing equation 12, 26, 37, 400Dynamical system

-dissipative 104asymptotically compact 26, 249asymptotically smooth 34compact 27continuous 11discrete 11, 14, 365dissipative 24pointwise dissipative 34

��

416 I n d e x

EEEE

Equationnonlinear diffusion, explicitly solvable 118nonlinear heat 94, 117, 176, 187, 358oscillations of a plate 217, 402reaction-diffusion 96, 180, 328retarded 13, 138, 171, 327second order in time 189, 200, 209, 217, 350semilinear parabolic 77, 85, 182, 317wave 189, 201, 208, 351, 360

Evolutionary family 193, 321Evolutionary semigroup 11

FFFF

Flutter 217Frechét derivative 377Fredholm operator 113Function

Bochner integrable 219Bochner measurable 218

GGGG

Galerkin approximate solution 88, 190, 224, 233Galerkin method

non-linear 209traditional 87, 190, 224, 233

Generic property 117Gevrey regularity 347Green function of difference equation 372

HHHH

Hausdorff metric 32, 48Hodgkin-Huxley equations 339Hopf model of appearance of turbulence 40, 130

IIII

Ilyashenko attractor 22Inertial form 152Inertial manifold 151

approximate 182, 200, 276asymptotically complete 161exponentially asymptotically complete 161local 171, 276

Invariant torus 138KKKK

Kolmogorov width 302LLLL

Landau-Hopf scenario 138Lorentz equations 28Lyapunov function 36, 108Lyapunov-Perron method 153

Index 417

MMMM

Mappingcontinuously differentiable 377Frechét differentiable 113, 377proper 113

Melnikov function 411Milnor attractor 22Modes 300

determining 320, 331NNNN

Navier-Stokes equations 98, 335OOOO

Operatorevolutionary 11Frechét differentiable at a point 39hyperbolic 370monodromy 14, 397potential 108regular value 114with discrete spectrum 77, 218

Orbit see TrajectoryPPPP

Phase space 11Point

equilibrium 17fixed 17

hyperbolic 39, 116, 268heteroclinic 367homoclinic 367

transversal 381recurrent 49stationary 17

Poisson stability 49Process 322Projector 55

RRRR

Radius of dissipativity 24Reduction principle 49

SSSS

Scale of Hilbert spaces 79Semigroup property 11Semitrajectory 17Set

�-limit 18�-limit 18absorbing 24, 30fractal 53hyperbolic 377inertial 61, 65, 67, 103, 254

418 I n d e x

invariant 18negatively invariant 18of asymptotically determining functionals 286of determining functionals 292positively invariant 18

asymptotically stable 45Lyapunov stable 45uniformly asymptotically stable 45

unessential, with respect to measure 22unstable 35

Sobolev space 93, 306Solution

mild 85strong 82weak 82, 223, 232

Stabilityasymptotic 45

uniform 45Lyapunov 45Poisson 49

Star-like domain 307TTTT

Trajectory 17heteroclinic 367homoclinic 367

transversal 381, 402induced 161periodic 17Poisson stable 49

UUUU

Upper semicontinuity of attractors 46VVVV

Value of operator, regular 114Volume averages, determining 311, 331

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