Universite Badji Mokhtar d’AnnabaDepartement de Mathematiques

Thesis presented at Annaba University

in order to obtain the DOCTORATE OF SCIENCES

Presented by: Belkacem SAID-HOUARI

Speciality: Partial Differential Equations

November 29th, 2005.

Advisor: Prof. Salim A. Messaoudi (KFUPM)

Nonclassical Thermoelasticity(Global existence, Asymptotic Stability and Blow up in Finite Time).

Contents

Introduction 3

1 Classical Thermoelasticity 91.1 Derivation of the equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Longitudinal Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Global existence and asymptotic stability . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Blow up of smooth solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Thermoelasticity with second sound 192.1 Derivation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Asymptotic stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.3 Proof of Theorem 2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Blow up of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3.2 Blow up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Thermoelasticity of type III 633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Exponential stability in one-dimensional nonlinear system . . . . . . . . . . . . . 66

3.2.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.2 Proof of theorem 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Exponential stability in multi-dimensional nonlinear system with internal damping 88

3.4 Boundary stabilization for a multi-dimensional nonlinear system . . . . . . . . . . 92

4 Appendix 1014.1 Global non-existence of solutions of a class of wave equations with nonlinear

damping and source terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.1.2 Blow up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2 Global nonexistence result for the nonlinearly damped multi-dimensional Boussi-nesq equation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2.2 Blow up in the nonlinear damping case . . . . . . . . . . . . . . . . . . . 112

4.2.3 Blow up in the linear damping case . . . . . . . . . . . . . . . . . . . . . 117

CONTENTS 1

Abstract

In this thesis we study some hyperbolic problems arising in nonclassical thermoelasticity, namelysystems of thermoelasticity with second sound and of type III.

We establish several exponential decay and blow up results for classical and weak solutions inone-dimensional, as well as in multidimensional case. We also consider two problems related to aquasilinear wave equation and the Boussinesq equation.

Our technique of proof relies on the construction of the appropriate Lyaponov function equiv-alent to the energy, of the solution in consideration, and which satisfies a differential inequalityleading to the desired decay or blow up result.

2 CONTENTS

Introduction

The aim of this thesis is to study the long time behavior of solutions of certain hyperbolic systemsarising in both thermoelasticity and elasticity. In this regards, several results concerning decay, aswell as blow up, of solutions in nonclassical thermoelasticity have been proved. This study extendsand improves several earlier results.

This thesis is divided into four major chapters. In chapter one, we give a brief summary ofderivation of the equation in classical thermoelasticity and recall a general overview of resultsrelated to existence and nonexistence over the last two decades. In chapter two, a discussion of thelong time behavior (decay and finite-time blow up) of solutions to equations of thermoelasticitywith second sound is given and certain results in this directions have been established. In chapter3, we examined the asymptotic stability of solutions of another type of thermoelasticty, calledthermoelasticity of type III, in both the one-dimensional and multi-dimensional cases. Finally,chapter 4 was devoted to two blow up results for hyperbolic systems, we judged worth presentingfor the sake of completeness. We called this chapter an appendix since it is not in the core of ourstudy however it is related to non classical thermoelasticity in the sense that all the systems, weconsidered, are all of hyperbolic type.

Now, we give more details for our studyChapter 1: In this chapter, we give a short summary of the derivation of the equations of ther-moelasticity describe the reciprocal actions between elastic and temperature differences, we recallseveral results dealing with the mathematical questions arising in the study of initial value prob-lems where all classical boundary conditions are considered; rigidly clamped, constant temperature(Dirichlet type), or traction free, insulated (Neumann type), for instance.Chapter 2: This chapter is divided into two parts. In the first part we considered the followingsystem

utt − a (ux, θ, q) uxx + b (ux, θ, q) θx = α1 (ux, θ) qqx (1)

θt + g (ux, θ, q) qx + d (ux, θ, q) utx = α2 (ux, θ) qqt (2)

τ (ux, θ) qt + q + k (ux, θ) θx = 0, (3)

with initial and boundary conditions

u (0, .) = u0, ut (0, .) = u1, θ (0, .) = θ0, q (0, .) = q0 (4)

4 CONTENTS

u (t, 0) = u (t, 1) = θ (t, 0) = θ (t, 1) = 0. (5)

This system describes thermoelasticity with second sound, in which the heat flux is given by Cat-taneo’s law

τqt + q + kθx = 0

instead of the usual Fourier’s lawq + kθx = 0.

Results concerning existence, blow up, and asymptotic behavior of smooth, as well as weak solu-tions in thermoelasticity with second sound have been established over the past two decades.

Tarabek [79] treated problems related to (1)-(3) in both bounded and unbounded domains andestablished global existence results for small initial data. He also showed that these ” classical”solutions tend to equilibrium as t tends to infinity; however, no rate of decay has been discussed.In his work, Tarabek used the usual energy argument and exploited some relations from the secondlaw of thermodynamics to overcome the difficulty arising from the lack of Poincare’s inequality inthe unbounded domains. Saouli [76] used the nonlinear semigroup theory to prove a local existenceresult for a system similar to the one considered by Tarabek.

Concerning asymptotic behavior, Racke [71] discussed lately (1)-(3) and established exponen-tial decay results for several linear and nonlinear initial boundary value problems. In particular, hestudied the system (1)- (3), for a rigidly clamped medium with temperature held constant on theboundary. i.e

u (t, 0) = u (t, 1) = 0, θ (t, 0) = θ (t, 1) =−

θ, t ≥ 0,

and showed that, for small enough initial data and for α1 = α2 = 0, classical solutions decayexponentially to the equilibrium state. Although the dissipative effects of heat conduction inducedby Cattaneo’s law are usually weaker than those induced by Fourier’s law, a global existence aswell as exponential decay results for small initial data have been established.

In this thesis, we considered (1) - (3), for a rigidly clamped medium with temperature heldconstant at the boundary, and proved an exponential decay of the associated energy functional.Precisely, an estimate of the form

E (t) ≤ Ce−ηt, C > 0,

for classical solutions with small initial data, has been established. The method, we used, is basedon the construction of a Lyaponov functional F (t) equivalent to ”the third-order energy” E (t), forwhich an inequality of the form

ddt

F (t) ≤ −cF (t)

for c > 0, has been proved.

CONTENTS 5

In the second part, we dealt with the multi-dimensional problem

utt − µ∆u − (µ + λ)∇divu + β∇θ = |u|p−2u, x ∈ Ω, t > 0,

θt + γdivq + δdivut = 0, x ∈ Ω, t > 0,

τqt + q + κ∇θ = 0, x ∈ Ω, t > 0,

u(., 0) = u0, ut(., 0) = u1, θ(., 0) = θ0, q(., 0) = q0, x ∈ Ω,

u = θ = 0, x ∈ ∂Ω , t ≥ 0.

(6)

where Ω is a bounded domain of Rn, with a smooth boundary ∂Ω, u = u(x, t), q = q(x, t) ∈ Rn, andµ, λ, β, γ, δ, τ, κ are positive constants, where µ, λ are Lame moduli and τ is the relaxation time, asmall parameter compared to the others.

Racke [72] established an existence result for (6) in absence of the source term |u|p−2u andproved, under the conditions rotu = rotq = 0, an exponential decay result. This result appliesautomatically to the radially symmetric solution, since it is only a special case.

In [54] Messaoudi investigated the situation where the nonlinear source term is competingwith the damping caused by the heat conduction and established a local existence result. He alsoshowed that solutions with negative energy blow up in finite time. His work extended an earlier onein [50, 52] to thermoelasticity with second sound. In our work we extended the blow up result of[54] to situations, where the energy can be positive. The basic tool used in this part is the method ofGeorgiev and Todorova [19] combined with some techniques of Vitillaro [84] with the necessarymodifications imposed by the nature of our problem. This method consists of constructing an”energy equivalent” functional satisfying a differential inequality of the form

ddt

L (t) ≥ ξLq (t) , q > 1.

This, of course, lead to a blow up in finite time.Chapter 3: This chapter is divided in three parts and it is devoted to the analysis of the long timebehavior of solutions to systems of thermoelasticity of type III.

In the first part we considered the following one-dimensional nonlinear problem

utt − α (ux, θ) uxx + β (ux, θ) θx = 0, in [0,∞) × (0, 1)

θtt − δ (ux, θ) θxx + γ (ux, θ) uttx − κ (ux, θ) θtxx = 0, in [0,∞) × (0, 1)

u (0, .) = u0, ut (0, .) = u1, θ (0, .) = θ0, θt (0, .) = θ1, in (0, 1)

u (t, 0) = u (t, 1) = θ (t, 0) = θ (t, 1) , t ≥ 0

(7)

and established a decay result similar to the one for thermoelasticity with second sound. We recallhere that this theory introduced in the end of the last century, when Green and Naghdi [21, 22]introduced three types of thermoelastic theories based on an entropy equality instead of the usual

6 CONTENTS

entropy inequality. In each of these theories, the heat flux is given by a different constitutiveassumption. As a result, three theories are obtained and were called thermoelasticity type I, typeII, and type III respectively. This development is made in a rational way in order to obtain a fullyconsistent theory, which will incorporate thermal pulse transmission in a very logical manner andelevate the unphysical infinite speed of heat propagation induced by the classical theory of heatconduction. When the theory of type I is linearized the parabolic equation of the heat conductionarises. Whereas the theory of type II does not admit dissipation of energy and it is known asthermoelasticity without dissipation. In fact, it is a limiting case of thermoelasticity type III. See inthis regard [7, 8, 9, 62, 63, 64, 68] for more details.

In the second part, we considered the following multi-dimensional system of thermoelasticityof type III

utt − µ4u − (λ + µ)∇ (divu) + aut + β∇θ = 0, in [0,∞) ×Ω,

θtt − δ4θ + γdivutt − κ4θt = 0, in [0,∞) ×Ω,

u (0, .) = u0, ut (0, .) = u1, θ (0, .) = θ0, θt (0, .) = θ1, in Ω,

u = θ = 0, on (0,∞) × ∂Ω.

(8)

and showed that the presence of the linear damping term aut forced the following associated energy

E (t) = E (v, θ) =12

ˆΩ

(γv2

t + µγ |∇v|2 + (λ + µ) γ (divv)2 + βθ2t + δβ |∇θ|2

)dx

to decay exponentially as t → +∞.

The purpose of third part is to obtain an exponential stability for a multi-dimensional systemof thermoelasticity type III subject to a feedback on a part of the boundary. Precisely we studied

utt − µ4u − (µ + λ)∇(divu) + β∇θ = 0, in [0,∞) ×Ω,

cθtt − κ∆θ + βdivutt − δ∆θt = 0, in [0,∞) ×Ω,

u(., 0) = u0, ut(., 0) = u1, θ(., 0) = θ0, θt(., 0) = θ1, on Ω,

u = 0, x ∈ Γ0, t ≥ 0,

µ∂u∂ν

+ (µ + λ)(divu)ν = −aut, x ∈ Γ1, t ≥ 0,

θ = 0, x ∈ ∂Ω , t ≥ 0,

(9)

for c, δ, κ, β, λ, µ positive constants, Ω a bounded domain of Rn, with a smooth boundary ∂Ω. suchthat Γ0,Γ1 is a partition of ∂Ω, with meas(Γ1) > 0 and satisfying some extra geometry conditions,ν is the outward normal to ∂Ω, u = u(x, t) ∈ Rn is the displacement vector, and θ = θ(x, t) is thedifference temperature.

The technique, we use follows closely the method of Komornik and Zuazua [36] (see also [89])with the necessary modifications imposed by the nature of our problem.

CONTENTS 7

Chapter 4: In this chapter we studied two hyperbolic problems and established two importantresults on the global nonexistence for a special class of nonlinear wave equations with nonlineardamping and source terms as well as for the nonlinearly damped multi-dimensional Boussinesqequation. The main tool, used in both cases, is the techniques of Georgiev and Todorova [19].More precisely, in the first part we consider the nonlinear wave equation

utt − ∆ut − div(|∇u|α−2 ∇u

)− div

(|∇ut|

β−2 ∇ut)

+ a |ut|m−2 ut = b |u|p−2 u,

a, b > 0, associated with initial and Dirichlet-boundary conditions. We proved, under suitableconditions on α, β,m, p, that any weak solution with negative initial energy blows up in finite time.This improves a result by Yang [85], who requires that the initial energy be sufficiently negativeand relates the blow-up time to the size of Ω.

In the second part we considered a multi-dimensional nonlinear initial-boundary value problemrelated to the Boussinesq equation and proved a global nonexistence result. i.e we consider thefollowing nonlinearly damped problem

utt + ρ (x) |ut|m−2 ut + β42u − div (g (x)∇u) + Γ4

(|∇u|2 4u

)−div

(h (x) |∇u|p−2 ∇u

)− Γdiv

((4u)2 ∇u

)= |u|l−2 u, x ∈ Ω, t > 0,

u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ Ω,

u (x, t) =∂u∂η

(x, t) = 0, x ∈ ∂Ω, t > 0,

(10)

where Ω ⊂ Rn, n ≥ 1, is a bounded domain with sufficiently smooth boundary, η is the unit outernormal on ∂Ω, ρ ≥ 0, is a smooth bounded function given on Ω, g, h ∈ C1

(Ω,R+

), p, l,m ≥

1, and β and Γ are nonnegative constants. In addition to allowing the damping to be nonlinear, weestablished a blow up result under weaker conditions than those required in [20], on the initial dataas well as the constants p, l, and m. To achieve our goal we exploited the method of Georgiev andTodorova [19] (see also [51]) This work improves an earlier one by Gmira and Guedda [20].

8 CONTENTS

PublicationsThe following results were published / accepted, or submitted:

from chapter 2:

1. Exponential Stability in one-dimensional nonlinear thermoelasticity with second sound, toappear in Math Methods in Applied Sciences # 28 (2005), 205 - 232..

2. Blow up of solutions with positive energy in nonlinear thermoelasticity with second sound,J. Appl. Math methods 3 (2004) 201-211.

from chapter 3:

3. Boundary stabilization in thermoelasticity of type III, submitted to Discrete and ContinuousDynamical Systems (DCDS).

4. A decay result in system of thermoelasticity type III, to appear in the proceedings of theU.A.E. Math Day, Nova Publishing Company, New York.

5. Exponential stability in one-dimensional nonlinear thermoelasticity type III, submitted toNonlinear Analysis.

from chapter 4:

6. Blow up of solutions of a class of wave equations with a nonlinear damping and sourceterms, Math. Methods . Appl Sciences 27 no13 (2004), 1687-1696.

7. A global nonexistence result for the nonlinearly damped multi-dimensional Boussinesqequation, to appear in AJSE.

Chapter 1

Classical Thermoelasticity

1.1 Derivation of the equations

In this section we give a short summary of the derivation of the nonlinear equations describing thethermoelastic behavior of a body B. Consider a three-dimensional homogeneous body B with areference configuration Ω, a bounded region in Rn, n = 1, 2 or 3. A thermodynamic process for thisbody is given by the following functions of a material point with reference position x, and the timet :

1. The position X = X (x, t) at time t.

2. The Piola-Kirchhoff stress tensor S = S (x, t).

3. The body force b = b (x, t) .

4. The internal energy e = e (x, t) .

5. The heat flux vector q = q (x, t) .

6. The external heat supply r = r (x, t) .

7. The entropy η = η (x, t) .

8. The absolute temperature T (x, t)

The specific Helmhotz free energy is

ϕ = e − Tη (1.1)

and the displacement vector U (x, t) = X (x, t) − x. The equation of balance of linear momentumand balance of energy take the form

ρXtt (x, t) = divS (x, t) + ρb (x, t) . (1.2)

ρet (x, t) = tr(S T (x, t) Ft (x, t)

)− divq (x, t) + ρr (x, t) . (1.3)

10 Classical Thermoelasticity

where the superscript T denotes the transpose operator, F = 1+∇U the deformation gradient tensorgoes along with change of temperature T = T (x, t) , and ρ is the material density in Ω, whichassumed to be constant, and tr denotes the trace.

Under suitable smoothness assumptions the balance equations (1.2) , (1.3) are equivalent tothe following two balance equations in integral form respectively. For every part P of Ω we have

ˆP

Xttdm =

ˆP

bdm +

ˆ∂P

S .nds. (1.4)

12

ddt

ˆP

X2t dm +

ˆP

etdm =

ˆP

(Xtb + r) dm +

ˆ∂P

(Xt.S n − q.n) ds. (1.5)

Where dm denotes the element of mass in the body, ∂P the boundary of P, ds the element ofsurface area in the configuration at time t, and n the exterior unit normal to ∂P.

In Thermoelasticity the constitutive assumption is that the functions S , q, ϕ, η depend on thepresent values of F, T, and g = ∇T.

S = S (F,T, g) , q = q (F,T, g) .

ϕ = ϕ (F,T, g) , η = η (F,T, g) .

These functions are assumed to be smooth and

det F > 0, T > 0

Consistency with the second law of thermodynamic requires that the Clausius-Duhem inequality

ddt

ˆPρηdx ≥

ˆPρ

rT

dx −ˆ∂P

q.nT

ds

holds for all subbodies P of B. In particular the local form of the second law of thermodynamicstakes the form

ηt ≥ −divqT

+rT. (1.6)

Then combining (1.3) with (1.6) gives the free-energy inequality

ϕt + ηTt − tr(S T Ft

)+

q.gT≤ 0. (1.7)

The second law of thermodynamics (the Clausius- Duhem inequality) places restrictions on theform of constitutive relations for ϕ, S , η, q. Coleman and Mizel [12] and Coleman and Noll [13]showed that a necessary and sufficient conditions for the inequality (1.7) to hold is that

i) The response functions S , η, and ϕ are independent of the temperature gradient g :

S = S (F,T ) , ϕ = ϕ (F,T ) , η = η (F,T ) . (1.8)

1.1 Derivation of the equations 11

ii) ϕ determines S through the stress relation

S =∂ϕ

∂F(F,T ) (1.9)

and η through the entropy relation

η (F,T ) = −∂ϕ

∂T(F,T ) . (1.10)

iii) q obeys the heat conduction inequality

q (F,T, g) .g ≤ 0. (1.11)

In what follows, we shall use ϕ, η, q, S in place of ϕ, η, q, S , for simplicity.

As e = ϕ + Tη, we easily see that

et = ϕt + Ttη + Tηt

which, combined with (1.10), gives

ρet = ρtr∂ϕ

∂FFt

+ ρTηt.

This in turn yields from (1.3) the equation

ρTηt = −divq + ρr.

By using the chain rule we have

ηt = tr

(∂η

∂F

)T

Ft

+∂η

∂TTt. (1.12)

Combining (1.10) and (1.12), we obtain

ηt = tr

(∂2ϕ

∂F∂T

)T

Ft

+

(∂2ϕ

∂T 2

)Tt (1.13)

It follows from (1.13) that the evolution equations governing the thermoelastic body reads

ρXtt (x, t) = divS (x, t) + ρb (x, t)

ρT−∂2ϕ

∂T 2 Tt −∂2ϕ

∂F∂TFt

+ divq = ρr.

(1.14)

12 Classical Thermoelasticity

If θ = T − T0, denotes the difference temperature from a constant reference temperature T0 andU = X − x denotes the displacement, the linearized system in two or three-space dimension takesthe form

Utt − µ∆U − (λ + µ)∇divU + γ∇θ = b (1.15)

δθt − k∆θ + γdivUt = r, (1.16)

where µ, λ, γ, δ and k are positive constants , µ, λ are the Lame moduli. For a complete discussionfor the derivation of the equations (1.15) and (1.16) we refer the reader to Jiang and Racke [30] andLeis [40].

1.1.1 Longitudinal Motions

The motion of the body B is purely longitudinal if

X1 = w (t, x1) , X2 = x2, X3 = x3, T = ζ (t, x1) , (1.17)

and in the terms of the displacement U and the temperature difference θ :

U (t, x1) = w (t, x1) − x1, θ = ζ (t, x1) − T0, (1.18)

In this case the deformation gradient takes the form

F =

∂w∂x1

0 00 1 00 0 1

and the stress tensor S has the form (see [10])

S =

S 11 0 00 S 22 00 0 S 33

.We denote by

x := x1, ε := F11 − 1 = Ux, X := X1, σ := S 11

and observe that det F > 0 requires that ε > −1.

It follows from (1.2) , (1.3) that the evolution equations governing the purely longitudinal mo-tion take the form

ρXtt (x, t) = σx (x, t) + ρb (x, t)

ρet (x, t) − σ (x, t) εt (x, t) = −qx (x, t) + ρr (x, t)(1.19)

By specializing (1.19) to the case when the heat flux is given by Fourier’s law, i.e.

1.1 Derivation of the equations 13

q (x, t) = −κ (ε (x, t) ,T (x, t)) g (x, t) (1.20)

and by taking ρ = 1, (for simplicity) then (1.9)-(1.11) imply

σ (x, t) = ϕε (ε (x, t) ,T (x, t)) , (1.21)

η (x, t) = −ϕT (ε (x, t) ,T (x, t)) (1.22)

withκ ≥ 0. (1.23)

It follows from (1.1) and (1.23) that

ϕt + ηTt − σεt +qgT

=−κ (ε,T ) g2

T≤ 0.

By using (1.1), and (1.22) the internal energy is given by

e = ϕ (ε (x, t) ,T (x, t)) − TϕT (ε (x, t) ,T (x, t)) .

As consequence of (1.21) and (1.22) is the identity

ϕt + ηTt − σεt = 0, (1.24)

which is equivalent to Gibbs’s relation

et − Tηt − σεt = 0. (1.25)

By virtue of (1.1) and (1.25) , the equation of balance of energy (1.19)2 can be rewritten as

Tηt = −qx + r. (1.26)

Combining (1.19)1 and (1.26) (using u instead of U) we obtain

utt − [σ (ux + 1, θ + T0)]x = b

(1.27)(θ + T0) ηt (ux + 1, θ + T0) −

[κ (ε (x, t) ,T (x, t)) g (x, t)

]x = r,

(1.28)

in (0,∞) × (0, 1) where (0, 1) represents the reference configuration.

For simplicity, we assume that the heat supply r and the body force b are equal to zero, then byusing (1.21) and (1.22), the equations (1.27) and (1.28) becomes

utt = a(ux, θ)uxx + b(ux, θ)θx, x ∈ Ω, t > 0,

(θ + T0) c(ux, θ)θt + b(ux, θ)utx = [k (ux, θ) θx (x, t)]x , x ∈ Ω, t > 0.. (1.29)

where

a(ux, θ) = ϕεε (ux + 1, θ + T0) , b(ux, θ) = ϕεT (ux + 1, θ + T0)

c(ux, θ) = ϕTT (ux + 1, θ + T0) , k (ux, θ) θx (x, t) = κ (ux + 1, θ + T0)

We refer to the book of Day [16] for more information on the one-dimensional thermoelasticity.

14 Classical Thermoelasticity

1.2 Global existence and asymptotic stability

Over the past two decades, there has been a lot of work on local existence, global existence, well-posedeness, and asymptotic behavior of solutions to some initial-boundary value problems as wellas to Cauchy problems in both one-dimensional and multi-dimensional thermoelasticity.

Based on Matsumura’s refinement [49] of the Courant, Friedrichs and Lewy energy method,Slemrod [78] established global existence and decay of classical solution for small data when theboundary is either traction free and at a constant temperature or rigidly clamped and thermallyinsulated. (i.e. Neumann-Dirichlit or Dirichlit-Neumann boundary conditions). These boundaryconditions imply an additional one, i.e. if an end is clamped, the displacement u satisfies uxx = 0there. Thus, additional partial integrations were obtained and led to the desired a priori estimates.In that work the Poincare’s inequality played an essential role. See also in this direction the paperof Zeng [87], in which the author discussed the special boundary conditions studied by Slemrod[78].

For Dirichlet-Dirichlet boundary conditions, the problem of global existence of smooth so-lutions remained open for a long time. In 1991 Racke and Shibata [73] used the spectral analysismethod to obtain the polynomial decay rate of the solution for the linear case as well as for the non-linear case. They also showed that the rate of decay depends on higher regularity of the initial data,and therefore the global existence result depends on the initial data to be small in Hm (0, L) , form large. In [56] Rivera proved the exponential decay of solutions to the Dirichlet-Dirichlet initialboundary value problem in linear thermoelasticity, using the energy method and a tricky way1 todeal with the boundary terms. Racke, Shibata and Zheng [74] then extended Rivera’s results to thecase of nonlinear thermoelastic systems and improved the result of Racke and Shibata [73] for smalldata (u0, u1) in H3 (0, L)×H2 (0, L). Rivera and Barreto [57] showed how the energy method alonecan be used to improve the work in [73] by taking initial data (u0, u1) small in H2 (0, L)×H1 (0, L).In [48], Liu and Zheng showed that the semigroup associated with the one-dimensional thermoe-lasticity system with Dirichlet-Dirichlet boundary conditions is exponentially stable. Their methodwas based on an abstract theorem about the exponential stability of semigroups. By using thedecoupling technique, a similar result for the one-dimensional thermoelasticity was also obtainedby Hansen [24], in the case of Dirichlet-Neumann or Neumann-Dirichlet type boundary condi-tions and the Neumann-Neumann problem for a bounded domain by Kawashima and Shibata [34],Shibata [77] and Jian [27, 28] who also discussed the half-line case. Using the same method ofsemigroups, Burns, Liu and Zheng [6] further considered all other possible boundary conditionsfor the one-dimensional system of thermoelasticity and showed that the semigroups associated withthese boundary conditions are also exponentially stable.

The Cauchy problem, in which a thermoelastic body occupies the entire real line, was inves-tigated by Kawashima and Okada [33], Kawashima [32] Zheng and Shen [88], and Hrusa andTarabek [26] proving the global existence in time. In particular, Hrusa and Tarabek [26] com-

1This way of dealing with boundary terms was used by Racke [71] , and it will be used in this thesis(See; Chapter 2 and Chapter 3

).

1.2 Global existence and asymptotic stability 15

bined certain estimates of Slemrod that remain valid on unbounded intervals with some additionalones which exploited some relations associated with the second law of thermodynamics, to obtainthe energy estimate for lowest order terms, an estimate that cannot be obtained using Poincare’sinequality as in Slemrod’s paper.

Periodic solutions were studied by Feireisl [17] for Dirichlet-Neumann andNeumann-Dirichlet boundary conditions and by Racke, Shibata and Zheng [74] for Dirichlet-Dirichlet boundary conditions.

For the non-cylindrical domain (i.e. domain with moving boundary), Ferreira and Benabidalah[18] considered the one-dimensional linear thermoelasticity and established an exponential decayresult by combining the energy method with multiplier techniques.

The asymptotic stability of solutions in multi-spatial dimension has been less studied comparedwith one dimensional case. We have the pioneering work of Dafermos [14], in which he provedan asymptotic stability result; but no rate of decay has been given. He also pointed out that nodecaying solution can exist if the domain is a ball.

For the multi-dimensional case the situation is different from that in the one dimensional mod-els, where it was proved that the dissipation given by difference of temperature is strong enough toproduce uniform rate of decay to the solution. In fact, the degree of freedom of the displacementvector field is greater than the one of freedom of the temperature when the dimension is greaterthan n = 1. For this reason we do not have, in general, the uniform rate of decay for the solution inmulti-dimensional space.

The uniform rate of decay for the solution in two or three dimensional space, was obtained byJiang, Rivera and Racke [29] in special situation like radial symmetry.

Recently, Lebeau and Zuazua [39] proved that the decay rate is never uniform when the domainis convex. Thus, in order to solve this problem, additional damping mechanisms are necessary. Inthis aspect Pereira and Perla Menzala [60] introduced a linear internal damping, and establishedthe uniform decay rate. A similar result was obtained by Liu [46] for a linear boundary velocityfeedback acting on the elastic component of the system, and by Liu and Zuazua [47] for a nonlinearboundary feedback. Also, Aassila [1] proved a decay result and gave a precise decay rate. As hementioned, his result extends the one in Rivera and Oliviera [58] to the nonlinear damping case.

It is worth mentioning that the asymptotic behavior in unbounded domains with boundary,specially in exterior domains has been discussed by Racke [70] in special set of boundary con-ditions. Also, Rivera and Qin [59] established the global existence and exponential stability ofsmall solution to a nonlinear one- dimensional thermoelasticity with termal memory subject toDirichlet-Dirichlet boundary conditions.

16 Classical Thermoelasticity

1.3 Blow up of smooth solutions

We note that a global classical solution should not be expected in the case of large initial data.In fact, the work by Coleman and Gurtin [10] on the growth and decay of acceleration waves inone-dimensional nonlinear thermoelasticity provides a strong indication that, the damping effect ofthermal diffusion manages to restrain waves of small amplitude, but the nonlinear elastic responsetends to destabilize waves with large amplitude.

For specialized constitutive equations, (as we shall describe later) Dafermos and Hsiao [15]have shown that if the initial data are large then the solution to the Cauchy problem will developsingularities in finite time. Hrusa and Messaoudi [25] also showed that, for a special class ofnonlinear thermoelastic materials occupying the whole line, there are smooth initial data for whichthe solution break down in finite time. Precisely they considered special constitutive relations ofthe form :

S (ε,T ) : = p (ε) + γ (T − T0) . (1.30)

e (ε,T ) : = P (ε) + δ (T − T0) − γT0ε. (1.31)

q (ε,T,Tx) : = −κTx, (1.32)

where T0, κ, δ > 0, γ , 0, ε = ux, T0 is the reference temperature; p : (−1,∞) → R is a givenfunction,

P :=ˆ ε

0p (ζ) dζ, ε > −1. (1.33)

It is easy to see that these constitutive equations satisfy

eε = S − TS T gq(ε,T, g) ≤ 0

and thatϕ (ε,T ) = P (ε) + γε (T − T0) + δ log

(T0

T

)+ δ (T − T0) (1.34)

is a Helmholtz free energy and the corresponding entropy is given by

η (ε,T ) = −γε − δ log(T0

T

). (1.35)

Moreover the constitutive equations (1.30) − (1.32) are fully compatible with the second law ofthermodynamics. We should note that in [15], the authors supposed that the stress and the heat fluxare given by (1.30) and (1.32) respectively, and that

e (ε,T ) = P (ε) + δ (T − T0) .

As they pointed out, their constitutive relations comply with eε = S − TS T only if T0 = 0 or γ = 0.

1.3 Blow up of smooth solutions17

In [50], Messaoudi considered a special type of one dimensional nonlinear thermoelastic equa-tions and showed that, any weak solution with negative initial energy collapse in finite time. Themain tool used by the author was the concavity method of Levine [45]. Kirane and Tatar [35] im-proved Messaoudi’s result for a larger set of initial data and extended the result for more generalsituation, by allowing gradient terms in both equations. It should be noted that the method usedin [35] depends on a well-known lemma by Kalantarov and Ladyzhenskaya [31], which is in facta compact version of the concavity method. Recently, Qin and Rivera [61] extended the result of[35] and established the blow up result for a nonlinear one dimensional thermoelastic system witha non autonomous forcing term and thermal memory when the heat flux obeys both Fourier’s lawand Gurtin and Pipkin’s law [23].

In multi-space dimension, there are only a few results in this direction. The three-dimensionalcase was discussed by Racke [69], and more generally by Messaoudi [52] in multi-dimensionalcase. In his work, Messaoudi introduced a suitable functional, for which he could apply a methodby Georgiev and Todorova [19], and showed that any weak solution, with negative initial energy,blows up in finite time.

18 Classical Thermoelasticity

Chapter 2

Thermoelasticity with second sound

In classical thermoelasticity, the heat flux is given by Fourier’s law. As a result, this theory predictsan infinite speed of heat propagation; that is any thermal disturbance at one point has an instanta-neous effect elsewhere in the body. Experiments showed that heat conduction in some dielectriccrystals at low temperatures is free of this paradox and disturbances, which are almost entirelythermal, propagate in a finite speed. This phenomenon in dielectric crystals is called second sound.To overcome this physical paradox, many theories have merged. One of which suggests that weshould replace Fourier’ s law by so called Cattaneo’ s law.

In this chapter we present our main results concerning the time-asymptotic behavior of solu-tions to the nonlinear thermoelasticity with second sound in one space dimensional, as well as inmulti-spatial demensional. In section 2.1 we derive the problem, in section 2.2 we consider a one-dimensional nonlinear system of thermoelasticity with second sound, and establish an exponentialdecay result for solution with small ”enough” initial data. In section 2.3 we state and prove theblow up result for certain solutions with positive initial energy.

20 Thermoelasticity with second sound

2.1 Derivation of the Model

In this section we shall give a short summary in the derivation of the nonlinear one- dimensionalthermoelasticity with second sound.

For a one-dimensional homogeneous body occupying , in its reference configuration, an inter-val I = (0, 1) the laws of balance of momentum, balance of energy, and growth of entropy have theforms

ρutt = σx + b (2.1)

et + qx = σεt + r (2.2)

ut ≥rθ−

(qθ

)x

(2.3)

where the displacement u, the strain ε = ux, the stress σ, the difference ”absolute” temperatureθ, the heat flux q, the internal energy e, the body force b, and the external heat supply r are allfunctions of (x, t) (t ≥ 0, x ∈ I = (0, 1)) . Moreover the strain and the difference temperature arerequired to satisfy

ε > −1 θ > 0.

We then define the free energy byψ = e − θη. (2.4)

For thermoelasticity with second sound, the constitutive relations are

ψ = ψˆ (ux, θ, q) , η = ηˆ (ux, θ, q)

σ = σˆ (ux, θ, q) , e = eˆ (ux, θ, q) (2.5)

and the heat conduction is given by Cattaneo’s law instead of Fourier’s law

τ (ux, θ) qt + q = −k (ux, θ) θx (2.6)

where ψˆ, ηˆ, σˆ, eˆ, τ, and k are smooth functions. We note here that τ is the thermal relaxation timeand κ is the thermal conductivity.

Using the second law of thermodynamics (See [13], [12], and [79]), one can show that

ψˆ (ε, θ, q) = ψ0 (ε, θ) +12χ (ε, θ) q2

χ (ε, θ) =τ (ε, θ)θk (ε, θ)

. (2.7)

σˆ (ε, θ, q) = ψˆε (ε, θ, q)

ηˆ (ε, θ, q) = −ψˆθ (ε, θ, q) .

It then follows from (2.5) and (2.7) that

eˆ (ε, θ, q) = ψˆ (ε, θ, q) − θψˆθ (ε, θ, q) , (2.8)

2.1 Derivation of the Model 21

which gives, in turn,

eˆθ = −θψˆ

θθ,σˆ − eˆ

ε

σˆθ

= θ. (2.9)

In the absence of the body force b and the external heat supply r, assuming that the materialdensity ρ equal to one, and taking in consideration (2.7)-(2.9), equations (2.1), (2.2), together withCattaneo’s law (2.6) take the form

utt − a (ux, θ, q) uxx + b (ux, θ, q) θx = α1 (ux, θ) qqx (2.10)

θt + g (ux, θ, q) qx + d (ux, θ, q) utx = α2 (ux, θ) qqt (2.11)

τ (ux, θ) qt + q + k (ux, θ) θx = 0, (2.12)

where

a = σˆε, b = −σˆ

θ, α1 = χε,

g =−1θψˆ

θθ

, d =σˆθ

ψˆθθ

, α2 =χ − θχθ

θψˆθθ

.

22 Thermoelasticity with second sound

2.2 Asymptotic stability

2.2.1 Introduction

Result concerning existence, blow up, and asymptotic behavior of smooth, as well as weak solu-tions in thermoelasticity with second sound have been established over the past two decades.

Tarabek [79] treated problems related to (2.10)-(2.12) in both bounded and unbounded situa-tions and established global existence results for small initial data. He also showed that these ”classical” solutions tend to equilibrium as t tends to infinity; however, no rate of decay has beendiscussed. In his work, Tarabek used the usual energy argument and exploited some relations fromthe second law of thermodynamics1 to overcome the difficulty arising from the lack of Poincare’sinequality in the unbounded domains. Saouli [76] used the nonlinear semigroup theory to prove alocal existence result for a system similar to the one considered by Tarabek.

Concerning asymptotic behavior, Racke [71] discussed lately (2.10)-(2.12) and established ex-ponential decay results for several linear and nonlinear initial boundary value problems. In par-ticular he studied the system (2.10)-(2.12), for a rigidly clamped medium with temperature holdconstant on the boundary. i.e

u (t, 0) = u (t, 1) = 0, θ (t, 0) = θ (t, 1) =−

θ, t ≥ 0

and showed that, for small enough initial data and for α1 = α2 = 0, classical solutions decayexponentially to the equilibrium state. It is interesting to observe that taking α1 = α2 = 0 makesχ (ε, θ) = c0θ, and consequently τ = c0θk by virtue of (2.7) . Although the dissipative effects ofheat conduction induced by Cattaneo’s law are usually weaker than those induced by Fourier’s law,a global existence as well as exponential decay results for small initial data have been established.For a discussion in this direction, see Reference [71].

For the multi-dimensional case (n = 2, 3) Racke [72] established an existence result for thefollowing n-dimensional problem

utt − µ∆u − (µ + λ)∇divu + β∇θ = 0, x ∈ Ω, t > 0,

θt + γdivq + δdivut = 0, x ∈ Ω, t > 0,

τqt + q + κ∇θ = 0, x ∈ Ω, t > 0,

u(., 0) = u0, ut(., 0) = u1, θ(., 0) = θ0, q(., 0) = q0, x ∈ Ω,

u = θ = 0, x ∈ ∂Ω , t ≥ 0.

(2.13)

where Ω is a bounded domain of Rn, with a smooth boundary ∂Ω, u = u(x, t), q = q(x, t) ∈ Rn,and µ, λ, β, γ, δ, τ, κ are positive constants, where µ, λ are Lame moduli and τ is the relaxation time,

1Relations from thermodynamics have been also used by Hrusa & Tarabek [26] to prove a global existence for theCauchy problem to a classical thermoelasticity system and then by Hrusa & Messaoudi [25] to establish a blow up resultfor a thermoelastic system.

2.2 Asymptotic stability 23

a small parameter compared to the others. In particular if τ = 0, (2.13) reduces to the system ofclassical thermoelasticity, in which the heat flux is given by Fourier’s law instead of Cattaneo’s law.He also proved, under the conditions rotu = rotq = 0, an exponential decay result for (2.13). Thisresult applies automatically to the radially symmetric solution, since it is only a special case.

In the following section we consider (2.10)-(2.12), for a rigidly clamped medium with temper-ature hold constant at the boundary, and show that a similar argument to the one in [71] is still validto prove the exponential decay for classical solutions with small initial data.

2.2.2 Statement of the problem

We consider the problem

utt − auxx + bθx = α1qqx (2.14)

θt + gqx + dutx = α2qqt (2.15)

τqt + q + kθx = 0 (2.16)

u (0, .) = u0, ut (0, .) = u1, θ (0, .) = θ0, q (0, .) = q0 (2.17)

u (t, 0) = u (t, 1) = θ (t, 0) = θ (t, 1) = 0, (2.18)

where

a = a (ux, θ, q) , b = b (ux, θ, q) ,

g = g (ux, θ, q) , d = d (ux, θ, q) ,

τ = τ (ux, θ) , k = k (ux, θ) ,

α1 = α1 (ux, θ) , α2 = α2 (ux, θ) .

We assume that there exists positive constants β > 0 such that

β ≤ a (ux, θ, q) , β ≤ g (ux, θ, q) , (2.19)

β ≤ k (ux, θ) , β ≤ τ (ux, θ)

d (ux, θ, q) , 0, b (ux, θ, q) , 0. (2.20)

In order to make this section self contained we state, without proof, a local existence result. Theproof can be established by a classical energy argument [11]. For this purpose we set

u2 = a (u0x, θ0, q0) u0xx − b (u0x, θ0, q0) θ0x + α1 (u0x, θ0) q0q0x

θ1 = −g (u0x, θ0, q0) q0x − d (u0x, θ0, q0) u1x + α2 (u0x, θ0) q0q1

q1 =−1

τ (u0x, θ0)q0 −

k (u0x, θ0)τ (u0x, θ0)

θ0x.

24 Thermoelasticity with second sound

Theorem 2.2.1 Assume that a, b, α1, g, d, α2, τ, k are C3 functions satisfying (2.19) and (2.20) .Then for any initial data

u0 ∈ H3 (I) ∩ H10 (I) , u1, θ0 ∈ H2 (I) ∩ H1

0 (I) , q0 ∈ H2 (I)

u2 ∈ H10 (I) , θ1 ∈ H1

0 (I) , θ0 > 0

problem (2.14)−(2.18) has a unique local solution (u, θ, q) , on a maximal time interval [0,T ) , satisfying

u ∈2∩

m=0Cm

([0,T ) ,H3−m (I) ∩ H1

0 (I)), ∂3

t u ∈ C([0,T ) , L2 (I)

)θ ∈

1∩

m=0Cm

([0,T ) ,H2−m (I) ∩ H1

0 (I)), ∂2

t θ ∈ C([0,T ) , L2 (I)

)q ∈

1∩

m=0Cm

([0,T ) ,H2−m (I) ∩ H1

0 (I)), ∂2

t q ∈ C([0,T ) , L2 (I)

).

To state our main result, we denote by

Λ (t) =

ˆ 1

0(u2

ttt + u2xxx + u2

ttx + u2xxt + u2

tt + u2xx + u2

tx + u2t + u2

x

+θ2tt + θ2

xx + θ2xt + θ2

t + θ2x + θ2 + q2

tt + q2xx + q2

xt + q2x

+q2t + q2)dx, (2.21)

α (t) = sup0≤x≤1

(|θ| + |θx| + |θt| + |q| + |qt| + |qx| + |ut| + |ux|

+ |utx| + |uxx| + |utt| + |u|

), (2.22)

andE (t) = E1 (t) + E2 (t) + E3 (t) , (2.23)

where

E1 (t) =12

ˆ 1

0

[κdu2

t + κdau2x + κbθ2 + bgτq2

](t, x) dx = E1 (t, u, θ, q) (2.24)

and

E2 (t) = E1 (t, ut, θt, qt) ,

E3 (t) = E1 (t, utt, θtt, qtt) ,

Theorem 2.2.2 Assume that a, b, α1, g, d, α2, τ, k are C2 functions satisfying (2.19) and (2.20).Then there exists a small positive constant δ such that if

Λ0 = ‖u0‖2H3 + ‖u1‖

2H2 + ‖θ0‖

2H2 + ‖q0‖

2H2 < δ (2.25)

the solution of (2.14)- (2.18) decays exponentially as t → +∞.

Remark 2.2.1 C3b denotes the set of bounded functions with bounded derivatives up to the third

2.2 Asymptotic stability 25

order. This condition is only made for the simplicity of the proof. However, the decay result stillholds if (2.19) and (2.20) are satisfied only in a neighborhood of the equilibrium state and the func-tions are taken in C3. In this case a slight modification in the proof, as in References [11, 53], isneeded.

2.2.3 Proof of Theorem 2.2.2

Given a local solution, we multiply (2.14) by κdut, (2.15) , by κbθ and (2.16) by gbq, integrate overI,using integration by parts and add equations, to obtain

dE1 (t)dt

= −

ˆ 1

0bgq2dx + R1 (2.26)

with

R1 =

ˆ 1

0

12

(κd)t u2t − (aκd)x uxut +

12

(aκd)t u2x + (bκd)x θut

+12

(κb)t θ2 + (κbg)x qθ +

12

(bgτ)t q2 + (κdα1) qqxut

+ (κdα2) qqtθ)dx.

By using (2.19), it is easy to see|R1| ≤ Γα (t) Λ (t) (2.27)

where Γ denotes a positive (possibly large) generic constant independent of u, θ, q, t. Then (2.26)becomes

dE1 (t)dt

≤ −

ˆ 1

0bgq2dx + Γα (t) Λ (t) . (2.28)

Differentiating (2.14) -(2.16) with respect to t,we get

uttt − auxxt − atuxx + bθxt + btθx = α1qtqx + α1qqxt + α1tqqx. (2.29)

θtt + gqxt + gtqx + duttx + dtutx = α2q2t + α2tqtq + α2qttq. (2.30)

τqtt + τtqt + qt + κθxt + κtθx = 0. (2.31)

In the same manner, multiplying (2.29) by κdutt, (2.30) by κbθt, and (2.31) by gbqt we obtain

dE2 (t)dt

= −

ˆ 1

0bgq2

t dx + R2, (2.32)

with

26 Thermoelasticity with second sound

R2 =

ˆ 1

0(κdatuttuxx − κdbtθxutt + α1κdqtqxutt + α1κdqqxtutt

+α1tκdqqtutt − κbgtqxθt − κbdtutxθt

+α2κbq2t θt + α2tκbqtqθt + α2κbqqttθt − bgτtq2

t

−κtbgθxqt − (aκd)xuttutx + (bκd)xθtutt + (κbg)xθtqt)dx

+12

ˆ 1

0(κd)tu2

tt + (aκd)tu2tx + (κb)tθ

2t + (bgτ)tq2

t dx

For additional estimates, we differentiate (2.29)-(2.31), with respect to t, to get

utttt − auttxx − 2atutxx + bθttx + 2btθtx − attuxx + bttθx

= 2α1tqtqx + α1qttqx + 2α1qtqtx + 2α1tqqtx + α1qqttx.+α1ttqqx. (2.33)

θttt + gqxtt + 2gtqtx + dutttx + 2dtuttx + gttqx + dttutx

= 2α2tq2t + 3α2qtqtt + α2ttqtq + 2α2tqttq + α2qtttq

. (2.34)

τqttt + 2τtqtt + τttqt + qtt + kθttx + 2ktθtx + kttθx = 0. (2.35)

We then multiply (2.33) by κduttt,(2.34) by κbθtt and (2.35) by bgqtt to have, by similar calculations

dE3 (t)dt

= −

ˆ 1

0bgq2

ttdx + R3, (2.36)

where

R3 =

ˆ 1

0

[(bkd)x utttθtt + (kbg)x θttqtt − (adk)x uttxuttt

]+

ˆ 1

0kduttt(2atutxx − 2btθtx + attuxx − bttθx + 2α1tqtqx + α1qttqx

+2α1qtqtx + 2α1tqqtx + α1qqttx + α1ttqqx)dx

+

ˆ 1

0kbθtt(−2gtqtx − 2dtuttx − gttqx − dttutx

+2α2tq2t + 3α2qtqtt + α2ttqtq + 2α2tqttq + α2qtttq)dx

−

ˆ 1

0bgqtt(2τtqtt + τttqt + 2ktθtx + kttθx)dx

+12

ˆ 1

0

[(kd)t u2

ttt + (akd)t uttx + (kb)t θ2tt + (bgτ)t q2

tt

]dx. (2.37)

By the same manner as in (2.28) and (2.32) we arrive at

dE3 (t)dt

≤ −

ˆ 1

0bgq2

ttdx + Γ(α (t) + α2 (t) + α3 (t)

)Λ (t) . (2.38)

2.2 Asymptotic stability 27

We then use (2.16) to get

ˆ 1

0(kθx)2 dx =

ˆ 1

0(τqt + q)2 dx

=

ˆ 1

0

(τ2q2

t + q2 + 2τqqt)

dx

and Young’s inequality to obtain

ˆ 1

0θ2

xdx ≤ˆ 1

0

2τ2

k2 q2t dx +

ˆ 1

0

2k2 q2dx. (2.39)

A differentiation of (2.16) with t then leads to

τqtt + qt + kθxt + τtqt + ktθx = 0. (2.40)

Multiply (2.40) by θxt and integrate over I to find

ˆ 1

0kθ2

xtdx = −

ˆ 1

0(τqtt + qt + τtqt + ktθx) θxtdx (2.41)

and thanks to Young inequality and (2.19) , we obtain

ˆ 1

0kθ2

xtdx ≤ Cˆ 1

0

(q2

tt + q2t

)dx + R1, (2.42)

where

R1 =

ˆ 1

0(τtqtθxt + ktθxθxt) dx.

Then from (2.39) and (2.42) we can write

ˆ 1

0

(θ2

x + θ2xt

)dx ≤ C

ˆ 1

0

(q2

tt + q2t + q2

)dx + R1. (2.43)

28 Thermoelasticity with second sound

Multiplying equation (2.14) by 1a uxx integrate over I and using integration by parts, we have

ˆ 1

0u2

xxdx =

ˆ 1

0

1a

uttuxxdx +

ˆ 1

0

baθxuxxdx −

ˆ 1

0

α1

aqqxuxxdx

= −

ˆ 1

0

(1a

utt

)x

uxdx +

ˆ 1

0

baθxuxxdx −

ˆ 1

0

α1

aqqxuxxdx

= −

ˆ 1

0

1a

uttxuxdx −ˆ 1

0

(1a

)x

uttuxdx +

ˆ 1

0

baθxuxxdx

−

ˆ 1

0

α1

aqqxuxxdx

= −ddt

ˆ 1

0

1a

utxuxdx +

ˆ 1

0

(1a

)tutxuxdx +

ˆ 1

0

1a

u2txdx

−

ˆ 1

0

(1a

)x

uttuxdx +

ˆ 1

0

baθxuxxdx −

ˆ 1

0

α1

aqqxuxxdx.

Young’s inequality then gives

ˆ 1

0u2

xxdx ≤ −ddt

ˆ 1

0

1a

utxuxdx +

ˆ 1

0

1a

u2txdx +

ˆ 1

0

3b2

4a2 θ2xdx

+13

ˆ 1

0u2

xxdx +

ˆ 1

0

(1a

)tutxuxdx −

ˆ 1

0

(1a

)x

uttuxdx

−

ˆ 1

0

α1

aqqxuxxdx,

which implies

23

ˆ 1

0u2

xxdx +ddt

ˆ 1

0

1a

utxuxxdx

≤

ˆ 1

0

1a

u2txdx +

ˆ 1

0

3b2

4a2 θ2xdx +

ˆ 1

0

(1a

)tutxuxdx

−

ˆ 1

0

(1a

)x

uttuxdx −ˆ 1

0

α1

aqqxuxxdx. (2.44)

2.2 Asymptotic stability 29

Multiplying (2.15) by 3ad utx, integrating over I and using integration by parts, we get

ˆ 1

0

3a

u2txdx = −

ˆ 1

0

3adθtutxdx −

ˆ 1

0

3gad

qxutxdx +

ˆ 1

0

3α2

adqqtutxdx

=

ˆ 1

0

(3

adθt

)x

utdx −[3gad

qutx

]x=1

x=0+

ˆ 1

0

(3gad

utx

)x

qdx

+

ˆ 1

0

3α2

adqqtutxdx

=

ˆ 1

0

(3

ad

)xθtutdx +

ˆ 1

0

3adθtxutdx −

[3gad

qutx

]x=1

x=0

+

ˆ 1

0

(3gad

)x

utxqdx +ddt

ˆ 1

0

3gad

quxxdx −ˆ 1

0

(3gad

)tuxxqdx

−

ˆ 1

0

3gad

uxxqtdx +

ˆ 1

0

3α2

adqqtutxdx

then we have

ˆ 1

0

3a

u2txdx =

ˆ 1

0

(3

ad

)xθtutdx +

ddt

ˆ 1

0

3adθxutdx −

ˆ 1

0

(3

ad

)tθxutdx

−

ˆ 1

0

3adθxuttdx −

[3gad

qutx

]x=1

x=0+

ˆ 1

0

(3gad

)x

utxqdx +ddt

ˆ 1

0

3gad

quxxdx

−

ˆ 1

0

(3gad

)tuxxqdx −

ˆ 1

0

3gad

uxxqtdx +

ˆ 1

0

3α2

adqqtutxdx

=

ˆ 1

0

(3

ad

)xθtutdx +

ddt

ˆ 1

0

3adθxutdx −

ˆ 1

0

(3

ad

)tθxutdx

−

ˆ 1

0

3dθxuxxdx +

ˆ 1

0

3badθ2

xdx −ˆ 1

0

3α1

adθxqqx −

[3gad

qutx

]x=1

x=0

+

ˆ 1

0

(3gad

)x

utxqdx −ˆ 1

0

(3gad

)tuxxqdx −

ˆ 1

0

3gad

uxxqtdx

+

ˆ 1

0

3α2

adqqtutxdx +

ddt

ˆ 1

0

3ga2d

qutt +3gba2d

qθx −3gα1

a2dq2qx

dx

30 Thermoelasticity with second sound

Then we obtain, with the help of (2.16) and Young’s inequality,

ˆ 1

0

3a

u2txdx ≤

ddt

ˆ 1

0

3

adθxut +

3ga2d

quttdx −3gbτa2dk

qqtdx −3gα1

a2dq2qx −

3gba2dk

q2

dx

+

ˆ 1

0

(3

ad

)xθtutdx −

ˆ 1

0

(3

ad

)tθxutdx +

ˆ 1

0

3badθ2

xdx −ˆ 1

0

3α1

adθxqqx

−

[3gad

qutx

]x=1

x=0+

ˆ 1

0

(3gad

)x

utxqdx −ˆ 1

0

(3gad

)tuxxqdx +

112

ˆ 1

0u2

xxdx

+

ˆ 1

0

27g2

a2d2 q2t dx +

112

ˆ 1

0u2

xxdx +

ˆ 1

0

27d2 θ

2xdx +

ˆ 1

0

3α2

adqqtutxdx. (2.45)

A combination of (2.44) and (2.45) then yields

ˆ 1

0

2a

u2txdx +

ˆ 1

0

12

u2xxdx +

ddt

ˆ 1

0(1a

utxux −3ga2d

quttdx

+3gbτa2dk

qqt +3gba2dk

q2 +3

akdqut +

3τakd

qtut)dx (2.46)

≤

ˆ 1

0

(3b2

4a2 +3bad

+27d2

)θ2

xdx +

ˆ 1

0

27g2

a2d2 q2t dx −

[3gad

qutx

]x=1

x=0+ R4,

where

R4 =

ˆ 1

0

(1a

)tutxuxdx −

ˆ 1

0

(1a

)x

uttuxdx −ˆ 1

0

α1

aqqxuxxdx

+

ˆ 1

0

(3

ad

)xθtutdx −

ˆ 1

0

(3

ad

)tθxutdx −

ˆ 1

0

3α1

adθxqqx

+

ˆ 1

0

(3gad

)x

utxqdx −ˆ 1

0

(3gad

)tuxxqdx +

ddt

ˆ 1

0

3gα1

a2dq2qxdx

+

ˆ 1

0

3α2

adqqtutxdx. (2.47)

Similarly, multiplying equation (2.29) by 1a uxxt we get, after integration over I, and use of Young’s

2.2 Asymptotic stability 31

inequality,

ˆ 1

0u2

xxtdx =

ˆ 1

0

1a

utttuxxtdx −ˆ 1

0

at

auxxuxxtdx +

ˆ 1

0

baθxtuxxtdx

+

ˆ 1

0

bt

aθxuxxt −

ˆ 1

0

α1

aqtqxuxxtdx −

ˆ 1

0

α1

aqqxtuxxtdx

−

ˆ 1

0

α1t

aqqxuxxtdx

= −

ˆ 1

0

(1a

)x

utttuxtdx −ddt

ˆ 1

0

1a

uttxuxtdx +

ˆ 1

0

(1a

)tuttxuxtdx

+

ˆ 1

0

1a

u2ttxdx −

ˆ 1

0

at

auxxuxxtdx +

ˆ 1

0

baθxtuxxtdx +

ˆ 1

0

bt

aθxuxxtdx

−

ˆ 1

0

α1

aqtqxuxxtdx −

ˆ 1

0

α1

aqqxtuxxtdx −

ˆ 1

0

α1t

aqqxuxxtdx.

Then we obtain

ˆ 1

0u2

xxtdx ≤ −ddt

ˆ 1

0

1a

uttxuxtdx +

ˆ 1

0

1a

u2ttxdx +

ˆ 1

0

3b2

4a2 θ2xtdx +

13

ˆ 1

0u2

xxtdx

+

ˆ 1

0

(1a

)tuttxuxtdx −

ˆ 1

0

(1a

)x

utttuxtdx −ˆ 1

0

at

auxxuxxtdx

+

ˆ 1

0

bt

aθxuxxtdx −

ˆ 1

0

α1

aqtqxuxxtdx −

ˆ 1

0

α1

aqqxtuxxtdx

−

ˆ 1

0

α1t

aqqxuxxtdx (2.48)

Also multiply (2.30) by 3ad uttx to get

ˆ 1

0

3adθttuttxdx +

ˆ 1

0

3gad

uttxqxtdx +

ˆ 1

0

3gt

adqxuttxdx +

ˆ 1

0

3a

u2ttxdx

ˆ 1

0

3dt

adutxuttxdx

=

ˆ 1

0

3α2

adq2

t uttxdx +

ˆ 1

0

3α2t

adqtquttxdx +

ˆ 1

0

3α2

adqttquttxdx,

32 Thermoelasticity with second sound

which impliesˆ 1

0

3a

u2ttxdx = −

ˆ 1

0

3adθttuttxdx −

ˆ 1

0

3gad

uttxqxtdx −ˆ 1

0

3gt

adqxuttxdx

−

ˆ 1

0

3dt

adutxuttxdx +

ˆ 1

0

3α2

adq2

t uttxdx

+

ˆ 1

0

3α2t

adqtquttxdx +

ˆ 1

0

3α2

adqttquttxdx. (2.49)

By multiplying (2.29) by 3adθtx and integrating over I,

−

ˆ 1

0

3adθttuttxdx =

ˆ 1

0

(3

ad

)xθttuttdx +

ddt

ˆ 1

0

3adθtxuttdx

−

ˆ 1

0

(3

ad

)tθtxuttdx −

ˆ 1

0

3dθtxuxxtdx

−

ˆ 1

0

3at

adθtxuxxdx +

ˆ 1

0

3badθ2

txdx

+

ˆ 1

0

3bt

adθtxθxdx −

ˆ 1

0

3α1

adθtxqtqxdx

−

ˆ 1

0

3α1

adθtxqqxtdx −

ˆ 1

0

3α1t

adθtxqqxdx. (2.50)

Using integration by parts, we easily see

−

ˆ 1

0

3gad

uttxqxtdx = −

[3gad

qtuttx

]x=1

x=0+

ˆ 1

0

(3gad

)x

uttxqtdx

+ddt

ˆ 1

0

3gad

utxxqtdx −ˆ 1

0

(3gad

)tutxxqtdx

−

ˆ 1

0

3gad

utxxqttdx.

Again multiply (2.29) by 3ga2d qt and integrating over I to obtain

−

ˆ 1

0

3gad

uttxqxtdx = −

[3gad

qtuttx

]x=1

x=0+

ˆ 1

0

(3gad

)x

uttxqtdx

−

ˆ 1

0

(3gad

)tutxxqtdx −

ˆ 1

0

3gad

utxxqttdx

+ddt

ˆ 1

0(

3ga2d

qtuttt −3gba2d

qtθxt −3α1ga2d

qtqqxt

−3gat

a2dqtuxx +

3gbt

a2dqtθx −

3α1ga2d

q2t qx −

3α1tga2d

qtqqx)dx. (2.51)

2.2 Asymptotic stability 33

Combining (2.49)-(2.51) we arrive at

ˆ 1

0

3a

u2ttxdx =

ˆ 1

0

(3

ad

)xθttuttdx +

ddt

ˆ 1

0

3adθtxuttdx −

ˆ 1

0

(3

ad

)tθtxuttdx

−

ˆ 1

0

3dθtxuxxtdx −

ˆ 1

0

3at

adθtxuxxdx +

ˆ 1

0

3badθ2

txdx

+

ˆ 1

0

3bt

adθtxθxdx −

ˆ 1

0

3α1

adθtxqtqxdx −

ˆ 1

0

3α1

adθtxqqxtdx

−

ˆ 1

0

3α1t

adθtxqqxd −

[3gad

qtuttx

]x=1

x=0+

ˆ 1

0

(3gad

)x

uttxqtdx

−

ˆ 1

0

(3gad

)tutxxqtdx −

ˆ 1

0

3gad

utxxqttdx −ˆ 1

0

3gt

adqxuttxdx

−

ˆ 1

0

3dt

adutxuttxdx +

ˆ 1

0

3α2

adq2

t uttxdx +

ˆ 1

0

3α2t

adqtquttxdx

+

ˆ 1

0

3α2

adqttquttxdx +

ddt

ˆ 1

0(

3ga2d

qtuttt −3gba2d

qtθxt −3α1ga2d

qtqqxt

−3gat

a2dqtuxx +

3gbt

a2dqtθx −

3α1ga2d

q2t qx −

3α1tga2d

qtqqx)dx. (2.52)

Using equation (2.31) we can write

ˆ 1

0

3gba2d

qtθxtdx = −

ˆ 1

0

3gbτa2dk

qtqttdx −ˆ 1

0

3gbτt

a2dkq2

t dx

−

ˆ 1

0

3gba2dk

q2t dx −

ˆ 1

0

3gbkt

a2dkqtθxdx. (2.53)

Young’s inequality gives

−

ˆ 1

0

3dθtxuxxtdx ≤

112

ˆ 1

0u2

txxdx +

ˆ 1

0

27d2 θ

2txdx (2.54)

and

−

ˆ 1

0

3gad

uxxtqttdx ≤112

ˆ 1

0u2

txxdx +

ˆ 1

0

27g2

a2d2 q2ttdx. (2.55)

34 Thermoelasticity with second sound

Taking into account (253)- (2.55), estimate (2.52) becomes

ˆ 1

0

3a

u2ttxdx =

ˆ 1

0(16

u2txx +

27g2

a2d2 q2ttdx + (

27d2 +

3bad

)θ2tx −

(3

ad

)tθtxutt

−3at

adθtxuxx +

3bt

adθtxθx −

3α1

adθtxqtqx −

3α1

adθtxqqxt

−3α1t

adθtxqqx +

(3gad

)x

uttxqt −

(3gad

)tutxxqt

−3gt

adqxuttx −

3dt

adutxuttx +

3α2

adq2

t uttx +3α2t

adqtquttx

+3α2

adqttquttx)dx −

[3gad

qtuttx

]x=1

x=0+

ddt

ˆ 1

0(−

3τadk

uttqtt

−3τt

adkuttqt −

3adk

uttqt −3kt

adkuttθx +

3ga2d

qtuttt

−3gbτa2dk

qtqtt −3gbτt

a2dkq2

t −3gba2dk

q2t −

3gbkt

a2dkqtθx

−3α1ga2d

qtqqxt −3gat

a2dqtuxx +

3gbt

a2dqtθx −

3α1ga2d

q2t qx

−3α2tga2d

qtqqx)dx (2.56)

Combining (2.48) and (2.56) we obtain

ˆ 1

0

2a

u2ttxdx +

12

ˆ 1

0u2

xxtdx +ddt

ˆ 1

0(1a

uttxuxtdx −3ga2d

qtuttt

+3gbτa2dk

qtqtt +3bga2dk

q2t +

3τadk

uttqtt +3

adkuttqt

−3α1

a2dqtqθtt −

3α1

a2 qtquttx −3α1α2

a2dqtq2qtt)dx

≤ −

[3gad

qtuttx

]x=1

x=0+

ˆ 1

0

(3b2

4a2 +3bad

+27d2

)θ2

xtdx

+

ˆ 1

0

27g2

a2d2 q2ttdx + R5, (2.57)

2.2 Asymptotic stability 35

where

R5 =

ˆ 1

0

(1a

)tuttxuxtdx −

ˆ 1

0

(1a

)x

utttuxtdx −ˆ 1

0

at

auxxuxxtdx

+

ˆ 1

0

bt

aθxuxxtdx −

ˆ 1

0

α1

aqtqxuxxtdx −

ˆ 1

0

α1

aqqxtuxxtdx

−

ˆ 1

0

α1t

aqqxuxxtdx −

ˆ 1

0

(3

ad

)tθtxuttdx −

ˆ 1

0

3at

adθtxuxxdx

+

ˆ 1

0

3bt

adθtxθxdx −

ˆ 1

0

3α1

adθtxqtqxdx

−

ˆ 1

0

3α1

adθtxqqxtdx −

ˆ 1

0

3α1t

adθtxqqxdx +

ˆ 1

0

(3gad

)x

uttxqtdx

−

ˆ 1

0

(3gad

)tutxxqtdx −

ˆ 1

0

3gt

adqxuttxdx −

ˆ 1

0

3dt

adutxuttxdx

+

ˆ 1

0

3α2

adq2

t uttxdx +

ˆ 1

0

3α2t

adqtquttxdx +

ˆ 1

0

3α2

adqttquttxdx

+ddt

ˆ 1

0(−

3τt

adkuttqt −

3kt

adkuttθx −

3bgτt

a2dkq2

t −3bgkt

a2dkqtθx

−3gat

a2dqtuxx +

3gbt

a2dqtθx −

3α1ga2d

q2t qx −

3α2tga2d

qtqqx

+3α1

a2dqtq(−gtqx − dtutx + α2q2

t + α2tqtq −3α2tga2d

qtqqx)dx.

Now we multiply (2.14) by utt, integrating over I and use Young’s inequality to arrive atˆ 1

0u2

ttdx =

ˆ 1

0(auxxutt − buttθx + α1qqxutt)dx

≤ 2ˆ 1

0a2u2

xxdx + 2ˆ 1

0b2θ2

xdx + 2ˆ 1

0α1qqxuttdx

and from (2.16), ˆ 1

0θ2

xdx ≤ˆ 1

0

(2τ2

k2 q2t +

2k2 q2

)dx.

At this point we exploit Poincare’s inequality :ˆ 1

0θ2dx ≤

ˆ 1

0θ2

xdx

ˆ 1

0u2

t dx ≤ˆ 1

0u2

txdx

36 Thermoelasticity with second sound

to obtainˆ 1

0

(u2

tt + u2t + θ2

)dx ≤ 2

ˆ 1

0a2u2

xxdx +

ˆ 1

0

(2b2 + 1

) (2τ2

k2 q2t +

2k2 q2

)dx

+2ˆ 1

0α1qqxuttdx +

ˆ 1

0u2

txdx. (2.58)

From (2.29) we see thatˆ 1

0u2

tttdx =

ˆ 1

0(auxxtuttt − bθxtuttt + atuxxuttt − btθxuttt + α1qtqxuttt

+α1qqxtuttt + α1tqqxuttt)dx

≤ 2ˆ 1

0(a2u2

xxt + b2θ2xt + atuxxuttt − btθxuttt + α1qtqxuttt

+α1qqxtuttt + α1tqqxuttt)dx.

Using (2.31) we get

ˆ 1

0θ2

txdx ≤ 2ˆ 1

0(τ2

k2 q2tt +

1k2 q2

t −τt

kqtθtx −

kt

kθxθtx)dx.

Again use of Poincare’s inequality yields

ˆ 1

0

(u2

ttt + u2tt + θ2

t

)dx ≤ 2

ˆ 1

0a2u2

xxtdx +

ˆ 1

0

(2b2 + 1

) (2τ2

k2 q2tt +

2k2 q2

t

)dx

+

ˆ 1

0u2

ttxdx + 2ˆ 1

0(atuxxuttt − btθxuttt + α1qtqxuttt

+α1qqxtuttt + α1tqqxuttt −τt

kqtθtx −

kt

kθxθtx)dx. (2.59)

Therefore (2.58) together with (2.59) give

ˆ 1

0

(u2

ttt + u2tt + u2

t + +θ2 + θ2t

)dx

≤ Cˆ 1

0

(u2

xx + u2xxt + u2

tx + u2ttx + q2

tt + q2t + q2

)dx + R6, (2.60)

where C is a constant and

R6 = 2ˆ 1

0(α1qqxutt + atuxxuttt − btθxuttt + α1qtqxuttt + α1qqxtuttt

+α1tqqxuttt −τt

kqtθtx −

kt

kθxθtx)dx.

2.2 Asymptotic stability 37

Multiplying (2.14) by u we obtainˆ 1

0au2

xdx =

ˆ 1

0(−uttu − bθxu − axuux + α1qqxu)dx.

Using Poincare’s inequality and Young’s inequality we easily arrive atˆ 1

0u2

xdx ≤ Cˆ 1

0(u2

tt + θ2x)dx + R7, (2.61)

where

R7 =

ˆ 1

0(−axuuxx + α1qqxu)dx.

Also, multiplication of (2.14) by θt integrating over I, and use of Young’s inequality leads toˆ 1

0θ2

t dx =

ˆ 1

0(−gqxθt − dutxθt + α2qqtθt)dx

=

ˆ 1

0((gθt)xq − dutxθt + α2qqtθt)dx

=

ˆ 1

0(gqθtx + gxθtq − dutxθt + α2qqtθt)dx

=ddt

ˆ 1

0gqθxdx +

ˆ 1

0(−gtqθx − gqtθx − dutxθt

+gxθtq + α2qqtθt)dx

≤ddt

ˆ 1

0gqθxdx +

ˆ 1

0(g2

2q2

t +12θ2

x +d2

2u2

tx

+12θ2

t − gtqθx + gxθtq + α2qqtθt)dx (2.62)

A similar treatement to (2.30) givesˆ 1

0θ2

ttdx ≤ddt

ˆ 1

0gθtxqtdx +

ˆ 1

0(−gtθtxqt + gxθttqt +

d2

2u2

ttx

+12θ2

tt +g2

2q2

tt +12θ2

tx − gtθttqx − dtθttutx

+α2q2t θtt + α2tqqtθtt)dx. (2.63)

From (2.62), (3.38), (2.19) and (2.20)we obtainˆ 1

0(θ2

tt + θ2t )dx −

ddt

ˆ 1

0(2gqθx + 2gqtθxt)dx

≤ Cˆ 1

0(q2

t + θ2x + q2

tt + θ2tx)dx +

ˆ 1

0d2(u2

ttx + u2tx)dx + R8, (2.64)

38 Thermoelasticity with second sound

where

R8 = 2ˆ 1

0(−gtqθx + gxθtq + α2qqtθt − dtθttutx + α2q2

t θtt

−gtθtxqt + gxθttqt − gtθttqx + α2tqqtθtt)dx. (2.65)

The boundary terms in (2.46) and (2.56) are treated as in Reference [71]. In fact, by Young’sinequality we have

[3gad

qutx

]x=1

x=0≤

[9g2

4a3d2εq2 + εau2

tx

]x=1

x=0

≤

∣∣∣∣∣∣∣[

9g2

4a3d2εq2

]x=1

x=0

∣∣∣∣∣∣∣ +

∣∣∣∣∣[εau2tx

]x=1

x=0

∣∣∣∣∣≤

Ct

ε

(q2(1) + q2(0)

)+

∣∣∣∣∣[εau2tx

]x=1

x=0

∣∣∣∣∣where

Ct = Max[

9g2

4a3d2

]x=0

,

[9g2

4a3d2

]x=1

.

By the imbedding of W1,1 in L∞ we have

|q (x)|2 ≤ˆ 1

0

(q2 + (q2)x

)dx, 0 ≤ x ≤ 1.

We exploit Young’s inequality to get

|q (x)|2 ≤ 2(1 +

1ε2

)ˆ 1

0q2dx + 2ε2

ˆ 1

0q2

xdx, 0 ≤ x ≤ 1.

Therefore,Ct

ε

(q2(1) + q2(0)

)≤

4Ct

ε

(1 +

1ε2

)ˆ 1

0q2dx + 4Ctε

ˆ 1

0q2

xdx.

Using (2.15) to obtain

Ct

ε

(q2(1) + q2(0)

)≤

4Ct

ε

(1 +

1ε2

) ˆ 1

0q2dx + 4Ctε

ˆ 1

0(−

1gθtqx

−dg

utxqx +α2

gqqtqx)dx.

2.2 Asymptotic stability 39

Again Young’s inequality allows us to write

[3gad

qutx

]x=1

x=0≤

4Ct

ε3 (1 + ε2)ˆ 1

0q2dx +

ˆ 1

0

36Ctε

a2d2 θ2t dx

+

ˆ 1

0

36Ctε

a2 u2txdx +

ˆ 1

0

2Ctεa2d2

9q2

xdx

+

ˆ 1

0

4Ctεα2

gqqtqx +

∣∣∣∣[εau2tx

]x=0

∣∣∣∣ +∣∣∣∣[εau2

tx

]x=1

∣∣∣∣ . (2.66)

By the same manner we can estimate the boundary term in (2.56) as follows

[3gad

qtuttx

]x=1

x=0≤

[9g2

4a3d2εq2

t + εau2ttx

]x=1

x=0

≤

∣∣∣∣∣∣∣[

9g2

4a3d2εq2

t

]x=1

x=0

∣∣∣∣∣∣∣ +

∣∣∣∣∣[εau2tx

]x=1

x=0

∣∣∣∣∣≤

Ct

ε

(q2

t (1) + q2t (0)

)+

∣∣∣∣∣[εau2ttx

]x=1

x=0

∣∣∣∣∣then we obtain

Ct

ε

(q2

t (1) + q2t (0)

)≤

4Ct

ε

(1 +

1ε2

)ˆ 1

0q2

t dx + 4Ctε

ˆ 1

0q2

txdx.

Using (2.30) and Young’s inequality, we get

[3gad

qtuttx

]x=1

x=0≤

4Ct

ε3 (1 + ε2)ˆ 1

0q2

t dx +

ˆ 1

0

36Ctε

a2d2 θ2ttdx

+

ˆ 1

0

36Ctε

a2 u2ttxdx +

ˆ 1

0

2Ctεa2d2

9q2

txdx

+∣∣∣∣[εau2

ttx

]x=0

∣∣∣∣ +∣∣∣∣[εau2

ttx

]x=1

∣∣∣∣+4Ctε

ˆ 1

0(−

gt

gqxqxt −

dt

gutxqtx +

α2

gq2

t qtx

+α2t

gqqtqtx +

α2

gqqttqtx)dx. (2.67)

40 Thermoelasticity with second sound

Addition of (2.66) and (2.67) leads to∣∣∣∣∣∣∣[3gad

(qutx + qtuttx)]x=1

x=0

∣∣∣∣∣∣∣≤

2Ct

ε3 (1 + ε2)ˆ 1

0

(q2 + q2

t

)dx +

ˆ 1

0

36Ctε

a2d2

(θ2

tt + θ2t

)dx

+

ˆ 1

0

36Ctε

a2

(u2

ttx + u2tx

)dx +

∣∣∣∣[εau2ttx + εau2

tx

]x=0

∣∣∣∣+

∣∣∣∣[εau2ttx + εau2

tx

]x=1

∣∣∣∣ +

ˆ 1

0

2Ctεa2d2

9

(q2

tx + q2x

)dx

+

ˆ 1

0

4Ctεα2

gqqtqxdx + 4Ctε

2ˆ 1

0(−

gt

gqxqxt

−dt

gutxqtx +

α2

gq2

t qtx +α2t

gqqtqtx +

α2

gqqttqtx)dx.

Using (2.15) and Young’s inequality we findˆ 1

0

2Ctεa2d2

9q2

xdx ≤ˆ 1

0

4Ctεa2d2

9(

1g2 θ

2t +

d2

g2 u2tx +

α2

gqqtqx)dx

By the same manner using equation (2.30) we obtainˆ 1

0

2Ctεa2d2

9q2

txdx ≤

ˆ 1

0

4Ctεa2d2

9(

1g2 θ

2tt +

d2

g2 u2ttx −

gt

gqxqtx

−dt

gutxqtx +

α2

gq2

t qtx +α2t

gqtqqtx

+α2

gqttqqtx)dx

Then, we conclude ∣∣∣∣∣∣∣[3gad

(qutx + qtuttx)]x=1

x=0

∣∣∣∣∣∣∣≤

2Ct

ε3 (1 + ε2)ˆ 1

0

(q2 + q2

t

)dx +

ˆ 1

0

36Ctε

a2d2

(θ2

tt + θ2t

)dx

+

ˆ 1

0

36Ctε

a2

(u2

ttx + u2tx

)dx +

∣∣∣∣[εau2ttx + εau2

tx

]x=0

∣∣∣∣+

∣∣∣∣[εau2ttx + εau2

tx

]x=1

∣∣∣∣ +

ˆ 1

0

4Ctεa2d2

9(

1g2 θ

2t +

d2

g2 u2tx)dx

+

ˆ 1

0

4Ctεa2d2

9(

1g2 θ

2tt +

d2

g2 u2ttx)dx + R9, (2.68)

2.2 Asymptotic stability 41

where

R9 =

ˆ 1

0

4Ctεα2

gqqtqxdx + 4Ctε

2ˆ 1

0(−

gt

gqxqxt

−dt

gutxqtx +

α2

gq2

t qtx +α2t

gqqtqtx +

α2

gqqttqtx)dx

+

ˆ 1

0

4Ctεa2d2

9(α2

gqqtqx −

gt

gqxqtx

−dt

gutxqtx +

α2

gq2

t qtx +α2t

gqtqqtx +

α2

gqttqqtx)dx. (2.69)

As in Reference [71], we multiply (2.29) by φutx, for φ (x) = 1 − 2x, to obtain

ˆ 1

0(utttφutx − auxxtφutx − atuxxφutx + bθxtφutx + btθxφutx)dx

=

ˆ 1

0(α1qtqxφutx + α1qqxtφutx + α1tqqxφutx)dx,

which implies

ddt

ˆ 1

0uttφutx +

ˆ 1

0(−uttφuttx − autxxφutx + bθxtφutx)dx

=

ˆ 1

0(atuxxφutx − btθxφutx + α1qtqxφutx + α1qqtxφutx + α1tqqxφutx)dx.

An integration by parts yields

ddt

ˆ 1

0uttφutxdx +

12

(u2

tt (1) + u2tt (0)

)+

[a2

u2tx

]x=0

+

[a2

u2tx

]x=1

−

ˆ 1

0u2

ttdx +

ˆ 1

0(−au2

tx + bθtxφutx)dx

=

ˆ 1

0(−

12

axφu2tx + atuxxφutx − btθxφutx

+α1qtqxφutx + α1qqxtφutx + α1tqqxφutx). (2.70)

If we multiply (2.15) by −bd φθxt we get

ˆ 1

0(bdφθtθxt +

gbdφqxθxt + bφθxtutx)dx =

ˆ 1

0

α2bdφθxtqqtdx.

42 Thermoelasticity with second sound

Again an integration by parts yields[b

2dθ2

t

]x=1

+

[b

2dθ2

t

]x=0−

ˆ 1

0

bdθ2

t dx −ddt

ˆ 1

0(bgdφqxθx)dx

+

ˆ 1

0(bgdφqxtθx − bφθxtutx)dx

= −

ˆ 1

0(α2bdφqqtθtx −

12

(bd

)xφθ2t − (

bgd

)tφqxθxdx

which implies, using equation resulting from differentiation of (2.16) with respect to x,[b

2dθ2

t

]x=1

+

[b

2dθ2

t

]x=0−

ˆ 1

0

bdθ2

t dx −ddt

ˆ 1

0

bgdφqxθxdx

−

ˆ 1

0

bgdτφθx(qx + kθxx + τxqt + kxθx)dx −

ˆ 1

0bφθtxutxdx

= −

ˆ 1

0(α2bdφqqtθtx +

12

(bd

)xφθ2

t +

(bgd

)tφqxθxdx.

Then we have [b

2dθ2

t

]x=1

+

[b

2dθ2

t

]x=0

+

ˆ 1

0(−

bdθ2

t dx −bgdτφθxqx −

bgkdτ

θ2x − butxφθtx)dx

−ddt

ˆ 1

0(bgdφqxθx)dx +

[bgk2dτ

θ2x

]x=1

+

[bgk2dτ

θ2x

]x=0

=

ˆ 1

0(−

12

(bgdτ

)xφθ2x +

bgτx

dτφθxqt +

bgkx

dτφθ2

x)dx

−

ˆ 1

0(α2bdφqqtθtx −

12

(bd

)xφθ2t + (

bgd

)tφqxθxdx. (2.71)

Combining (2.70), (2.71) and usingˆ 1

0

bgdτφθxqxdx ≤ sup

0≤x≤1|φ|

ˆ 1

0

∣∣∣∣∣bgdτθxqx

∣∣∣∣∣ dx

≤

ˆ 1

0

∣∣∣∣∣bgdτθxqx

∣∣∣∣∣ dx

we arrive at

ddt

ˆ 1

0(uttφutx −

bgdφqxθx)dx +

[a2

u2tx

]x=0

+

[a2

u2tx

]x=1

≤

ˆ 1

0(u2

tt + au2tx +

bdθ2

t + (bgkdτ

+b2g2

4d2τ2 )θ2x + q2

x)dx + ∆1, (2.72)

2.2 Asymptotic stability 43

where

∆1 =

ˆ 1

0(−

12

axφu2tx + atuxxφutx − btθxφutx + α1qtqxφutx + α1qqxtφutx

+α1qqxφutx −12

(bgdτ

)xφθ2x +

bgτx

dτφθxqt +

bgkx

dτφθ2

x

−α2bdφqqtθtx −

12

(bd

)xφθ2t − (

bgd

)tφθxqx)dx. (2.73)

Next, multiply (2.15) by qxˆ 1

0q2

xdx =

ˆ 1

0(−

1gθtqx −

dg

utxqx +α2

gqqtqx)dx

and Young’s inequality to getˆ 1

0q2

xdx ≤ˆ 1

0(

2g2 θ

2t +

2d2

g2 u2tx +

2α2

gqqtqx)dx.

Thus (2.72) takes the form

ddt

ˆ 1

0(uttφutx −

bgdφqxθx)dx +

[a2

u2tx

]x=0

+

[a2

u2tx

]x=1

≤

ˆ 1

0(u2

tt + (a +2d2

g2 )u2tx + (

bd

+2g2 )θ2

t + (bgkdτ

+b2g2

4d2τ2 )θ2x

+2α2

gqqtqx)dx + ∆1. (2.74)

Similarly, multiplying (2.33) by φuttx, we obtainˆ 1

0(uttttφuttx − auxxttφuttx + bθxttφuttx)dx = ∆2,

where

∆2 =

ˆ 1

0(2atuxxtφuttx − 2btθxtφuttx + attuxxφuttx − bttθxφuttx

+φuttx(α1qttqx + 2α1qtqtx + 2α1tqtqx + α1qqttx

+2α1tqqtx + α1ttqqx)dx.

Hence

ddt

ˆ 1

0utttφuttxdx +

[a2

u2ttx

]x=0

+

[a2

u2ttx

]x=1

+12

(u2

ttt(1) + u2ttt(0)

)+

ˆ 1

0(−au2

ttx − u2ttt + bθxttφuttx)dx

= ∆2 +12

ˆ 1

0axφu2

ttxdx. (2.75)

44 Thermoelasticity with second sound

The multiplication of (2.30) by − bdφθttx yields,

ˆ 1

0(−

bdφθttxθtt −

bgdφθttxqxt − bφθttxuttx)dx

=

ˆ 1

0−

bdφuttx(−gtqx − dtutx + α2q2

t + α2tqtq

+α2qttq)dx;

hence

ˆ 1

0−

bdθ2

ttdx −ddt

ˆ 1

0

bgdφθtxqxtdx +

[b

2dθ2

tt

]x=1

+

[b

2dθ2

tt

]x=0

+

ˆ 1

0(bgd

qttxφθtx − bφuttxθttx)dx

= −12

ˆ 1

0

(bd

)xφθ2

ttdx −ˆ 1

0

(bgd

)tqtxφθtxdx

−

ˆ 1

0

bdφuttx(−gtqx − dtutx + α2q2

t + α2tqtq + α2qttq)dx,

which implies, using equation resulting from differentiation of (2.31) with respect to x,

ˆ 1

0−

bdθ2

ttdx −ddt

ˆ 1

0

bgdφθtxqxtdx −

ˆ 1

0bφuttxθttxdx +

[b

2dθ2

tt

]x=1

+

[b

2dθ2

tt

]x=0

+

ˆ 1

0

bgdτφθtx(−τtqtx − qtx − kθxxt − ktθxx − τxqtt − τxtqt − kxθxt − ktxθx)dx

= −12

ˆ 1

0

(bd

)xθ2

ttdx −ˆ 1

0

(bgd

)tqtxφθtxdx

+

ˆ 1

0−

bdφuttx(−gtqx − dtutx + α2q2

t + α2tqtq + α2qttq)dx.

We then integrate by parts to get

ˆ 1

0−

bdθ2

ttdx −ddt

ˆ 1

0

bgdφθtxqxtdx +

[b

2dθ2

tt

]x=1

+

[b

2dθ2

tt

]x=0ˆ 1

0(−bφθttxuttx −

bgdτφθtxqxt −

bgkdτ

θ2txdx +

[bgk2dτ

θ2tx

]x=1

+

[bgk2dτ

θ2tx

]x=0

= ∆3, (2.76)

2.2 Asymptotic stability 45

where

∆3 =

ˆ 1

012

(bgdτ

)xφθ2

tx +bgτt

dτφθtxqtx +

bgkt

dτφθtxθxx +

bgτx

dτφθtxqtt

+bgτxt

dτφθtxqt +

bgkx

dτφθ2

tx +bgktx

dτφθxθtx −

12

(bd

)xθ2

tt

−

(bgd

)tqtxφθtxdx +

ˆ 1

0−

bdφuttx(−gtqx − dtutx + α2q2

t + α2tqtq + α2qttq)dx.

Combining (2.75) and (2.76) we obtain

ddt

ˆ 1

0(utttφuttx −

bgdφθtxqxt)dx +

[a2

u2ttx

]x=0

+

[a2

u2ttx

]x=1

≤

ˆ 1

0(u2

ttt + au2ttx +

bdθ2

tt +bgkdτ

θ2tx +

b2g2

4d2τ2 θ2tx + q2

tx)dx + ∆2 + ∆3

+12

ˆ 1

0axφu2

ttxdx. (2.77)

Also multiplying (2.30) by −1g qtx we have

ˆ 1

0q2

txdx =

ˆ 1

0(−

1gθttqtx −

gt

gqxqtx −

dg

uttxqtx −dt

gutxqtx

+α2

gq2

t qtx +α2t

gqtqqtx +

α2

gqttqqtx)dx.

Thanks to Young’s inequality we find

ˆ 1

0q2

txdx ≤

ˆ 1

0(

2g2 θ

2tt +

2d2

g2 u2ttx)dx

+

ˆ 1

02(−

gt

gqxqtx −

dt

gutxqtx +

α2

gq2

t qtx

+α2t

gqtqqtx +

α2

gqttqqtx)dx.

Therefore (2.77) becomes

ddt

ˆ 1

0(utttφuttx −

bgdφθtxqxt)dx +

[a2

u2ttx

]x=0

+

[a2

u2ttx

]x=1

(2.78)

≤

ˆ 1

0(u2

ttt + (a +2d2

g2 )u2ttx + (

bd

+2g2 )θ2

tt + (bgkdτ

+b2g2

4d2τ2 )θ2tx)dx + ∆4,

46 Thermoelasticity with second sound

where

∆4 = ∆2 + ∆3 +

ˆ 1

0(12

axφu2ttx −

2gt

gqxqtx −

2dt

gutxqtx

+2α2

gq2

t qtx +2α2t

gqtqqtx +

2α2

gqttqqtx)dx.

Adding (2.74) and (2.78) we obtain, after multiplication by 2ε,

ddt

ˆ 1

02ε(uttφutx + utttφuttx −

bgd

(φθxqx + φθtxqxt))dx

+ε[a(u2

ttx + u2tx)

]x=0

+ ε[a(u2

ttx + u2tx)

]x=1

≤ 2εˆ 1

0(u2

tt + u2ttt)dx + 2ε

ˆ 1

0(a + 2

d2

g2 )(u2tx + u2

ttx)dx

+2εˆ 1

0(bd

+2g2 )(θ2

t + θ2tt)dx (2.79)

+2εˆ 1

0(bgkdτ

+b2g2

4d2τ2 )(θ2tx + θ2

x)dx + R10,

where

R10 =

ˆ 1

0

2εα2

gqqtqtx + 2ε (∆1 + ∆4) . (2.80)

A combination of (2.68) and (2.69) yields∣∣∣∣∣∣∣[3gad

(qutx + qtuttx)]x=1

x=0

∣∣∣∣∣∣∣≤

2Ct

ε3 (1 + ε2)ˆ 1

0

(q2 + q2

t

)dx +

ˆ 1

0

36Ctε

a2d2

(θ2

tt + θ2t

)dx

+

ˆ 1

0

36Ctε

a2

(u2

ttx + u2tx

)dx +

ˆ 1

0

4Ctεa2d2

9(

1g2 θ

2t +

d2

g2 u2tx)dx

+

ˆ 1

0

4Ctεa2d2

9(

1g2 θ

2tt +

d2

g2 u2ttx)dx −

ddt

ˆ 1

02ε(uttφutx + utttφuttx

−bgd

(φθxqx + φθtxqxt))dx + 2εˆ 1

0(u2

tt + u2ttt)dx

+2εˆ 1

0(a + 2

d2

g2 )(u2tx + u2

ttx)dx + 2εˆ 1

0(bd

+2g2 )(θ2

t + θ2tt)dx

+2εˆ 1

0(bgkdτ

+b2g2

4d2τ2 )(θ2x + θ2

tx)dx + R9 + R10 (2.81)

2.2 Asymptotic stability 47

Combining (2.46), (2.57), (2.81) and using (2.19) and (2.20) we obtain for sufficiently small ε

ˆ 1

0

2a

(u2tx + u2

ttx)dx +

ˆ 1

0

12

(u2xx + u2

xxt)dx +ddt

G1 (t)

≤

ˆ 1

0

(3b2

4a2 +3bad

+27d2

)(θ2

x + θ2tx)dx +

ˆ 1

0

27g2

a2d2 (q2t + q2

tt)dx

+2Ct

ε3 (1 + ε2)ˆ 1

0

(q2 + q2

t

)dx +

ˆ 1

0

36Ctε

a2d2

(θ2

tt + θ2t

)dx

+

ˆ 1

0

4Ctεa2d2

9(

1g2 θ

2t +

d2

g2 u2tx)dx +

ˆ 1

0

4Ctεa2d2

9(

1g2 θ

2tt +

d2

g2 u2ttx)dx

+

ˆ 1

0

36Ctε

a2

(u2

ttx + u2tx

)dx + 2ε

ˆ 1

0(u2

tt + u2ttt)dx

+2εˆ 1

0(a + 2

d2

g2 )(u2tx + u2

ttx)dx + 2εˆ 1

0(bd

+2g2 )(θ2

t + θ2tt)dx

+2εˆ 1

0(bgkdτ

+b2g2

4d2τ2 )(θ2x + θ2

tx)dx + R4 + R5 + R9 + R10

where

G1 (t) =

ˆ 1

0(1a

(utxux + uttxutx)dx −3ga2d

(qutt + qtuttt) +3gbτa2dk

(qqt + qtqtt)

+3bga2dk

(q2 + q2t ) +

3τadk

(utqt + uttqtt) +3

adk(utq + uttqt)

−3α1

a2dqtqθtt −

3α1

a2 qtquttx −3α1α2

a2dqtq2qtt

+2ε(uttφutx + utttφuttx −bgd

(φθxqx + φθtxqxt)))dx.

Then we have, by (2.19) and (2.20),

c1

ˆ 1

0(u2

tx + u2ttx)dx + c2

ˆ 1

0(u2

xx + u2xxt)dx +

ddt

G1 (t)

≤ c3

ˆ 1

0(θ2

x + θ2tx)dx + c4

ˆ 1

0(q2

t + q2tt)dx +

c5

ε3

ˆ 1

0(q2 + q2

t )dx

+c6ε

ˆ 1

0

(θ2

tt + θ2t

)dx + c7ε

ˆ 1

0

(u2

ttx + u2tx

)dx (2.82)

+c8ε

ˆ 1

0(u2

tt + u2ttt)dx + c9ε

ˆ 1

0(θ2

x + θ2tx)dx + R4 + R5 + R9 + R10.

48 Thermoelasticity with second sound

Adding σ (2.64) to (2.82) we get

c1

ˆ 1

0(u2

tx + u2ttx)dx + c2

ˆ 1

0(u2

xx + u2xxt)dx +

ddt

G1 (t) + σ

ˆ 1

0

(θ2

tt + θ2t

)dx

−σddt

ˆ 1

02g(qθx + qtθxt)dx

≤ c3

ˆ 1

0(θ2

x + θ2tx)dx + c4

ˆ 1

0(q2

t + q2tt)dx +

c5

ε3

ˆ 1

0(q2 + q2

t )dx

+c6ε

ˆ 1

0

(θ2

tt + θ2t

)dx + c7ε

ˆ 1

0

(u2

ttx + u2tx

)dx + c8ε

ˆ 1

0(u2

tt + u2ttt)dx

+c9ε

ˆ 1

0(θ2

x + θ2tx)dx + c10

ˆ 1

0(q2

t + q2tt + θ2

x + θ2tx)dx + σ

ˆ 1

0d2(u2

tx + u2ttx)dx

+R4 + R5 + c10R8 + R9 + R10,

where c1 - c10 are constants depending on β and the upper of the function a, b, g, d, τ, k, α1, α2 andtheir derivatives only. Using (2.43), (2.57), (2.63), (2.81) and choosing ε small and σ such thatc1 − σ||d||2∞ > 0, we have

c11

ˆ 1

0(u2

tx + u2ttx)dx + c12

ˆ 1

0(u2

xx + u2xxt)dx + c13

ˆ 1

0

(θ2

tt + θ2t

)dx +

ddt

G2 (t)

≤ c14

ˆ 1

0(q2 + q2

t + q2tt)dx + c15

(R4 + R5 + R8 + R9 + R10 + R1

), (2.83)

where

G2 (t) = G1 (t) − σˆ 1

02g(qθx + qtθxt)dx.

For γ > 0 we define the Lyapunov function

F (t) =1γ

E (t) + G2 (t) . (2.84)

Lemma 2.2.1. There exists constant c16 > 0 such that(1γ− c16

(1 + α (t) + α2 (t)

))E (t) ≤ F (t) ≤

(1γ

+ c16(1 + α (t) + α2 (t)

))E (t) . (2.85)

Proof. LetG2 (t) = F (t) −

1γ

E (t) .

It suffices to show for a constant c16 > 0 that

|G2 (t)| ≤ c16(1 + α (t) + α2 (t)

)E (t) . (2.86)

2.2 Asymptotic stability 49

To do this, we estimate all terms of G2 (t) , indeed by Young’s inequality we have

ˆ 1

0

1a

(utxux + uttxutx)dx ≤ˆ 1

0

12a

(u2tx + u2

x + u2tx + u2

ttx)dx.

Then using (2.19) and (2.20) we have

ˆ 1

0

1a

(utxux + uttxutx)dx ≤ Γ1

ˆ 1

0(u2

tx + u2x + u2

tx + u2ttx)dx

≤ Γ1E (t) .

By the same manner we obtain

ˆ 1

0−

3ga2d

(qutt + qtuttt) +3bgτa2dk

(qqt + qtqtt) +3bga2dk

(q2 + q2t )

+3τ

adk(qtut + qttutt) +

3adk

(qut + qtutt) − 2g(qθx + qtθtx)dx

≤ Γ (1 + α (t)) E(t),

and we can also write ∣∣∣∣∣∣ˆ 1

0(uttφutx + utttφuttx −

bgd

(φθxqx + φθtxqxt)

∣∣∣∣∣∣≤ Γ| |φ| |∞ (1 + α (t)) E (t)

≤ Γ (1 + α (t)) E (t) ,

ˆ 1

0

(3α1

a2dqtqθtt +

3α1

a2 qtquttx

)dx ≤ Cα (t) E (t) ,

and ˆ 1

0

3α1α2

a2dqtq2qttdx ≤ Cα2 (t) E (t) .

Where C depends only on a, b, g, d, τ, k, α1, α2.

Therefore

|G2 (t)| =

∣∣∣∣∣F (t) −1γ

(E (t))∣∣∣∣∣

≤ c16(1 + α (t) + α2 (t)

)E (t) .

Hence, (2.86) is established and the assertion of the lemma is proved.Lemma 2.2.2 There exist c17, c18 > 0, such that

c17E (t) ≤ Λ (t) ≤ c18(1 + α (t) + α2 (t) + α3 (t) + α4 (t)

)E (t) (2.87)

50 Thermoelasticity with second sound

Proof. The First inequality in (2.87) is obvious, the second is established as follows. By Poincare’sinequality we have ˆ 1

0u2dx ≤

ˆ 1

0u2

xdx.

Using (2.39) and (2.19) and (2.20) we obtainˆ 1

0θ2

xdx ≤ Cˆ 1

0

(q2

t + q2)

dx

≤ CE (t) .

Using equation (2.14) we getˆ 1

0u2

xxdx =

ˆ 1

0

(1a

utt +baθx −

α1

aqqx

)uxxdx

By Young’s inequality and (2.19) and (2.20) we haveˆ 1

0u2

xxdx ≤ C(ˆ 1

0u2

tt + θ2x + α (t) q2

)≤ C (1 + α (t)) E (t) .

We then multiply (2.15) by qx to obtain by similar calculationsˆ 1

0q2

xdx ≤ Cˆ 1

0

(θ2

t + u2tx + α (t) q2

)dx

≤ C (1 + α (t)) E (t) .

The multiplication of equation (3.30) by qxt yieldsˆ 1

0q2

txdx =

ˆ 1

0

qxt

g(−θtt − gtqx − duttx − dtutx + α2q2

t + α2tqtq + α2qttq)dx

and by the same manner, using (2.19), (2.20) , and Young’s inequality, we obtainˆ 1

0q2

txdx ≤ C(ˆ 1

0θ2

tt + α (t) q2x + u2

ttx + α (t) u2tx + α (t) q2

t + α2 (t) q2 + α (t) q2tt

)≤ C

(1 + α (t) + α2 (t)

)E (t) .

Differentiate (2.16) with respect to x and multiply the result by θxx to getˆ 1

0θ2

xxdx = −

ˆ 1

0

1k

(τxqt + τqtx + kxθx) θxxdx

≤ Cˆ 1

0

(α (t) q2

t + q2tx + α (t) θ2

x

)dx

≤ C(1 + α (t) + α2 (t)

)E (t)

2.2 Asymptotic stability 51

Using (2.42) we haveˆ 1

0θ2

xtdx ≤ Cˆ 1

0

(q2

tt + q2t + θ2

x

)dx

≤ C (1 + α (t)) E (t) .

Using equation (2.33) we obtainˆ 1

0u2

xxt ≤ Cˆ 1

0

(u2

ttt + α (t) u2xx + θ2

xt + α (t) θ2x + α (t) q2

t + α (t) q2tx + α2 (t) q2

)dx

≤ C(1 + α (t) + α2 (t) + α3 (t)

)E (t) .

Differentiate (2.15) with respect to x and multiply the result by qxx to get, after some calculations,ˆ 1

0q2

xxdx ≤ Cˆ 1

0(θ2

xt + α (t) q2x + α (t) u2

tx + u2xxt + α2 (t) q2

+α (t) q2t + α (t) q2

tx)dx

≤ C(1 + α (t) + α2 (t) + α3 (t)

)E (t) .

Differentiate (2.14) with respect to x and multiply the result by uxxx to getˆ 1

0u2

xxxdx =

ˆ 1

0

1a

(uttx + bθxx − axuxx + bxθx − α1xqxq

−α1q2x + α1qqxx)uxxxdx

≤ Cˆ 1

0(u2

ttx + θ2xx + α (t) u2

xx + α (t) θ2x + α2 (t) q2 + α (t) q2

x + α (t) q2xx)dx

≤ C(1 + α (t) + α2 (t) + α3 (t) + α4 (t)

)E (t) .

Then we have

Λ (t) ≤ c18(1 + α (t) + α2 (t) + α3 (t) + α4 (t)

)E (t) . (2.88)

This completes the proof of lemma 2.2.2.

Next, using (2.26), (2.32), (2.36), (2.43), (2.60), (2.61) and (2.83) we obtain

ddt

F (t) ≤ −c19E (t) +∣∣∣R1 + R2 + ... + R10 + R1

∣∣∣ . (2.89)

Lemma 2.2.3 ∣∣∣R1 + R2 + ... + R10 + R1∣∣∣ ≤ c20

(α (t) + α2 (t) + α3 (t)

)Λ (t) . (2.90)

Proof. The proof is also similar to the one of Lemma 2.2.1. We only consider those of highestorder; namely

´ 10 α1ttkdqqxutttdx,

´ 10 α2ttkdqqtθttdx. We first compute

α1tt = α1uxuxu2tx + α1uxuxtt + 2α1uxθθtutx + α1θθθ

2t + α1uxθtt.

52 Thermoelasticity with second sound

By using the fact that α1, k, d ∈ C3b, we obtain∣∣∣∣∣∣

ˆ 1

0α1ttkdqqxutttdx

∣∣∣∣∣∣ ≤ Cˆ 1

0

∣∣∣qqxutttu2xt

∣∣∣ dx +

ˆ 1

0|qqxutttuxt| dx

+

ˆ 1

0|qqxutttuxtθt| dx +

ˆ 1

0

∣∣∣qqxutttθ2t

∣∣∣ dx

+

ˆ 1

0|qqxutttθtt| dx

≤ Cα2 (t) + α3 (t)

E (t) .

Similarly the other integral is treated. By carrying all calculation, the proof of the lemma is com-pleted.

By combining (2.85), (2.87) and (2.89) we have

ddt

F (t) ≤ −γc19F (t)

+c18

(1 + α (t) + α2 (t) + α3 (t) + α4 (t)

) (α (t) + α2 (t) + α3 (t)

)(1/γ) − c16

(1 + α (t) + α2 (t)

) F (t) .

(2.91)

At this point, we chose γ so that 1/γ > 2c16. Once is fixed we pick δ, in (2.25) , small that enoughso that

c18(1 + α (0) + α2 (0) + α3 (0) + α4 (0)

) (α (0) + α2 (0) + α3 (0)

)(1/γ) − c16

(1 + α (0) + α2 (0)

) ≤14γc19,

hence (2.91) yields, for some t0 > 0,

ddt

F (t) ≤ −c5F (t) , ∀t ∈ [0, t0)

Direct integration then leads to

F (t) ≤ F (0) e−c17t, ∀t ∈ [0, t0) . (2.92)

Since F (t) ≤ F (0) we extend (2.92) beyond t0. By repeating the same procedure, taking δ evensmaller if necessary, and using the continuity of F, (2.92) is established for all t ≥ 0.This completesthe proof of the theorem.Remark 2.2.2 The proof show that the initial data can be taken in a neighbourhood of the equi-librium state (0, 0, 0), in which the solution remains for over. Therefore the result is also valid fora, b, g, d, τ, k, α1, α2 in C3 instead in C3

b as mentioned in Remark 2.2.1.

2.3 Blow up of solutions 53

2.3 Blow up of solutions

This section is concerned with a blow up result for certain solutions with positive initial energy.Our technique of proof is based on a method used by Vitillaro with the necessary modificationimposed by the nature of our problem.

2.3.1 Introduction

In [54] Messaoudi considered the multi-dimensional nonlinear thermoelasticity with second sound,and investigated the situation where a nonlinear source term is competing with the damping causedby the heat conduction and established a local existence result. He also showed that solutions withnegative energy blow up in finite time. His work extended an earlier one in [50, 52] to thermoelas-ticity with second sound.

In our studies we are concerned with the nonlinear problem

utt − µ∆u − (µ + λ)∇divu + β∇θ = |u|p−2u, x ∈ Ω, t > 0,

θt + γdivq + δdivut = 0, x ∈ Ω, t > 0,

τqt + q + κ∇θ = 0, x ∈ Ω, t > 0,

u(., 0) = u0, ut(., 0) = u1, θ(., 0) = θ0, q(., 0) = q0, x ∈ Ω,

u = θ = 0, x ∈ ∂Ω , t ≥ 0.

(2.93)

for p > 2. This is a similar problem to (2.13) with a nonlinear source term competing with thedamping factor. We will extend the blow up result of [54] to situations, where the energy can bepositive. Our technique of proof follows carefully the techniques of Vitillaro [84] with the neces-sary modifications imposed by the nature of our problem.

In the sake of completeness, we state here the local existence of [54]. For this purpose, weintroduce the following functional spaces

Π : =[H1

0(Ω) ∩ H2(Ω)]n×

[H1

0(Ω)]n× H1

0(Ω) × D

D : = q ∈[L2(Ω)

]n/divq ∈ L2(Ω)

H : =[H1

0(Ω)]n×

[L2(Ω)

]n× L2(Ω) ×

[L2(Ω)

]n

Theorem 2.3.1 Assume that

2 < p ≤2(n − 3)

n − 4, n ≥ 5

2 < p, n ≤ 4

54 Thermoelasticity with second sound

holds. Then given any (u0, u1, θ0, q0) ∈ Π, there exists a positive number T small enough such thatproblem (2.93) has a unique strong solution satisfying

(u, ut, θ, q) ∈ C1([0,T ); Π) ∩C([0,T ); H). (2.94)

2.3.2 Blow up

The following lemma will play an important role in the proof of our main result.Lemma 2.3.1 Let L (t) be a solution of the ordinary differential inequality

dL (t)dt≥ ξL1+ν (t) (2.95)

defined in [0,∞) , where ν > 0. If L (0) > 0, then the solution ceases to exist for t ≥ L (0)−ν ξ−νν−1.

Proof. From (2.95) we have

L−(1+ν) (t)dL (t)

dt≥ ξ.

Direct integration then givesL−ν (0) − L−ν (t) ≥ ξνt,

from which follows the estimate

Lν (t) ≥[L−ν (0) − ξνt

]−1. (2.96)

It is clear that the right- hand side of (2.96) is unbounded when

ξνt = L−ν (0) .

In order to state and prove our result we introduce the following :Let B1 be the best constant of the Sobolev embedding

[H1

0 (Ω)]n→ [Lp (Ω)]n and B2 = B1/µ. We

set

α1 = B−p/(p−2)2 , E1 =

(12−

1p

)α2

1, (2.97)

and

E(t) =12‖ut‖

22 +

µ

2‖∇u‖22 +

λ + µ

2‖divu‖22 +

β

2δ‖θ‖22

+γβτ

2δk‖q‖22 −

1p‖u‖pp. (2.98)

Lemma 2.3.2 Let (u, θ, q) be solution of (2.93). Assume that E (0) < E1 and[µ‖∇u0‖

22 + (λ + µ) ‖divu0‖

22 +

β

δ‖θ0‖

22 +

γβτ

δk‖q0‖

22

]1/2> B−p/(p−2

2 .

2.3 Blow up of solutions 55

Then there exists a constant α2 > B−p/(p−2)2 such that

[µ‖∇u‖22 + (λ + µ) ‖divu‖22 +

β

δ‖θ‖22 +

γβτ

δk‖q‖22

]1/2≥ α2 (2.99)

and‖up ≥ B2α2, ∀t ∈ [0,T ) . (2.100)

Proof. We first note that, by (2.98) and the Sobolev embedding, we have

E (t) ≥µ

2‖∇u‖22 +

λ + µ

2‖divu‖22 +

β

2δ‖θ‖22

+γβτ

2δk‖q‖22 −

1p‖u‖pp

≥12

[µ‖∇u‖22 + (λ + µ) ‖divu‖22 +

β

δ‖θ‖22 +

γβτ

δk‖q‖22

]−

Bp2

p

[µ‖∇u‖22 + (λ + µ) ‖divu‖22 +

β

δ‖θ‖22 +

γβτ

δk‖q‖22

]p/2

=12α2 −

Bp2

pαp = g (α) , (2.101)

where

α =

[µ‖∇u‖22 + (λ + µ) ‖divu‖22 +

β

δ‖θ‖22 +

γβτ

δk‖q‖22

]1/2.

It is easy to verify that g is increasing for 0 < α < α1, and decreasing for α > α1,where g (α)→ −∞as α→ +∞, and

g (α1) =

(12−

1p

)B−2p/(p−2)

2 = E1,

where α1 is given in (2.97). Therefore, since E(0) < E1, there exists α2 > α1 such that g (α2) =

E (0) .

If we set

α0 =

[µ ‖∇u0‖

22 + (λ + µ) ‖divu0‖

22 +

β

δ‖θ0‖

22 +

γβτ

δk‖q0‖

22

]1/2,

then by (2.101) we haveg (α0) ≤ E (0) = g (α2) ,

which implies that α0 ≥ α2.

Now to establish (2.99), we suppose by contradiction that[µ ‖∇u (t0)‖22 + (λ + µ) ‖divu (t0)‖22 +

β

δ‖θ (t0)‖22

γβτ

δk‖q (t0)‖22

]1/2< α2,

56 Thermoelasticity with second sound

for some t0 > 0 and by the continuity of

µ ‖∇u (.)‖22 + (λ + µ) ‖divu (.)‖22 +β

δ‖θ (.)‖22

γβτ

δk‖q (.)‖22

we can choose t0 such that[µ ‖∇u (t0)‖22 + (λ + µ) ‖divu (t0)‖22 +

β

δ‖θ (t0)‖22

γβτ

δk‖q (t0)‖22

]1/2> α1.

Again the use of (2.101) leads to

E (t0) ≥ g(µ ‖∇u (t0)‖22 + (λ + µ) ‖divu (t0)‖22 +

β

δ‖θ (t0)‖22

γβτ

δk‖q (t0)‖22

)> g (α2) = E (0) .

This is impossible since E(t) ≤ E (0), for all t ∈ [0,T ). Hence (2.99) is established.

To prove (2.100), we exploit (2.98) to see

12

[µ ‖∇u‖22 + (λ + µ) ‖divu‖22 +

β

δ‖θ‖22

γβτ

δk‖q‖22

]≤ E (0) +

1p‖u‖pp .

Consequently

1p‖u‖pp ≥

12

[µ ‖∇u‖22 + (λ + µ) ‖divu‖22 +

β

δ‖θ‖22 +

γβτ

δk‖q‖22

]− E (0)

≥12α2

2 − E (0) (2.102)

≥12α2

2 − g (α2) =Bp

2

pα

p2 .

Therefore (2.102) and (2.97) yield the desired result.Theorem 2.3.2 Suppose that

2 < p ≤2n

n − 2, n ≥ 3

andβτδ

κγ< 8. (2.103)

Then any solution of (2.93) , with initial data satisfying[µ ‖∇u0‖

22 + (λ + µ) ‖divu0‖

22 +

β

δ‖θ0‖

22 +

γβτ

δk‖q0‖

22

]> B−2p/(p−2)

2

andE (0) < E1,

2.3 Blow up of solutions 57

blows up in finite time.Remark 2.3.1. The condition (2.103) is ’physically’ reasonable due to the very small value of τ.For instance in [71], for the isotropic silicon and a medium temperature of 300K we have

β ≈ 391.62[

m2

s2K

], τ ≈ 10−12[[s], δ ≈ 163.82[K],

γ ≈ 5.99 × 10−7[ms2K

kg

], κ ≈ 148

[ WmK

]consequently we get

βτδ

κγ≈ 72.367 × 10−7 < 8

Proof. A multiplication of equations (2.93) by ut,βδθ,

βγδτq respectively and integration over Ω,

using integration by parts, and addition of equalities yields

E′

(t) = −γβ

δk‖q‖22 ≤ 0. (2.104)

We then setH (t) = E1 − E (t) . (2.105)

By using (2.98) and (2.104), we get

0 < H (0) ≤ H (t)

≤ E1 −12

(‖ut‖22 + µ ‖∇u‖22 + (λ + µ) ‖divu‖22 (2.106)

+β

δ‖θ‖22 +

γβτ

δk‖q‖22) +

1p‖u‖pp .

and from (2.97) and (2.99) we obtain

E1 −12

(‖ut‖22 + µ ‖∇u‖22 + (λ + µ) ‖divu‖22 +

β

δ‖θ‖22 +

γβτ

δk‖q‖22)

< E1 −12α2

1 = −1pα2

1 < 0,∀t ≥ 0.

Hence0 < H (0) ≤ H (t) ≤

1p‖u‖pp , ∀t ≥ 0. (2.107)

We then defineL (t) = H1−σ (t) + ε

ˆΩ

[u.ut +

βτ

ku.q

](x, t) dx, (2.108)

for ε small to be chosen later andσ =

p − 22p

(2.109)

58 Thermoelasticity with second sound

By taking a derivative of (2.108) and using equations (2.93) we obtain

L′

(t) = (1 − σ) H−σ (t) H′

(t) + ε(‖u‖pp + ‖ut‖22 − µ ‖∇u‖22

− (λ + µ) ‖divu‖22) −εβ

k

ˆΩ

u.qdx +εβτ

k

ˆΩ

ut.qdx. (2.110)

By exploiting (2.98) and (2.105), the estimate (2.110) takes the form

L′

(t) = (1 − σ) H−σ (t) H′

(t) + ε

(1 −

2p

)‖u‖pp + 2ε ‖ut‖

22

−εβ

k

ˆΩ

u.qdx +εβτ

k

ˆΩ

ut.qdx + 2εH (t) (2.111)

−2εE1 +εβ

δ‖θ‖22 +

εγβτ

δk‖q‖22 .

Then using (2.100) we obtain

L′

(t) ≥ (1 − σ) H−σ (t) H′

(t) + ε

(1 −

2p− 2E1(B2α2)−p

)‖u‖pp

+2ε ‖ut‖22 −

εβ

k

ˆΩ

u.qdx +εβτ

k

ˆΩ

ut.qdx (2.112)

+2εH (t) +εβ

δ‖θ‖22 +

εγβτ

δk‖q‖22 ,

which implies

L′

(t) ≥ (1 − σ) H−σ (t) H′

(t) + εc0 ‖u‖pp + 2ε ‖ut‖

22

+2εH (t) +εβ

δ‖θ‖22 +

εγβτ

δk‖q‖22 (2.113)

−εβ

k

ˆΩ

u.qdx +εβτ

k

ˆΩ

ut.qdx,

where c0 = 1 − 2/p − 2E1(B2α2)−p > 0 since α2 > B−p/(p−2)2 .

Next we exploit the Young’s inequality to estimate the last two terms in (2.113) as follows∣∣∣∣∣ˆΩ

ut.qdx∣∣∣∣∣ ≤ a

2‖ut‖

22 +

12a‖q‖22 ,∀a > 0

ˆΩ

u.qdx ≤b2‖q‖22 +

12b‖u‖22 ,∀b > 0.

Thus (2.113) yields

L′

(t) ≥ (1 − σ) H−σ (t) H′

(t) + εc0 ‖u‖pp + ε

(2 −

aβτ2k

)‖ut‖

22

+2εH (t) +εβ

δ‖θ‖22 + ε

(γβτ

δk−βτ

2ak

)‖q‖22 (2.114)

−εβ

k

[b2‖q‖22 +

12b‖u‖22

].

2.3 Blow up of solutions 59

At this point we choose a so that

A1 := 2 −aβτ2k

> 0, A2 :=βτ

2k

(2γδ−

1a

)> 0.

This is possible by virtue of (2.103), consequently (2.114) becomes

L′

(t) ≥ (1 − σ) H−σ (t) H′

(t) + εA1 ‖ut‖22

+εA2 ‖q‖22 + εc0 ‖u‖pp + εA3 ‖θ‖

22 (2.115)

+2εH (t) −εβ

k

[b2‖q‖22 +

12b‖u‖22

],

where A1, A2, A3 are strictly positive constants.

We also set b = 2MγH−σ (t) /δ, for M a large constant to be determined, to deduce from (2.115)

L′

(t) ≥ [(1 − σ) − εM] H−σ (t) H′

(t) + εA1 ‖ut‖22

+εA2 ‖q‖22 + εc0 ‖u‖pp + εA3 ‖θ‖

22 (2.116)

+2εH (t) −Cε4M

Hσ (t) ‖u‖2p ,

where C, here and in the sequel, is a positive generic constant depending on Ω, p, β, γ, δ, k, λ, µ,τ, only.

We then use (2.107), to get

L′

(t) ≥ [(1 − σ) − εM] H−σ (t) H′

(t) + εA1 ‖ut‖22

+εA2 ‖q‖22 + εc0 ‖u‖pp + εA3 ‖θ‖

22 (2.117)

+2εH (t) −Cε

4pM‖u‖2+σp

p .

By using (2.109) and the inequality

zν ≤ (z + 1) ≤(1 +

1a

)(z + a) ,∀z ≥ 0, 0 < ν ≤ 1, a ≥ 0, (2.118)

we have the following

‖u‖2+σpp ≤ d

(‖u‖pp + H (0)

)≤ d

(‖u‖pp + H (t)

),∀t ≥ 0 (2.119)

where d = 1 + 1/H (0) .

60 Thermoelasticity with second sound

Inserting the estimate (2.119) into (2.117) we arrive at

L′

(t) ≥ [(1 − σ) − εM] H−σ (t) H′

(t) + εA1 ‖ut‖22

+εA2 ‖q‖22 + ε

(c0 −

Cd4pM

)‖u‖pp + εA3 ‖θ‖

22 (2.120)

+ε

(2 −

Cd4pM

)H (t) .

At this point, we choose M large enough so that (2.120) becomes, for some positive constant A0,

L′

(t) ≥ [(1 − σ) − εM] H−σ (t) H′

(t)

+εA0[‖ut‖

22 + ‖q‖22 + ‖u‖pp + H (t)

](2.121)

Once M is fixed (hence A0), we pick ε small enough so that (1 − σ) − εM ≥ 0 and

L (0) = H1−σ (0) + ε

ˆΩ

[u0.u1 +

βτ

ku0.q0

](x, t) dx > 0.

Therefore (2.121) yields

L′

(t) ≥ εA0[‖ut‖

22 + ‖q‖22 + ‖u‖pp + H (t)

]. (2.122)

Consequently we haveL (t) ≥ L (0) > 0, ∀t ≥ 0.

Next we estimate ∣∣∣∣∣ˆΩ

uut(x, t)dx∣∣∣∣∣ ≤ ||u||2||ut||2 ≤ C||u||p||ut||2,

which implies ∣∣∣∣∣ˆΩ

uut(x, t)dx∣∣∣∣∣1/(1−σ)

≤ C||u||1/(1−σ)p ||ut||

1/(1−σ)2 .

Again Young’s inequality gives us∣∣∣∣∣ˆΩ

uut(x, t)dx∣∣∣∣∣1/(1−σ)

≤ C[||u||r/(1−σ)

p + ||ut||s/(1−σ)2

], (2.123)

for 1/r + 1/s = 1. We take s = 2(1 − σ), to get r/(1 − σ) = 2/(1 − 2σ) = p by virtue of (2.109).Therefore (2.123) becomes∣∣∣∣∣ˆ

Ω

uut(x, t)dx∣∣∣∣∣1/(1−σ)

≤ C[||u||pp + ||ut||

22

], ∀t ≥ 0. (2.124)

Similarly we have ∣∣∣∣∣ˆΩ

uq(x, t)dx∣∣∣∣∣1/(1−σ)

≤ C[||u||pp + ||q||22

], ∀t ≥ 0. (2.125)

2.3 Blow up of solutions 61

Finally by noting that

L1/(1−σ)(t) =

(H1−σ(t) + ε

ˆΩ

u(ut +βτ

κq)(x, t)dx

)1/(1−σ)

≤ C(H(t) +

∣∣∣∣∣ˆΩ

uut(x, t)dx∣∣∣∣∣1/(1−σ)

+

∣∣∣∣∣ˆΩ

uq(x, t)dx∣∣∣∣∣1/(1−σ))

≤ C[H(t) + ||u||pp + ||ut||

22 + ||q||22

], ∀t ≥ 0

and combining it with (2.122) we obtain

L′(t) ≥ a0L1/(1−σ)(t), ∀ t ≥ 0 (2.126)

where a0 is a positive constant depending on εA0 and C. By using lemma 2.3.1 our conclusionfollows, indeed a simple integration of (2.126) over (0, t) then yields

L (p−2)/(p+2)(t) ≥1

L −(p−2)/2(0) − a0t(p − 2)/2.

Therefore L(t) blows up in a time

T ∗ ≤1 − α

αa0[L (0)](p−2)/(p+2) . (2.127)

Remark 2.3.1 .The estimate (2.127) shows that the larger L(0) is the quicker the blow up takesplace.

62 Thermoelasticity with second sound

Chapter 3

Thermoelasticity of type III

This chapter is devoted to the analysis of the long time behavior of solutions to systems of ther-moelasticity of type III. In section 3.1, this new theory is introduced. In section 3.2, we study aone-dimensional nonlinear system and establish a decay result similar to the one for thermoelas-ticity with second sound. The last two sections are devoted to systems with internal and boundarydampings. We will also prove exponential decay results.

64 Thermoelasticity of type III

3.1 Introduction

By the end of the last century, Green and Naghdi [21, 22] introduced three types of thermoelastictheories based on an entropy equality instead of the usual entropy inequality. In each of these the-ories, the heat flux is given by a different constitutive assumption. As a result, three theories areobtained and were called thermoelasticity type I, type II, and type III respectively. This develop-ment is made in a rational way in order to obtain a fully consistent theory, which will incorporatethermal pulse transmission in a very logical manner and elevate the unphysical infinite speed ofheat propagation induced by the classical theory of heat conduction. When the theory of type I islinearized the parabolic equation of the heat conduction arises. Whereas the theory of type II doesnot admit dissipation of energy and it is known as thermoelasticity without dissipation. In fact, itis a limiting case of thermoelasticity type III. See in this regard [7, 8, 9, 62, 63, 64, 68] for moredetails.

To understand these new theories and their applications, several mathematical and physicalcontributions have been made; see for example [62, 63, 64, 65, 66, 67, 68]. In particular, we mustmention the survey paper of Chandrasekharaiah [9], in which the author has focussed attentionon the work done during the last 10 or 12 years. He reviewed the theory of thermoelasticity withthermal relaxation and the temperature rate depend thermoelasticity. He also described the ther-moelasticity without dissipation and clarified its properties. By the end of his paper, he made abrief discussion to the new theories, including what is called dual-phase-large effects.

Zhang and Zuazua [86] have recently analyzed the long time behavior of the solution of thesystem

utt − µ4u − (µ + λ)∇(divu) + β∇θ = 0, in (0,∞) ×Ω,

θtt − ∆θ + divutt − ∆θt = 0, in (0,∞) ×Ω,

u(., 0) = u0, ut(., 0) = u1, θ(., 0) = θ0, x ∈ Ω,

u = θ = 0, on (0,∞) × ∂Ω

(3.1)

and they concluded the following:

• For most domains, the energy of the system does not decay uniformly.

• Under suitable conditions on the domain, which might described in terms of Geometric Op-tics, the energy of the system decays exponentially

• For most domains in two space dimension, the energy of smooth solutions decays in a poly-nomial rate.

In [67] Quintanilla and Racke considered a system similar to (3.1) and used the spectral analy-sis method and the energy method to obtain the exponential stability in one dimension for differentboundary conditions; (Dirichlet-Dirichlet or Dirichlet-Neumann). They also proved a decay of en-ergy result in the radially symmetric situations, in multi-dimensional case (n = 2, 3) . These recent

3.1 Introduction 65

results of theirs are similar, to some extent, to the ones obtained for systems of thermoelasticity withsecond sound [71, 72]. We also recall the contribution of Quintanilla [64], in which he proved thatsolutions of thermoelasticity of type III converge to solutions of the classical thermoelasticity aswell as to the solution of thermoelasticity without energy dissipation and Quintanilla [66], in whichhe established a structural stability result on the coupling coefficients and continuous dependenceon the external data in thermoelasticity type III.

66 Thermoelasticity of type III

3.2 Exponential stability in one-dimensional nonlinear system

In this section we consider a one-dimensional nonlinear system of thermoelasticity type III. Weestablish an exponential decay result for solutions with small ’enough’ initial data. This workextends the result of [67], in which a similar, but linear, problem has been discussed. Our problemis

utt − αuxx + βθx = 0, in [0,∞) × (0, 1) ,

θtt − δθxx + γuttx − κθtxx = 0, in [0,∞) × (0, 1) ,(3.2)

where α, β, δ, γ, κ are functions of (ux, θ) . We will show that a similar argument to the one in [55]and [71] is still valid to prove the exponential decay for classical solutions with small initial data.

3.2.1 Statement of the problem

We consider the following one-dimensional nonlinear problem

utt − αuxx + βθx = 0, in [0,∞) × (0, 1) ,

θtt − δθxx + γuttx − κθtxx = 0, in [0,∞) × (0, 1) ,

u (0, .) = u0, ut (0, .) = u1, θ (0, .) = θ0, θt (0, .) = θ1, in (0, 1),

u (t, 0) = u (t, 1) = θ (t, 0) = θ (t, 1) = 0, t ≥ 0

(3.3)

where

α = α (ux, θ) , β = β (ux, θ) , δ = δ (ux, θ) , γ = γ (ux, θ) , κ = κ (ux, θ)

are smooth functions satisfying, for two positive constants λ1, λ2 > 0,

λ1 ≤ α (ux, θ) ≤ λ2, λ1 ≤ β (ux, θ) ≤ λ2, λ1 ≤ δ (ux, θ) ≤ λ2

λ1 ≤ γ (ux, θ) ≤ λ2, λ1 ≤ κ (ux, θ) ≤ λ2.(3.4)

We denote by

σ (t) = sup0≤x≤1

|θ| + |θx| + |θt| + |θtt| + |θtx|+ (3.5)

|v| + |vt| + |vx| + |vtt| + |vtx| + |vxx| ,

E (t) = E1 (t) + E2 (t) + E3 (t) , (3.6)

where

E1 (t) = E1 (v, θ) =12

ˆ 1

0

(γv2

t + αγv2x + βθ2

t + δβθ2x

), (3.7)

E2 (t) = E1 (vt, θt) , E3 (t) = E1 (vtt, θtt) ,

3.2 Exponential stability in one-dimensional nonlinear system 67

and

Λ (t) =

ˆ 1

0(v2

ttt + v2xxx + v2

ttx + v2xxt + v2

tt + v2xx + v2

tx + v2t + v2

x

+θ2ttt + θ2

ttx + θ2xxt + θ2

tt + θ2xx + θ2

tx + θ2t + θ2

x)dx (3.8)

Remark 3.2.1: By the Sobolev embedding inequalities we have

σ (t) ≤ C√

Λ (t), ∀t ≥ 0. (3.9)

Theorem 3.2.1 : Assume than α, β, δ, γ, and κ are C3 functions satisfying (3.4) . Then there existsa small positive constant η such that if

Λ0 = ‖u0‖2H3 + ‖u1‖

2H2 + ‖θ0‖

2H3 + ‖θ1‖

2H2 < η, (3.10)

the energy term

H (t) = (E1(t) + E2 (t) + E3 (t)) +

ˆ 1

0θ2

xxdx

decays exponentially as t → +∞.

Remark 3.2.2 : The decay result still holds if (3.4) is only satisfied in a neighborhood of the equi-librium state. In this case a slight modification in the proof, as in [55] is needed.

3.2.2 Proof of theorem 3.2.1

Our goal is to find a Lyapunov functional F (t) equivalent to the third-order energy H (t) , for whichwe have

ddt

F (t) ≤ −cF (t) ,

for some positive constant c.

If v = ut denotes the velocity field, we obtain from (3.3)

vtt − αvxx + βθtx = αtuxx − βtθx (3.11)

θtt − δθxx + γvtx − κθtxx = 0 (3.12)

with initial and boundary conditions

v(., 0) = u1, vt(., 0) = αu0xx − βθ0x, θ(., 0) = θ0, x ∈ Ω

v = θ = 0, x ∈ ∂Ω , t ≥ 0,(3.13)

We multiply (3.11) by γvt, (3.12) by βθt integrate over I = (0, 1) using integrations by parts andadd equations, to obtain

dE1 (t)dt

= −

ˆ 1

0βκθ2

txdx + R1, (3.14)

68 Thermoelasticity of type III

where

R1 =12

ˆ 1

0γtv

2t + (αγ)t v2

x + (δβ)t θ2x + βtθ

2t dx

+

ˆ 1

0[− (αγ)x vtvx + γαtvtuxx − βtγθxθt (3.15)

− (δβ)x θtθx + (βγ)x vtθt − (κβ)x θtθtx)dx

Differentiating (3.11) and (3.12) with respect to t, we get

vttt − αvtxx + βθttx = αttuxx − βttθx + 2αtvxx − 2βtθtx (3.16)

θttt − δθtxx + γvttx − κθttxx = δtθxx − γtvtx + κtθtxx (3.17)

In the same manner, multiplying (3.16) by γvtt and (3.17) by βθtt we obtain

dE2 (t)dt

= −

ˆ 1

0βκθ2

ttxdx + R2 (3.18)

for

R2 =12

ˆ 1

0γtv

2tt + (αγ)t v2

tx + (δβ)t θ2tx + βtθ

2ttdx

+

ˆ 1

0(− (αγ)x vttvtx − (δβ)x θttθtx + (βγ)x vttθtt − (κβ)x θttθttx)dx

+

ˆ 1

0γvtt

(αttuxx − βttθtx + 2αtvxx − 2βtθtx

)dx (3.19)

+

ˆ 1

0βθtt

(δtθxx − γtvtx + κtθtxx

)dx.

Remark 3.2.3: We note that the term uxx, appearing in R1, R2, and what follows, will be estimatedby vt and θx using (3.3).

For additional estimates, we differentiate (3.16), (3.17) with respect to t, to get

vtttt − αvttxx + βθtttx = αtttuxx − βtttθx + 3αttvxx

+3αtvtxx − 3βttθtx − 3βtθttx (3.20)

θtttt − δθttxx + γvtttx − κθtttxx = 2δtθtxx − 2γtvttx + 2κtθttxx

+δttθxx − γttvtx + κttθtxx (3.21)

3.2 Exponential stability in one-dimensional nonlinear system 69

We then multiply (3.20) by γvttt and (3.21) by βθttt and use Young’s inequality to obtain

dE3 (t)dt

≤ −12

ˆ 1

0βκθ2

tttxdx + R3, (3.22)

where

R3 =12

ˆ 1

0γtv

2ttt + (αγ)t v2

ttx + (δβ)t θ2ttx + βtθ

2tttdx (3.23)

+

ˆ 1

0(− (αγ)x vtttvttx − (δβ)x θtttθttx + (βγ)x vtttθttt −

(κβ)x

2βκθ2

ttt)dx

+

ˆ 1

0γvttt(αtttuxx − βtttθx + 3αttvxx + 3αtvtxx − 3βttθtx − 3βtθttx)dx

+

ˆ 1

0βθttt(2δtθtxx − 2γtvttx + 2κtθttxx + δttθxx − γttvtx + κttθtxx)dx.

The multiplication of (3.11) by −vxx/α yields

ˆ 1

0v2

xxdx = −ddt

ˆ 1

0

1α

vtxvxdx +

ˆ 1

0

1α

v2txdx +

ˆ 1

0

β

αθtxvxxdx + R4, (3.24)

with

R4 =

ˆ 1

0

βt

αθxvxx −

αt

αuxxvxx +

(1α

)tvtxvx −

(1α

)x

vxvttdx. (3.25)

Thanks of Young’s inequality (3.24) becomes

23

ˆ 1

0v2

xxdx +ddt

ˆ 1

0

1α

vtxvxdx ≤ˆ 1

0

1α

v2txdx +

ˆ 1

0

3β2

4α2 θ2txdx + R4 (3.26)

Multiplying (3.12) by 3vtx/αγ and using integration by parts we easily get

ˆ 1

0

3α

v2txdx = −

ˆ 1

0

3αγθttvtxdx +

ˆ 1

0

3δαγ

vtxθxxdx +

ˆ 1

0

3καγθtxxvtxdx (3.27)

Therefore, exploiting (3.11) , we obtain

−

ˆ 1

0

3αγθttvtxdx =

ˆ 1

0

(3αγ

)xθttvtdx +

ddt

ˆ 1

0

3αγθtxvtdx −

ˆ 1

0

(3αγ

)tθtxvtdx

−

ˆ 1

0

3γθtxvxxdx +

ˆ 1

0

3βαγθ2

txdx −ˆ 1

0

3αt

αγθtxuxxdx +

ˆ 1

0

3βt

αγθtxθxdx. (3.28)

Using integration by parts, we easily see that

70 Thermoelasticity of type III

ˆ 1

0

3δαγ

vtxθxxdx = −

ˆ 1

0

(3δαγ

)x

vtxθxdx −ˆ 1

0

3δαγ

vtxxθxdx

= −

ˆ 1

0

(3δαγ

)x

vtxθxdx −ddt

ˆ 1

0

3δαγθxvxxdx

+

ˆ 1

0

(3δαγ

)tθxvxxdx +

ˆ 1

0

3δαγθtxvxxdx +

[3δαγ

vtxθx

]x=1

x=0.

Again the use of (3.11) yields

ˆ 1

0

3δαγ

vtxθxxdx = −

ˆ 1

0

(3δαγ

)x

vtxθxdx −ddt

ˆ 1

0(

3δα2γ

θxvtt +3δβα2γ

θxθtx

−3δαt

α2γuxxθx +

3δβt

α2γθ2

x)dx +

ˆ 1

0

(3δαγ

)tθxvxxdx

+

ˆ 1

0

3δαγθtxvxxdx +

[3δαγ

vtxθx

]x=1

x=0(3.29)

andˆ 1

0

3καγθtxxvtxdx = −

ddt

ˆ 1

0

3καγθtxvxxdx +

ˆ 1

0

3καγθttxvxxdx (3.30)

−

ˆ 1

0

(3καγ

)xθtxvtxdx +

ˆ 1

0

(3καγ

)tθtxvxxdx +

[3καγθtxvtx

]x=1

x=0.

By inserting (3.28) − (3.30) into (3.27) we arrive at

ˆ 1

0

3α

v2txdx = −

ddt

ˆ 1

0

3καγθtxvxxdx +

ˆ 1

0

3καγθttxvxxdx −

ˆ 1

0

(3καγ

)xθtxvtxdx +

ddt

ˆ 1

0

3αγθtxvtdx −

ˆ 1

0

3γθtxvxxdx +

ˆ 1

0

3βαγθ2

txdx +

ˆ 1

0

(3αγ

)xθttvtdx

−

ˆ 1

0

(3αγ

)tθtxvtdx −

ˆ 1

0

3αt

αγθtxuxxdx +

ˆ 1

0

3βt

αγθtxθxdx (3.31)

−

ˆ 1

0

(3δαγ

)x

vtxθxdx +

ˆ 1

0

(3δαγ

)tθxvxxdx +

ˆ 1

0

3δαγθtxvxxdx

−ddt

ˆ 1

0(

3δα2γ

θxvtt +3δβα2γ

θxθtx −3δαt

α2γuxxθx +

3δβt

α2γθ2

x)dx

+

ˆ 1

0

(3καγ

)tθtxvxxdx +

[3καγ

vtxθtx

]x=1

x=0+

[3δαγ

vtxθx

]x=1

x=0

3.2 Exponential stability in one-dimensional nonlinear system 71

Young’s inequality gives ∣∣∣∣∣∣ˆ 1

0

3γθtxvxxdx

∣∣∣∣∣∣ ≤ 112

ˆ 1

0v2

xxdx +

ˆ 1

0

27γ2 θ

2txdx, (3.32)

ˆ 1

0

3δαγθtxvxxdx ≤

112

ˆ 1

0v2

xxdx +

ˆ 1

0

27δ2

α2γ2 θ2txdx, (3.33)

and ˆ 1

0

3καγθttxvxxdx ≤

112

ˆ 1

0v2

xxdx +

ˆ 1

0

27κ2

α2γ2 θ2ttxdx (3.34)

Taking into account (3.32) − (3.34), estimate (3.31) becomes

ˆ 1

0

3α

v2txdx ≤

14

ˆ 1

0v2

xxdx +

ˆ 1

0

(3βαγ

+27γ2 +

27δ2

α2γ2

)θ2

txdx (3.35)

+

ˆ 1

0

27κ2

α2γ2 θ2ttxdx +

ddt

ˆ 1

0(1α

vtxvx −3αγθtxvt +

3δα2γ

θxvtt

+3δβα2γ

θxθtx +3καγθtxvxx)dx +

[3δαγ

vtxθx

]x=1

x=0+

[3καγ

vtxθtx

]x=1

x=0+ ∆,

where

∆ =

ˆ 1

0

(3αγ

)xθttvtdx −

ˆ 1

0

(3αγ

)tθtxvtdx −

ˆ 1

0

3αt

αγθtxuxxdx

+

ˆ 1

0

3βt

αγθtxθxdx −

ˆ 1

0

(3δαγ

)x

vtxθxdx +

ˆ 1

0

(3δαγ

)tθxvxxdx (3.36)

−ddt

ˆ 1

0(−

3δαt

α2γuxxθx +

3δβt

α2γθ2

x)dx −ˆ 1

0

(3καγ

)xθtxvtxdx +

ˆ 1

0

(3καγ

)tθtxvxxdx

Combining (3.26) and (3.35) we obtain

ˆ 1

0

2α

v2txdx +

512

ˆ 1

0v2

xxdx −ddt

ˆ 1

0(

3αγθtxvt −

3δα2γ

θxvtt (3.37)

−3δβα2γ

θxθtx −3καγθtxvxx)dx ≤

ˆ 1

0

(3βαγ

+27γ2 +

27δ2

α2γ2 +3β2

4α2

)θ2

txdx

+

ˆ 1

0

27κ2

α2γ2 θ2ttxdx +

[3δαγ

vtxθx

]x=1

x=0+

[3καγ

vtxθtx

]x=1

x=0+ R5,

whereR5 = ∆ + R4

72 Thermoelasticity of type III

Similarly, multiplying equation (3.16) by vtxx/α we get, after integration over I,ˆ 1

0v2

txxdx = −

ˆ 1

0

(1α

)x

vtttvtxdx −ddt

ˆ 1

0

1α

vttxvtxdx

+

ˆ 1

0βθttxvtxxdx +

ˆ 1

0

(1α

)tvttxvtxdx +

ˆ 1

0

1α

v2ttxdx

+

ˆ 1

0

1α

vtxx(αttuxx − βttθx + 2αtvxx − 2βtθtx

)dx,

which gives, using Young’s inequality,ˆ 1

0v2

txxdx ≤ −ddt

ˆ 1

0

1α

vttxvtxdx +

ˆ 1

0

1α

v2ttxdx +

ˆ 1

0

3β2

4α2 θ2ttxdx

+13

ˆ 1

0v2

txxdx +

ˆ 1

0

(1α

)tvttxvtxdx −

ˆ 1

0

(1α

)x

vtttvtxdx

+

ˆ 1

0

1α

vtxx(αttuxx − βttθx + 2αtvxx − 2βtθtx

)dx. (3.38)

We also multiply (3.17) by 3vttx/αγ to getˆ 1

0

3α

v2ttxdx = −

ˆ 1

0

3αγθtttvttxdx +

ˆ 1

0

3δαγθtxxvttxdx +

ˆ 1

0

3καγθttxxvttxdx

+

ˆ 1

0

3αγ

vttx(δtθxx − γtvtx + κtθtxx

)dx (3.39)

and exploit (3.16) to obtain

−

ˆ 1

0

3αγθtttvttxdx =

ˆ 1

0

(3αγ

)xθtttvttdx +

ddt

ˆ 1

0

3αγθttxvttdx

−

ˆ 1

0

(3αγ

)tθttxvttdx −

ˆ 1

0

3γθttxvtxxdx +

ˆ 1

0

3βαγθ2

ttxdx −ˆ 1

0

3αtt

αγθttxuxxdx

+

ˆ 1

0

3βtt

αγθttxθxdx −

ˆ 1

0

6αt

αγθttxvxx +

ˆ 1

0

6βt

αγθttxθtxdx (3.40)

Using integration by parts, we easily seeˆ 1

0

3δαγθtxxvttxdx =

[3δαγ

vttxθtx

]x=1

x=0−

ˆ 1

0

(3δαγ

)xθtxvttxdx

−ddt

ˆ 1

0

3δαγθtxvtxxdx +

ˆ 1

0

(3δαγ

)tθtxvtxxdx +

ˆ 1

0

3δαγθttxvtxxdx.

3.2 Exponential stability in one-dimensional nonlinear system 73

Again the use of (3.16) leads to

ˆ 1

0

3δαγθtxxvttxdx =

[3δαγ

vttxθtx

]x=1

x=0−

ˆ 1

0

(3δαγ

)xθtxvttxdx

+

ˆ 1

0

(3δαγ

)tθtxvtxxdx +

ˆ 1

0

3δαγθttxvtxxdx −

ddt

ˆ 1

0(

3δα2γ

θtxvttt +3δβα2γ

θtxθttx

−3δαtt

α2γθtxuxx +

3δβtt

α2γθtxθx −

6δαt

α2γvxxθtx +

6δβt

α2γθ2

tx)dx (3.41)

and the integration by parts gives

ˆ 1

0

3καγθttxxvttxdx = −

ddt

ˆ 1

0

3καγθttxvtxxdx +

ˆ 1

0

(3καγ

)tθttxvtxxdx (3.42)

By inserting (3.40) − (3.42) into (3.39) we get

ˆ 1

0

3α

v2ttxdx =

ˆ 1

0

(3αγ

)x

vttθtttdx +ddt

ˆ 1

0

3αγ

vttθttxdx −ˆ 1

0

(3αγ

)tvttθttxdx

−

ˆ 1

0

3γθttxvtxxdx +

ˆ 1

0

3βαγθ2

ttxdx −ˆ 1

0

3αtt

αγθttxuxxdx +

ˆ 1

0

3βtt

αγθttxθxdx

−

ˆ 1

0

6αt

αγθttxvxxdx +

ˆ 1

0

6βt

αγθttxθtxdx −

ˆ 1

0

(3δαγ

)xθtxvttxdx

+

ˆ 1

0

(3δαγ

)tθtxvtxxdx +

ˆ 1

0

3δαγθttxvtxxdx −

ddt

ˆ 1

0(

3δα2γ

θtxvttt +3δβα2γ

θtxθtxx

−3δαtt

α2γθtxuxx −

3δβtt

α2γθtxθx −

6δαt

α2γvxxθtx −

6δβt

α2γθ2

tx)dx (3.43)

−ddt

ˆ 1

0

3καγθtxvtxxdx +

ˆ 1

0

(3καγ

)tθtxvtxxdx

ˆ 1

0

(3καγ

)xθttxvttxdx

+

ˆ 1

0

3καγθtxvttxxdx +

ˆ 1

0

3αγ

vttx(δtθxx − γtvtx + κtθtxx

)dx

+

[3καγθttxvttx

]x=1

x=0+

[3καγθttxvttx

]x=1

x=0+

[3δαγ

vttxθtx

]x=1

x=0

Similarly to (3.32) - (3.34), we have∣∣∣∣∣∣ˆ 1

0

3γθttxvtxxdx

∣∣∣∣∣∣ ≤ 112

ˆ 1

0v2

txxdx +

ˆ 1

0

27γ2 θ

2ttxdx, (3.44)

74 Thermoelasticity of type III

ˆ 1

0

3δαγθttxvtxxdx ≤

112

ˆ 1

0v2

txxdx +

ˆ 1

0

27δ2

α2γ2 θ2ttxdx, (3.45)

and ˆ 1

0

3καγθtttxvtxxdx ≤

112

ˆ 1

0v2

txxdx +

ˆ 1

0

27κ2

α2γ2 θ2tttxdx (3.46)

Taking into account (3.44) − (3.46), estimate (3.43) becomes

ˆ 1

0

3α

v2ttxdx ≤

14

ˆ 1

0v2

txxdx +

ˆ 1

0(

3βαγ

+27δ2

α2γ2 +27γ2 )θ2

ttxdx +

ˆ 1

0

27κ2

α2γ2 θ2tttxdx

+ddt

ˆ 1

0(

3αγ

vttθttx −3δα2γ

θtxvttt −3δβα2γ

θtxθttx −3καγθtxvtxx)dx

+

[3δαγ

vttxθtx

]x=1

x=0+

[3καγθttxvttx

]x=1

x=0+ R6, (3.47)

where

R6 =

ˆ 1

0

(3αγ

)x

vttθtttdx −ˆ 1

0

(3αγ

)tvttθttxdx −

ˆ 1

0

3αtt

αγθttxuxxdx

+

ˆ 1

0

3βtt

αγθttxθxdx −

ˆ 1

0

6αt

αγθttxvxxdx +

ˆ 1

0

6βt

αγθttxθtxdx (3.48)

−

ˆ 1

0

(3δαγ

)xθtxvttxdx +

ˆ 1

0

(3δαγ

)tθtxvtxxdx +

ˆ 1

0

(3καγ

)tθtxvtxxdx

−ddt

ˆ 1

0(−

3δαtt

α2γθtxuxx −

3δβtt

α2γθtxθx −

6δαt

α2γvxxθtx −

6δβt

α2γθ2

tx)dx

+

ˆ 1

0

(3καγ

)xθttxvttxdx +

ˆ 1

0

3αγ

vttx(δtθxx − γtvtx + κtθtxx

)dx.

A combination (3.38) and (3.47) leads to

ˆ 1

0

2α

v2ttxdx +

512

ˆ 1

0v2

txxdx +ddt

ˆ 1

0(1α

vttxvtx −3αγ

vttθttx

−3δα2γ

θtxvttt −3δβα2γ

θtxθtxx +3καγθtxvtxx)dx

≤

[3δαγ

vttxθtx

]x=1

x=0+

[3καγθttxvttx

]x=1

x=0(3.49)

+

ˆ 1

0(

3βαγ

+27δ2

α2γ2 +27γ2 +

3β2

4α2 )θ2ttxdx +

ˆ 1

0

27κ2

α2γ2 θ2tttxdx + R7,

3.2 Exponential stability in one-dimensional nonlinear system 75

where

R7 = R6 +

ˆ 1

0

(1α

)tvttxvtxdx −

ˆ 1

0

(1α

)x

vtttvtxdx

+

ˆ 1

0

1α

vtxx(αttuxx − βttθx + 2αtvxx − 2βtθtx

). (3.50)

Now we conclude from (3.11)ˆ 1

0v2

ttdx ≤ 2ˆ 1

0α2v2

xxdx + 2ˆ 1

0β2θ2

txdx + 2ˆ 1

0αtuxxvttdx − 2

ˆ 1

0βtθxvttdx.

At this point we exploit Poincare’s inequalityˆ 1

0θ2

t dx ≤ˆ 1

0θ2

txdx,ˆ 1

0v2

t dx ≤ˆ 1

0v2

txdx

to obtain ˆ 1

0(v2

tt + v2t + θ2

t )dx ≤ 2ˆ 1

0α2v2

xxdx +

ˆ 1

0(2β2 + 1)θ2

txdx +

ˆ 1

0v2

txdx

+2ˆ 1

0(αtuxxvtt − βtθxvtt)dx. (3.51)

¿From (3.16) we see thatˆ 1

0v2

tttdx ≤ 2ˆ 1

0α2v2

txxdx + 2ˆ 1

0β2θ2

ttxdx

+2ˆ 1

0(αttuxxvttt − βttθxvttt + 2αtvxxvttt − 2βtθtxvttt)dx.

Again, use of Poincare’s inequality yields

ˆ 1

0(v2

ttt + v2tt + θ2

tt)dx ≤ 2ˆ 1

0α2v2

txxdx +

ˆ 1

0(2β2 + 1)θ2

ttxdx +

ˆ 1

0v2

ttxdx

+2ˆ 1

0(αttuxxvttt − βttθxvttt + 2αtvxxvttt

−2βtθtxvttt)dx.

A combination of (3.4), (3.51) and the last inequality givesˆ 1

0(v2

ttt + v2tt + v2

t + θ2tt + θ2

t )dx

≤ Cˆ 1

0(v2

xx + v2txx + v2

tx + v2ttx + θ2

ttx + θ2tx)dx + R8. (3.52)

76 Thermoelasticity of type III

for C a positive constant and

R8 = 2ˆ 1

0(αtuxxvtt − βtθxvtt + αttuxxvttt

−βttθxvttt + 2αtvxxvttt − 2βtθtxvttt)dx. (3.53)

Multiplying (3.11) by v/α and using Poincare’s inequality and Young’s inequality we obtain

ˆ 1

0v2

xdx ≤ Cˆ 1

0(v2

tt + θ2tx)dx + R9, (3.54)

where

R9 =

ˆ 1

0(αt

αuxxv −

βt

αθxv)dx. (3.55)

A multiplication of (3.12) by θtt yields

ˆ 1

0θ2

ttdx ≤ −ddt

ˆ 1

0δθtxθxdx +

ˆ 1

0δθ2

txdx +34

ˆ 1

0γ2v2

txdx +12

ˆ 1

0θ2

ttdx

+34

ˆ 1

0κ2θ2

txxdx −ˆ 1

0δxθttθxdx +

ˆ 1

0δtθtxθxdx. (3.56)

A multiplication of (3.17) by θttt gives

ˆ 1

0θ2

tttdx ≤ −ddt

ˆ 1

0δθttxθtxdx +

ˆ 1

0δθ2

ttxdx +34

ˆ 1

0γ2v2

ttxdx +12

ˆ 1

0θ2

tttdx

+34

ˆ 1

0κ2θ2

ttxxdx +

ˆ 1

0(−δxθtttθtx + δtθttxθtxx + δtθtttθxx

−γtvtxθttt + κtθtxxθttt)dx. (3.57)

We then combine (3.4), (3.56) and (3.57) to arrive at

ˆ 1

0(θ2

ttt + θ2tt)dx + 2

ddt

ˆ 1

0(δθtxθx + δθttxθtx)dx (3.58)

≤ Cˆ 1

0(θ2

tx + θ2ttx + θ2

txx + θ2ttxx)dx +

32

ˆ 1

0γ2(v2

tx + v2ttx)dx + R10,

where

R10 = 2ˆ 1

0(−δxθttθx + δtθtxθx + δtθtttθxx

−γtvtxθttt + κtθtxxθttt − δxθtttθtx + δtθttxθtxx)dx. (3.59)

3.2 Exponential stability in one-dimensional nonlinear system 77

Multiplying (3.12) by θ/δ we conclude, for ε > 0ˆ 1

0θ2

xdx ≤

ˆ 1

0

1δθ2

t dx + ε4ˆ 1

0v2

xdx +cε4

ˆ 1

0θ2

t dx

−ddt

ˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx + R11, (3.60)

where

R11 =

ˆ 1

0

(1δ

)tθtθ +

(γ

δ

)tvxθ −

(κ

δ

)xθθtxdx +

(κ

2δ

)tθ2

xdx. (3.61)

Multiplication of (3.12) by θxx/δ yieldsˆ 1

0θ2

xxdx ≤ −12

ddt

ˆ 1

0

κ

δθ2

xxdx +cε

ˆ 1

0θ2

xdx + ε

ˆ 1

0θ2

ttxdx (3.62)

+

ˆ 1

0

3γ2

δ2 θ2xdx +

112

ˆ 1

0v2

txxdx +

[γ

δvtxθx

]x=1

x=0+ R12,

where

R12 =12

ˆ 1

0

(κ

δ

)tθ2

xxdx −ˆ 1

0

(1δ

)xθttθxdx −

ˆ 1

0

(γ

δ

)x

vtxθxdx. (3.63)

By exploiting (3.4) and (3.12) we obtainˆ 1

0θ2

txxdx ≤ Cˆ 1

0(θ2

tt + θ2xx + v2

tx)dx. (3.64)

Using (3.4) and equation (3.17) to getˆ 1

0θ2

ttxxdx ≤ Cˆ 1

0(θ2

ttt + θ2txx + v2

ttx)dx + R13, (3.65)

where

R13 =

ˆ 1

0

(δtθxx − γtvtx + κtθtxx

)2 dx. (3.66)

The boundary terms in (3.37) and (3.47) are treated as follows:[3δαγ

vtxθx

]x=1

x=0≤

[9δ2

4α3γ2εθ2

x + εαv2tx

]x=1

x=0≤

∣∣∣∣∣∣∣[

9δ2

4α3γ2εθ2

x

]x=1

x=0

∣∣∣∣∣∣∣ +

∣∣∣∣∣[εαv2tx

]x=1

x=0

∣∣∣∣∣≤

Ct

ε

[θ2

x (1) + θ2x (0)

]+

∣∣∣∣∣[εαv2tx

]x=1

x=0

∣∣∣∣∣ ,where

Ct = Max[

9δ2

4α3γ2

]x=0

,

[9δ2

4α3γ2

]x=1

.

78 Thermoelasticity of type III

Using the embedding inequalities to get

Ct

ε

[θ2

x (1) + θ2x (0)

]≤

4Ct

ε

(1 +

1ε2

) ˆ 1

0θ2

xdx + 4Ctε

ˆ 1

0θ2

xxdx.

Then, [3δαγ

vtxθx

]x=L

x=0≤

4Ct

ε

(1 +

1ε2

) ˆ 1

0θ2

xdx + 4Ctε

ˆ 1

0θ2

xxdx

+∣∣∣∣[εαv2

tx

]x=0

∣∣∣∣ +∣∣∣∣[εαv2

tx

]x=1

∣∣∣∣ (3.67)

By the same argument we have[3δαγ

vttxθtx

]x=L

x=0≤

4Ct

ε

(1 +

1ε2

) ˆ 1

0θ2

txdx + 4Ctε

ˆ 1

0θ2

txxdx

+∣∣∣∣[εαv2

ttx

]x=0

∣∣∣∣ +∣∣∣∣[εαv2

ttx

]x=1

∣∣∣∣ . (3.68)

Also, [3καγ

vtxθtx

]x=1

x=0≤

4C′

t

ε

(1 +

1ε2

)ˆ 1

0θ2

txdx + 4Ctε

ˆ 1

0θ2

txxdx

+∣∣∣∣[εαv2

tx

]x=0

∣∣∣∣ +∣∣∣∣[εαv2

tx

]x=1

∣∣∣∣ (3.69)

and [3καγ

vttxθttx

]x=1

x=0≤

4C′

t

ε

(1 +

1ε2

)ˆ 1

0θ2

ttxdx + 4Ctε

ˆ 1

0θ2

ttxxdx

+∣∣∣∣[εαv2

ttx

]x=0

∣∣∣∣ +∣∣∣∣[εαv2

ttx

]x=1

∣∣∣∣ (3.70)

where

C′

t = Max[

9κ2

4α3γ2

]x=0

,

[9κ2

4α3γ2

]x=1

.

For further estimates we multiply (3.16) by φvtx, for φ (x) = 1 − 2x, to obtain

ˆ 1

0(vtttφvtx − αvtxxφvtx + βθttxφvtx)dx

=

ˆ 1

0φvtx(αttuxx − βttθx + 2αtvxx − 2βtθtx)dx.

3.2 Exponential stability in one-dimensional nonlinear system 79

This in turn gives

[αv2

tx

]x=0

+∣∣∣∣[αv2

tx

]x=1

∣∣∣∣≤ −2

ddt

ˆ 1

0vttφvtxdx + 2

ˆ 1

0v2

ttdx

−2ˆ 1

0βθttxφvtxdx + 2

ˆ 1

0αv2

txdx + R14, (3.71)

where

R14 = 2ˆ 1

0φvtx(αttuxx − βttθx + 2αtvxx − 2βtθtx)dx −

ˆ 1

0αxφv2

txdx. (3.72)

Similarly, multiplying (3.20) by φvttx to get

[αv2

ttx

]x=0

+∣∣∣∣[αv2

ttx

]x=1

∣∣∣∣ ≤ −2ddt

ˆ 1

0vtttφvttxdx

+2ˆ 1

0v2

tttdx − 2ˆ 1

0βθtttxφvttxdx + 2

ˆ 1

0αv2

ttxdx + R15 (3.73)

with

R15 =

ˆ 1

0φvttx

(αtttuxx − βtttθx + 3αttvxx + 3αtvtxx − 3βttθtx − 3βtθttx

)dx

+

ˆ 1

0αxφv2

ttxdx. (3.74)

Now, from (3.67) − (3.74) , we have

[3δαγ

vtxθx

]x=1

x=0+

[3δαγ

vttxθtx

]x=1

x=0+

[3καγ

vtxθtx

]x=1

x=0+

[3καγ

vttxθttx

]x=1

x=0

≤cε3

ˆ 1

0

(θ2

x + θ2tx + θ2

ttx

)dx + cε

ˆ 1

0

(θ2

xx + θ2txx + θ2

ttxx

)dx

−4εddt

ˆ 1

0(vttφvtx + vtttφvttx) dx + 4ε

ˆ 1

0

(v2

tt + v2ttt

)dx

+4εˆ 1

0

(αv2

tx + αv2ttx

)dx − 4ε

ˆ 1

0β (θttxφvtx + θtttxφvttx) dx

+2ε (R14 + R15) . (3.75)

80 Thermoelasticity of type III

Combining (3.37) , (3.49) , (3.75) we obtain for sufficiently small ε

ˆ 1

0

2α

(v2tx + v2

ttx)dx +5

12

ˆ 1

0(v2

xx + v2txx)dx +

ddt

G1 (t)

≤

ˆ 1

0(

3βαγ

+27δ2

α2γ2 +27γ2 +

3β2

4α2 )(θ2tx + θ2

ttx)dx +

ˆ 1

0

27κ2

α2γ2 (θ2ttx + θ2

tttx)dx

+4εˆ 1

0

(αv2

tx + αv2ttx

)dx − 4ε

ˆ 1

0β (θttxφvtx + θtttxφvttx) dx

+cε3

ˆ 1

0

(θ2

x + θ2tx + θ2

ttx + θ2tttx

)dx + cε

ˆ 1

0

(θ2

xx + θ2txx + θ2

ttxx

)dx

+4εˆ 1

0

(v2

tt + v2ttt

)dx + 2ε (R14 + R15) + R5 + R7, (3.76)

where

G1 (t) =

ˆ 1

0

1α

(vttxvtx + vtxvx) −ˆ 1

0

3αγ

(θtxvt + θttxvtt) dx

−

ˆ 1

0

3δβα2γ

(θxθtx + θxθttx) dx −ˆ 1

0

3δα2γ

(θxvtt + θtxvttt) dx

+

ˆ 1

0

3καγ

(θxvxx + θtxvtxx) dx + 4εddt

ˆ 1

0(vttφvtx + vtttφvttx) dx

We also have, from (3.4)

c1

ˆ 1

0(v2

tx + v2ttx)dx + c2

ˆ 1

0(v2

xx + v2txx)dx +

ddt

G1 (t)

≤ c3

ˆ 1

0(θ2

tx + θ2ttx + θ2

tttx)dx +c4

ε3

ˆ 1

0

(θ2

x + θ2tx + θ2

ttx + θ2tttx

)dx

+c5ε

ˆ 1

0

(θ2

xx + θ2txx + θ2

ttxx

)+ c6ε

ˆ 1

0

(v2

tx + v2ttx

)dx

+c7ε

ˆ 1

0

(v2

tt + v2ttt

)dx + 2ε (R14 + R15) + R5 + R7. (3.77)

3.2 Exponential stability in one-dimensional nonlinear system 81

Adding σ (3.58) to (3.77) we get

c1

ˆ 1

0(v2

tx + v2ttx)dx + c2

ˆ 1

0(v2

xx + v2txx)dx +

ddt

G2 (t) + σ

ˆ 1

0

(θ2

ttt + θ2tt

)dx

≤ c3

ˆ 1

0(θ2

tx + θ2ttx + θ2

tttx)dx +c4

ε3

ˆ 1

0

(θ2

x + θ2tx + θ2

ttx + θ2tttx

)dx

+c5ε

ˆ 1

0

(θ2

xx + θ2txx + θ2

ttxx

)+ c6ε

ˆ 1

0

(v2

tx + v2ttx

)dx

+c7ε

ˆ 1

0

(v2

tt + v2ttt

)dx + c8σ

ˆ 1

0

(θ2

tx + θ2ttx + θ2

txx + θ2ttxx

)dx

+32σ

ˆ 1

0γ2

(v2

tx + v2ttx

)dx + 2ε (R14 + R15) + R5 + R7 + σR10, (3.78)

where

G2 (t) = G1 (t) + 2σˆ 1

0δ (θtxθx + θttxθtx) dx.

Using (3.4) and the inequality below (3.51) :

ˆ 1

0v2

tttdx ≤ 2ˆ 1

0α2v2

txxdx + 2ˆ 1

0β2θ2

ttxdx

+2ˆ 1

0(αttuxxvttt − βttθxvttt + 2αtvxxvttt − 2βtθtxvttt)dx

≤ Cˆ 1

0v2

txxdx + Cˆ 1

0θ2

ttxdx

+2ˆ 1

0(αttuxxvttt − βttθxvttt + 2αtvxxvttt − 2βtθtxvttt)dx

By Poincare’s inequality, we have

c7ε

ˆ 1

0v2

ttt + v2ttdx ≤ c7C∗ε

ˆ 1

0v2

ttxdx + c7εCˆ 1

0v2

txxdx + c7εCˆ 1

0θ2

ttxdx

+2c7ε

ˆ 1

0(αttuxxvttt − βttθxvttt + 2αtvxxvttt − 2βtθtxvttt)dx

82 Thermoelasticity of type III

Inserting the last inequality in (3.78) we obtain

c1

ˆ 1

0(v2

tx + v2ttx)dx + c2

ˆ 1

0(v2

xx + v2txx)dx +

ddt

G2 (t) + σ

ˆ 1

0

(θ2

ttt + θ2tt

)dx

≤ c3

ˆ 1

0(θ2

tx + θ2tttx)dx + (c3 + c7εC)

ˆ 1

0θ2

ttxdx +c4

ε3

ˆ 1

0

(θ2

x + θ2tx + θ2

ttx + θ2tttx

)dx

+c5ε

ˆ 1

0

(θ2

xx + θ2txx + θ2

ttxx

)+ c6ε

ˆ 1

0

(v2

tx + v2ttx

)dx

+c7εC∗ˆ 1

0v2

ttxdx + c7Cεˆ 1

0v2

txxdx + c8σ

ˆ 1

0

(θ2

tx + θ2ttx + θ2

txx + θ2ttxx

)dx

+32σ

ˆ 1

0γ2

(v2

tx + v2ttx

)dx + c

15∑j=1

∣∣∣R j∣∣∣ ..

Consequently, (c1 −

32σγ2 − c6ε

)ˆ 1

0v2

txdx +

(c1 −

32σγ2 − c6ε − c7εC∗

)ˆ 1

0v2

ttxdx

+c2

ˆ 1

0v2

xxdx + (c2 − c7Cε)ˆ 1

0v2

txx)dx +ddt

G2 (t) + σ

ˆ 1

0

(θ2

ttt + θ2tt

)dx

≤cε3

ˆ 1

0

(θ2

tx + θ2ttx + θ2

tttx

)dx +

cε3

ˆ 1

0θ2

xdx + εc5

ˆ 1

0θ2

xxdx

+c (ε + σ)ˆ 1

0

(θ2

txx + θ2ttxx

)dx + c

15∑j=1

∣∣∣R j∣∣∣ (3.79)

where c denotes a generic positive constant.

Using (3.64) and (3.65) we get

ˆ 1

0

(θ2

txx + θ2ttxx

)dx ≤ C

ˆ 1

0

(θ2

tt + θ2ttt + θ2

xx + v2tx + v2

ttx

)dx + c

15∑j=1

∣∣∣R j∣∣∣ (3.80)

Inserting (3.80) in (3.79) , using Poincare’s inequality and choosing ε and σ small, we obtain

cˆ 1

0(v2

tx + v2ttx + v2

xx + v2txx)dx +

ddt

G2 (t) + σ

ˆ 1

0

(θ2

ttt + θ2tt

)dx

≤cε3

ˆ 1

0

(θ2

tx + θ2ttx + θ2

tttx

)dx +

cε3

ˆ 1

0θ2

xdx + c (ε + σ)ˆ 1

0θ2

xxdx

+c15∑j=1

∣∣∣R j∣∣∣ . (3.81)

3.2 Exponential stability in one-dimensional nonlinear system 83

By using (3.4) , (3.60) , (3.54) and Poincare’s inequality we get

ˆ 1

0θ2

xdx ≤ cˆ 1

0θ2

txdx + cε4ˆ 1

0

(v2

ttx + θ2tx

)dx +

cε4

ˆ 1

0θ2

txdx

−ddt

ˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx + c

15∑j=1

∣∣∣R j∣∣∣ . (3.82)

Inserting (3.82) in (3.81) and choosing ε small, we get

cˆ 1

0(v2

tx + v2ttx)dx + c

ˆ 1

0(v2

xx + v2txx)dx +

σ

ˆ 1

0

(θ2

ttt + θ2tt

)dx + ε

ˆ 1

0θ2

xxdx +ddt

G2 (t)

≤cε3

ˆ 1

0

(θ2

tx + θ2ttx + θ2

tttx

)dx −

cε3

ddt

ˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx

+c (ε + σ)ˆ 1

0θ2

xxdx + c15∑j=1

∣∣∣R j∣∣∣ . (3.83)

¿From (3.62) and (3.67) we have

ˆ 1

0θ2

xxdx ≤ −12

ddt

ˆ 1

0

κ

δθ2

xxdx +cε

ˆ 1

0θ2

xdx + ε

ˆ 1

0θ2

ttxdx

+1

12

ˆ 1

0v2

txxdx +cε3

ˆ 1

0θ2

xdx + cεˆ 1

0θ2

xxdx

+∣∣∣∣[εαv2

tx

]x=0

∣∣∣∣ +∣∣∣∣[εαv2

tx

]x=1

∣∣∣∣ + R12.

Using (3.71) and choosing ε small, we have

cˆ 1

0θ2

xxdx ≤ −12

ddt

ˆ 1

0

κ

δθ2

xxdx +cε3

ˆ 1

0θ2

xdx + ε

ˆ 1

0θ2

ttxdx

+112

ˆ 1

0v2

txxdx − 2εddt

ˆ 1

0vttφvtxdx + 2ε

ˆ 1

0v2

ttdx

−2εˆ 1

0βθttxφvtxdx + 2ε

ˆ 1

0αv2

txdx + c15∑j=1

∣∣∣R j∣∣∣ .

84 Thermoelasticity of type III

By (3.4), Young’s inequality and Poincare’s inequality we get

cˆ 1

0θ2

xxdx ≤ −12

ddt

ˆ 1

0

κ

δθ2

xxdx +cε3

ˆ 1

0θ2

xdx + εcˆ 1

0θ2

ttxdx

+112

ˆ 1

0v2

txxdx − 2εddt

ˆ 1

0vttφvtxdx + 2ε

ˆ 1

0v2

ttxdx

+εcˆ 1

0v2

txdx + c15∑j=1

∣∣∣R j∣∣∣ . (3.84)

Multiplying (3.84) by c (ε + σ) and inserting the result in (3.83) we obtain for small values of εand σ

cˆ 1

0(v2

tx + v2ttx)dx + c

ˆ 1

0(v2

xx + v2txx)dx +

σ

ˆ 1

0

(θ2

ttt + θ2tt

)dx + ε

ˆ 1

0θ2

xxdx +ddt

G3 (t)

≤cε3

ˆ 1

0

(θ2

tx + θ2ttx + θ2

tttx

)dx + c

15∑j=1

∣∣∣R j∣∣∣ , (3.85)

with

G3 (t) = G2 (t) +cε3

ˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx + cε

ˆ 1

0

κ

δθ2

xxdx + cεˆ 1

0vttφvtxdx

(Here the term cε3

´ 10 θ

2xdx can be analyzed as above).

For the Lyapunov function we take, for M and N large,

F (t) = M (E1 (t) + E2 (t) + E3 (t)) + G2 (t)

+Nˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx +

ε

2

ˆ 1

0

κ

δθ2

xxdx.

Then we conclude from (3.52), (3.54) and (3.85)

ddt

F (t) ≤ −dH (t) + c15∑j=1

∣∣∣R j∣∣∣ (3.86)

with a constant d > 0.Lemma 3.2.1 For T > 0 and sufficiently large M, there exist constants c1, c2 > 0, for which

c1H (t) ≤ F (t) ≤ c2H (t) , ∀t ≤ T. (3.87)

3.2 Exponential stability in one-dimensional nonlinear system 85

Proof : Let

F1 (t) = F (t) −ε

2

ˆ 1

0

κ

δθ2

xxdx,

then

F1 (t) − ME (t) = Nˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx + G2 (t)

By using (3.4) and Young’s inequality we can write∣∣∣∣∣∣Nˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx

∣∣∣∣∣∣ ≤ CE (t) . (3.88)

We determine now a constant C > 0 such that

|G2 (t)| ≤ C(1 + σ (t) + σ2 (t)

)E (t) .

By using (3.3)1 and (3.10) the termˆ 1

0

3καγ

(θxvxx + θtxvtxx) dx,

in G2 (t) , can be estimated as follows:∣∣∣∣∣∣ˆ 1

0

3καγθxvxxdx

∣∣∣∣∣∣ =

∣∣∣∣∣∣ˆ 1

0(

3κα2γ

θxvtt +3κβα2γ

θxθtx −3καt

α3γθxvt−

3καtβ

α3γθ2

x +3κβt

α2γθ2

x)dx

∣∣∣∣∣∣ ≤ C (1 + σ (t)) E (t) . (3.89)

By using equation (3.4) , (3.16) , and

αtt = αuxuxv2x + αuxvtx + αθθθ

2t + αθθtt,

we have ∣∣∣∣∣∣ˆ 1

0

3καγθtxvtxxdx

∣∣∣∣∣∣ =

ˆ 1

0(

3κα2γ

θtxvttt +3κβα2γ

θtxθttx −3καtt

α3γθtxvt −

3κβαtt

α3γθtxθx +

3κβtt

α2γθtxθx −

6καt

α2γvxxθtx +

6κβt

α2γθ2

tx)dx (3.90)

≤ C(1 + σ (t) + σ2 (t)

)E (t) .

Similarly, all the remaining terms of G2 (t) are estimated by CE (t) . Therefore, after carrying allcalculations, we arrive at∣∣∣∣∣∣G2 (t) + N

ˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx

∣∣∣∣∣∣ ≤ C(1 + σ (t) + σ2 (t)

)E (t) .

86 Thermoelasticity of type III

By the Sobolev embedding theorem we can easily get∣∣∣∣∣∣G2 (t) + Nˆ 1

0

(1δθtθ +

κ

2δθ2

x +γ

δvxθ

)dx

∣∣∣∣∣∣ ≤ CT E (t) .

where CT depends on sup0≤x≤1σ (t) + σ2 (t). This implies

(M −C) E (t) ≤ F1 (t) ≤ (M + C) ;

hence, by choosing M large enough, we have

C∗E (t) ≤ F1 (t) ≤ C∗∗E(t). (3.91)

This yields

C∗E (t) +12

ˆ 1

0

κ

δθ2

xxdx ≤ F (t) ≤ C∗∗E(t) +12

ˆ 1

0

κ

δθ2

xxdx (3.92)

consequentlyc1H (t) ≤ F (t) ≤ c2H (t) . (3.93)

Lemma 3.2.2 Λ (t) is equivalent to H (t) ; that is ∃ C3, C4 > 0 such that, ∀ t ≥ 0 we have

C3H (t) ≤ Λ (t) ≤ C4(1 + σ (t) + σ2 (t) + σ3 (t) + σ4 (t)

)H (t) . (3.94)

The proof of this lemma is similar to the lemma 2.2.2.Lemma 3.2.3 ∣∣∣∣∣∣∣∣

15∑j=1

R j

∣∣∣∣∣∣∣∣ ≤(σ (t) + σ2 (t) + σ3 (t)

)Λ (t) (3.95)

Proof : The proof is also similar to the one of lemma 3.2.1. We only consider those of highestorder; like

´ 10 γαtttvtttuxxdx for instance. We first compute

αttt = αuxuxuxv3x + αuxuxθθtv2

x + 3αuxuxvxvtx + αuxθθtvtx + αuxvttx

+αθθuxvxθ2t + αθθθθ

3t + 3αθθθtθtt + αθuxvxθtt + αθθttt,

soˆ 1

0γαtttvtttuxxdx =

ˆ 1

0γαtttvttt

(1α

vt +β

αθt

)dx

≤ C(σ (t) + σ2 (t) + σ3 (t)

)Λ (t) ,

and by the same method we obtainˆ 1

0γβtttvtttθxdx ≤ C

(σ (t) + σ2 (t) + σ3 (t)

)Λ (t) ,

3.2 Exponential stability in one-dimensional nonlinear system 87

ˆ 1

02βκtθtttθttxxdx ≤ Cσ (t)

(ˆ 1

0θ2

tttdx +

ˆ 1

0θ2

ttxxdx),

andˆ 1

02βκtθtttθttxxdx ≤ Cσ (t)

(ˆ 1

0(θ2

tttdx + θ2ttx + v2

ttx)dx)

+ Cσ (t) R13

≤ C(σ (t) + σ3 (t)

)Λ (t) .

After carrying all calculations, the proof of the lemma is completed.

To finish the proof of the theorem, we use (3.86) , (3.87) , (3.94), (3.95) to obtain

dF (t)dt

≤ −dH(t) + c∣∣∣∣ 15∑

j=1

R j

∣∣∣∣ (3.96)

≤ −dH(t) + c5(σ (t) + σ2 (t) + σ3 (t)

)Λ (t) .

≤ −dH(t) + Γ(σ (t) + σ2 (t) + σ3 (t)

) (1 + σ (t) + σ2 (t) + σ3 (t) + σ4 (t)

)H (t) .

So by choosing Λ0 small enough such that

Γ(σ (0) + σ2 (0) + σ3 (0)

) (1 + σ (0) + σ2 (0) + σ3 (0) + σ4 (0)

)<

d2

we obtain, from (3.96) ,

dF (t)dt

≤−d2

H (t) ≤−d2c1

F (t) , ∀t ≤ t0 ≤ T (3.97)

by virtue of lemma 3.2.1. Direct integration then leads to

F (t) ≤ F (0) e−7t, ∀t ≤ t0 ≤ T (3.98)

Since F(t) ≤ F(0) we then use the fact that σ (t) is majorized by F(t), hence (3.98) is extendedbeyond t0. By repeating the same procedure, taking η, in (3.10), even smaller if necessary, andusing the continuity of F, (3.98) is established for all t ≥ 0. The proof of the theorem is completed.

88 Thermoelasticity of type III

3.3 Exponential stability in multi-dimensional nonlinear system with

internal damping

In this section, we consider the following multi-dimensional system of thermoelasticity of type III

utt − µ4u − (λ + µ)∇ (divu) + aut + β∇θ = 0, in [0,∞) ×Ω,

θtt − δ4θ + γdivutt − κ4θt = 0, in [0,∞) ×Ω,

u (0, .) = u0, ut (0, .) = u1, θ (0, .) = θ0, θt (0, .) = θ1, in Ω,

u = θ = 0, on (0,∞) × ∂Ω.

(3.99)

and prove an exponential stability result. This problem is motivated by a similar one in the classicalthermoelasticity treated by Jiang and Racke [30].

If v = ut denote the velocity field, (3.99) becomes

vtt − µ∆v − (λ + µ)∇ (divv) + avt + β∇θt = 0, in [0,∞) ×Ω,

θtt − δ∆θ + γdivvt − κ∆θt = 0, in [0,∞) ×Ω,

v (0, .) = u1, vt (0, .) = µ∆u0 + (λ + µ)∇ (divu0) − au1 − β∇θ1, in Ω,

θ (0, .) = θ0, θt (0, .) = θ1,

v = θ = 0, on (0,∞) × ∂Ω

(3.100)

The associated energy is defined by

E (t) = E (v, θ) =12

ˆΩ

(γv2

t + µγ |∇v|2 + (λ + µ) γ (divv)2 + βθ2t + δβ |∇θ|2

)dx (3.101)

Theorem 3.3.1 There exist positive constants C and η independent of t, such that

E (t) ≤ Ce−ηt, ∀t ≥ 0. (3.102)

Proof. By multiplication of equations (3.100)1 by γvt, (3.100)2 by βθt respectively, integrationover Ω, using integration by parts and the boundary conditions, we get

ddt

E (t) = −aˆ

Ω

v2t dx − βκ

ˆΩ

|∇θt|2 dx ≤ 0, ∀t ≥ 0. (3.103)

Next, for ε > 0 we define

L (t) = E (t) + εβ

ˆΩ

θθtdx + εγ

ˆΩ

vvtdx

+εκβ

2

ˆΩ

|∇θ|2 dx + εβγ

ˆΩ

divvθdx. (3.104)

3.3 Exponential stability in multi-dimensional nonlinear system with internal damping89

Lemma 3.3.1 For ε small enough, L (t) and E (t) are equivalent; i.e. there exist C1,C2 > 0 suchthat

C1L (t) ≤ E (t) ≤ C2L (t) . (3.105)

Proof. By using Young’s inequality and Poincare’s inequality we obtain∣∣∣∣∣ˆΩ

θθtdx∣∣∣∣∣ ≤ C∗

2

ˆΩ

|∇θ|2 dx +12

ˆΩ

θ2t dx∣∣∣∣∣ˆ

Ω

vvtdx∣∣∣∣∣ ≤ C∗

2

ˆΩ

|∇v|2 dx +12

ˆΩ

v2t dx∣∣∣∣∣ˆ

Ω

divvθdx∣∣∣∣∣ ≤ C∗

2

ˆΩ

|∇θ|2 dx +12

ˆΩ

(divv)2 dx,

where C∗ is the Poincare’s constant.

We denote by

G (t) = βε

ˆΩ

θθtdx + εγ

ˆΩ

vvtdx +εκβ

2

ˆΩ

|∇θ|2 dx + εβγ

ˆΩ

divvθdx.

We have|G (t)| ≤ εCE (t) ,

where C is a positive constant. Consequently,

|L (t) − E (t)| ≤ εCE (t) .

This yields(1 − εC) E (t) ≤ L (t) ≤ (εC + 1) E (t) .

By choosing ε small, our conclusion follows.

To continue the proof of the theorem, we differentiate (3.104) and use equations (3.100) so weobtain

L′ (t) = −βκ

ˆΩ

‖∇θt‖2dx − a

ˆΩ

v2t dx + εβ

ˆΩ

θ2t dx − εβδ

ˆΩ

‖∇θ‖2dx

+εγβ

ˆΩ

∇vθtdx + εγβ

ˆΩ

divvθtdx + εγ

ˆΩ

v2t dx − εγµ

ˆΩ

‖∇v‖2dx

−εγ (λ + µ)ˆ

Ω

(divv)2 dx − εγaˆ

Ω

vvtdx. (3.106)

By using Young’s inequality we have the following estimates∣∣∣∣∣ˆΩ

vvtdx∣∣∣∣∣ ≤ 1

4c1

ˆΩ

v2t dx + c1

ˆΩ

v2dx, c1 > 0.

90 Thermoelasticity of type III

ˆΩ

∇vθtdx ≤ c2

ˆΩ

|∇v|2 dx +1

4c2

ˆΩ

θ2t dx, c2 > 0. (3.107)

ˆΩ

divvθtdx ≤ c3

ˆΩ

(divv)2dx +1

4c3

ˆΩ

θ2t dx, c3 > 0.

Taking into account the estimates (3.107), we have from (3.106)

L′ (t) ≤ −βκ

ˆΩ

|∇θt|2 dx − a

ˆΩ

v2t dx + εβ

ˆΩ

θ2t dx − εβδ

ˆΩ

|∇θ|2 dx

+εγβc2

ˆΩ

|∇v|2 dx + εγβ

(1

4c2+

14c3

) ˆΩ

θ2t dx (3.108)

+εγ

ˆΩ

v2t dx − εγµ

ˆΩ

|∇v|2 dx − εγ (λ + µ)ˆ

Ω

(divv)2 dx

+εγa4c1

ˆΩ

v2t dx + εγac1

ˆΩ

v2dx + εγβc3

ˆΩ

(divv)2 dx

Therefore,

L′ (t) ≤ −ME (t) +

(Mγ

2− a +

εγa4c1

+ εγ

) ˆΩ

v2t dx

+β

(ε +

M2

+εγ

4c2+εγ

4c3

)ˆΩ

θ2t dx

+

[µγ

( M2− ε

)+ εγβc2

]ˆΩ

|∇v|2 dx + εγac1

ˆΩ

v2dx (3.109)

+δβ( M

2− ε

)ˆΩ

|∇θ|2 dx − βκˆ

Ω

|∇θt|2 dx

+γ (λ + µ)( M

2− ε

) ˆΩ

(divv)2 dx + εγβc3

ˆΩ

(divv)2 dx

where M is positive constant. Using Poincare’s inequality, we haveˆ

Ω

θ2t dx ≤ C

ˆΩ

|∇θt|2 dx,

ˆΩ

v2dx ≤ Cˆ

Ω

|∇v|2 dx,

and choosing, c1 =µ

4aC , c2 =µ4β , c3 =

λ+µ2β ,M = ε and

ε ≤ Min

a

γ(

32 +

µ4c1

) , κ

C(

32 +

γ4c2 +

γ4c3

)the estimate (3.109) yields

L′ (t) ≤ −εE (t) . (3.110)

3.3 Exponential stability in multi-dimensional nonlinear system with internal damping91

Using (3.105) , the estimate (3.110) becomes

L′ (t) ≤ −ηL (t) . (3.111)

A simple integration of (3.111) then leads to

L (t) ≤ L (0) e−ηt. (3.112)

Consequently, using (3.105) again, we infer from (3.112) that

E (t) ≤ Ce−ηt.

This completes the proof.

92 Thermoelasticity of type III

3.4 Boundary stabilization for a multi-dimensional nonlinear system

The purpose of this section is to obtain the exponential stability for amulti-dimensional system of thermoelasticity type III subject to a feedback on a part of the bound-ary. The technique we use follows closely the method of Komornik and Zuazua [36] (see also [89])with the necessary modifications imposed by the nature of our problem.

We consider the problem

utt − µ4u − (µ + λ)∇(divu) + β∇θ = 0, in [0,∞) ×Ω,

cθtt − κ∆θ + βdivutt − δ∆θt = 0, in [0,∞) ×Ω,

u(., 0) = u0, ut(., 0) = u1, θ(., 0) = θ0, θt(., 0) = θ1, on Ω,

u = 0, x ∈ Γ0, t ≥ 0,

µ∂u∂ν

+ (µ + λ)(divu)ν = −aut, x ∈ Γ1, t ≥ 0,

θ = 0, x ∈ ∂Ω , t ≥ 0,

(3.113)

for c, δ, κ, β, λ, µ positive constants, Ω a bounded domain of Rn, with a smooth boundary ∂Ω. suchthat Γ0,Γ1 is a partition of ∂Ω, with meas (Γ1) > 0, ν is the outward normal to ∂Ω, u = u(x, t) ∈Rn is the displacement vector, and θ = θ(x, t) is the difference temperature.

In order to establish our result we shall make the following assumption(H) There exists x0 in Rn, for which m(x) = x − x0 satisfies

m(x).ν ≤ 0, ∀x ∈ Γ0,

and there exists η > 0 such thatm(x).ν > η on Γ1.

If v = ut denotes the velocity field, we obtain from (3.113)

vtt − µ4v − (µ + λ)∇(divv) + β∇θt = 0 (3.114)

cθtt − k∆θ + βdivvt − δ∆θt = 0 (3.115)

with initial and boundary conditions

v(., 0) = u1, vt(., 0) = (λ + µ)∇ (divu0) + µ4u0 − β∇θ0,

θ(., 0) = θ0, θt(., 0) = θ1, x ∈ Ω,

v (x, t) = 0, x ∈ Γ0, t ≥ 0,

µ∂v∂ν

+ (µ + λ)(divv)ν = −avt, x ∈ Γ1, t ≥ 0,

θ = 0, x ∈ ∂Ω , t ≥ 0.

(3.116)

3.4 Boundary stabilization for a multi-dimensional nonlinear system93

The associated energy is defined by

E (t) =12

[ˆΩ

|vt|2 dx + c

ˆΩ

θ2t dx + µ

ˆΩ

|∇v|2 dx]

+12

[(µ + λ)

ˆΩ

(divv)2dx + kˆ

Ω

|∇θ|2 dx]. (3.117)

A multiplication of equations (3.114) by vt and (3.115) by θt respectively, and integration over Ω,using integration by parts and the boundary conditions, gives

ddt

E (t) = −aˆ

Γ1

|vt|2 dσ − δ

ˆΩ

|∇θt|2 dx ≤ 0, ∀t ≥ 0. (3.118)

Theorem 3.4.1 Assume that (H) holds. Then there exist positive constants C and ζ independent oft such that

E(t) ≤ Ce−ζt. (3.119)

Proof. The proof is achieved by constructing a Lyaponov functional H (t) satisfying dH(t)dt ≤ 0, with

the aid of the energy multiplier method. Let

H (t) = E (t) + εF (t) (3.120)

for ε > 0 to be chosen later and

F (t) =

ˆΩ

[M (x) .vt + (n − 1)v.vt] dx + cˆ

Ω

θθtdx + β

ˆΩ

θdivvdx, (3.121)

where M =(M1,M2, ...,Mn

)T and Mi = 2m (x) .∇vi, i = 1, ..n.

It is easy to show that for ε small enough, H (t) and E (t) are equivalent.(See, lemma 3.4.1below)

By differentiating (3.121) we obtain

F′

(t) =

ˆΩ

[M (x) .vtt + M

′

.vt]

dx + (n − 1)ˆ

Ω

[|vt|

2 + v.vtt]

dx

+cˆ

Ω

θθttdx + cˆ

Ω

θ2t dx + β

ˆΩ

θtdivvdx + β

ˆΩ

θdivvtdx.

=

ˆΩ

[M (x) . ((µ + λ)∇(divv) + µ∆v − β∇θt) + M

′

.vt]

dx

+ (n − 1)ˆ

Ω

[|vt|

2 + v.vtt]

dx + cˆ

Ω

θθttdx (3.122)

+cˆ

Ω

θ2t dx + β

ˆΩ

θtdivvdx + β

ˆΩ

θdivvtdx.

94 Thermoelasticity of type III

By exploiting Green’s formula and the boundary conditions we estimate

I1 =

ˆΩ

M (x) .∇divvdx =

n∑i=1

ˆΩ

Mi (x)∂

∂xi(divv) dx

=

n∑i=1

ˆΩ

(2m (x) .∇vi)∂

∂xi(divv) dx

=

n∑i, j=1

ˆΩ

(2m j (x)

∂

∂x jvi

)∂

∂xi(divv) dx (3.123)

= −2n∑

i, j=1

ˆΩ

∂

∂xi

(m j (x)

∂

∂x jvi

)(divv) dx

+2n∑

i, j=1

ˆΓ

(m j (x)

∂

∂x jvi

)(divv) νidσ

= −2n∑

i=1

ˆΩ

∂

∂xivi (divv) dx − 2

n∑i, j=1

ˆΩ

m j (x)∂

∂xi

(∂

∂x jvi

)(divv) dx

+2n∑

i, j=1

ˆΓ

(m j (x)

∂

∂x jvi

)(divv) νidσ.

But,

−2n∑

i, j=1

ˆΩ

m j (x)∂

∂xi

(∂

∂x jvi

)(divv) dx

= 2n∑

i, j=1

ˆΩ

∂

∂x j

(m j (x) (divv)

) ∂

∂xividx − 2

n∑i, j=1

ˆΓ

(m j (x) divv

) ∂

∂xiviν jdσ

= 2nn∑

i=1

ˆΩ

divv∂

∂xividx + 2

n∑i, j=1

ˆΩ

m j (x)∂

∂x j(divv)

∂

∂xividx

−2n∑

i, j=1

ˆΓ

(m j (x) divv

) ∂

∂xiviν jdσ (3.124)

= nˆ

Ω

|divv|2 dx −n∑

i, j=1

ˆΓ

(m j (x) divv

) ∂

∂xiviν jdσ.

Substituting (3.124) in (3.123), we have

I1 = (n − 2)ˆ

Ω

|divv|2 dx −ˆ

Γ

(m (x) .ν) |divv|2 dσ

+2n∑

i=1

ˆΓ

(m (x) .∇vi) divvνidσ. (3.125)

3.4 Boundary stabilization for a multi-dimensional nonlinear system95

Note that

−

ˆΓ

(m (x) .ν) (divv)2 dσ = −

ˆΓ0

(m (x) .ν) |divν|2 dσ (3.126)

−

ˆΓ1

(m (x) .ν) (divν)2 dσ.

Since∂vi

∂xk= νk

∂vi

∂νon Γ0,

we obtain

2n∑

i=1

ˆΓ

(m (x) .∇vi) divvνidσ = 2n∑

i, j=1

ˆΓ0

(m j (x)

∂

∂x jvi

)(divv) νidσ

+ 2n∑

i=1

ˆΓ1

(m (x) .∇vi) divvνidσ (3.127)

= 2n∑

i, j=1

ˆΓ0

(m j (x) ν j

∂vi

∂ν

)(divv) νidσ + 2

n∑i=1

ˆΓ1

(m (x) .∇vi) divvνidσ

= 2ˆ

Γ0

(m (x) .ν) (divv)2 dσ + 2n∑

i=1

ˆΓ1

(m (x) .∇vi) divvνidσ.

We then exploit the definitions of Γ0, Γ1 and the boundary conditions to get, from (3.125)− (3.127)

I1 ≤ (n − 2)ˆ

Ω

|divv|2 dx − ηˆ

Γ1

|divv|2 dσ

+2n∑

i=1

ˆΓ1

(m (x) .∇vi) divvνidσ. (3.128)

By the same technique, we have

I2 =

ˆΩ

M (x) .∆vdx

= (n − 2)ˆ

Ω

|∇v|2 dx −ˆ

Γ

|∇v|2 (m (x) .ν) dσ + 2n∑

i=1

ˆΓ

(m (x) .∇vi) (ν.∇vi) dσ

≤ (n − 2)ˆ

Ω

|∇v|2 dx + 2n∑

i=1

ˆΓ1

∂vi

∂ν(m (x) .∇vi) dσ − η

ˆΓ1

|∇v|2 dσ. (3.129)

96 Thermoelasticity of type III

Also,

I3 =

ˆΩ

[M (x) . (β∇θt)

]dx = β

ˆΩ

n∑i=1

Mi (x)∂θt

∂xidx

= 2βn∑

i, j=1

ˆΩ

m j (x)∂vi

∂x j

∂θt

∂xidx = −2β

n∑i, j=1

ˆΩ

∂

∂xi

(m j (x)

∂vi

∂x j

)θtdx

= −2βn∑

i=1

ˆΩ

∂vi

∂xiθtdx + 2βn

n∑i=1

ˆΩ

∂vi

∂xiθtdx + 2β

n∑i, j=1

ˆΩ

∂vi

∂xim j (x)

∂θt

∂x jdx.

Therefore,

I3 = −2β (n − 1)ˆ

Ω

divvθtdx + 2βˆ

Ω

divv (m (x) .∇θt) dx. (3.130)

The next estimate is

I4 =

ˆΩ

M′

(x) .vtdx =

ˆΩ

n∑i=1

M′

i (x) v′

idx = 2n∑

i, j=1

ˆΩ

(m j (x)

∂

∂x jv′

i

)v′

idx

= −2n∑

i, j=1

ˆΩ

∂

∂x j

(m j (x) v

′

i

)v′

idx + 2n∑

i, j=1

ˆΓ

m j (x)(v′

i

)2ν jdσ

= −nˆ

Ω

|vt|2 dx +

ˆΓ

|vt|2 (m (x) .ν) dσ, (3.131)

The last integral is

I5 =

ˆΩ

v.vttdx =

ˆΩ

v. (µ∆v + (λ + µ)∇divv − β∇θt) dx

= − (λ + µ)ˆ

Ω

(divv)2 dx + (λ + µ)ˆ

Γ

((divv) ν) .vdσ (3.132)

−µ

ˆΩ

|∇v|2 + µ

ˆΓ

∂v∂ν.vdσ − β

ˆΩ

v.∇θtdx

= − (λ + µ)ˆ

Ω

(divv)2 dx − µˆ

Ω

|∇v|2 − aˆ

Γ1

v.vtdσ + β

ˆΩ

θtdivvdx

A combination of (3.128) − (3.132) and using the fact that

F′

(t) = (λ + µ) I1 + µI2 − I3 + I4 + (n − 1) I5 + (n − 1)ˆ

Ω

|vt|2 dx

+cˆ

Ω

θθttdx + cˆ

Ω

θ2t dx + β

ˆΩ

θtdivvdx + β

ˆΩ

θdivvtdx,

3.4 Boundary stabilization for a multi-dimensional nonlinear system97

we get

F′

(t) ≤ − (λ + µ)ˆ

Ω

(divv)2 dx −ˆ

Ω

|vt|2 dx + β(3n − 2)

ˆΩ

θtdivvdx

−2βˆ

Ω

divv (m (x) .∇θt) dx − µˆ

Ω

|∇v|2 dx − µηˆ

Γ1

|∇v|2 dσ (3.133)

+

ˆΓ1

|vt|2 (m (x) .ν) dσ − η(λ + µ)

ˆΓ1

(divv)2 dσ − a (n − 1)ˆ

Γ1

v.vtdσ

−2an∑

i=1

ˆΓ1

(m (x) .∇vi) v′

idσ + cˆ

Ω

θ2t dx − k

ˆΩ

|∇θ|2 dx − δˆ

Ω

∇θ∇θtdx.

Using Young’s inequality we arrive at

ˆΓ1

v.vtdσ ≤γ1

2

ˆΓ1

|v|2 dσ +1

2γ1

ˆΓ1

|vt|2 dσ

≤ C21γ1

2

ˆΩ

|∇v|2 dx +1

2γ1

ˆΓ1

|vt|2 dσ (3.134)

∣∣∣∣∣∣∣2an∑

i=1

ˆΓ1

(m (x) .∇vi) v′

idσ

∣∣∣∣∣∣∣ ≤ aR2γ2

ˆΓ1

|∇v|2 dσ +aγ2

ˆΓ1

|vt|2 dσ (3.135)

β(3n − 2)ˆ

Ω

θtdivvdx ≤β(3n − 2)γ3

2

ˆΩ

(divv)2 dx

+β(3n − 2)C1

2γ3

ˆΩ

|∇θt|2 dx (3.136)

∣∣∣∣∣2βˆΩ

divv (m (x) .∇θt) dx∣∣∣∣∣ ≤ βγ4

ˆΩ

(divv)2 dx +βR2

γ4

ˆΩ

|∇θt|2 dx (3.137)

∣∣∣∣∣δˆΩ

∇θ∇θtdx∣∣∣∣∣ ≤ δγ5

2

ˆΩ

|∇θ|2 dx +δ

2γ5

ˆΩ

|∇θt|2 dx, (3.138)

where C1 is the Poincare constant.

98 Thermoelasticity of type III

Inserting the estimates (3.134) − (3.138) into (3.133) we obtain

F′

(t) ≤ −

[(λ + µ) −

β(3n − 2)γ3

2− βγ4

]ˆΩ

(divv)2 dx

−

[µ −C2

1a (n − 1) γ1

2

]ˆΩ

|∇v|2 dx −ˆ

Ω

|vt|2 dx

+

[βC1 (3n − 2)

2γ3+βR2

γ4+

δ

2γ5

]ˆΩ

|∇θt|2 dx

+

[R +

aγ2

+a (n − 1)

2γ1

]ˆΓ1

|vt|2 dσ +

(aR2γ2 − µη

) ˆΓ1

|∇v|2 dσ

+cˆ

Ω

|θt|2 dx +

(δγ5

2− k

) ˆΩ

|∇θ|2 dx. (3.139)

At this point we choose γ1 − γ5 so small that (3.139) becomes

F′

(t) ≤ −k1

ˆΩ

(divv)2 dx − k2

ˆΩ

|∇v|2 dx −ˆ

Ω

|vt|2 dx + k3

ˆΩ

|∇θt|2 dx

+k4

ˆΓ1

|vt|2 dσ + c

ˆΩ

|θt|2 dx − k5

ˆΩ

|∇θ|2 dx (3.140)

where k1, k2, k3, k4 and k5 are strictly positive constants.

Combining (3.118) , (3.120) and (3.140) we conclude

H′

(t) ≤ −εk1

ˆΩ

(divv)2 dx − εk2

ˆΩ

|∇v|2 dx − εˆ

Ω

|vt|2 dx − (δ − εk3)

ˆΩ

|∇θt|2 dx

− (a − k4ε)ˆ

Γ1

|vt|2 dσ + εc

ˆΩ

|θt|2 dx

−εk5

ˆΩ

|∇θ|2 dx. (3.141)

We choose ε small so that (3.141) becomes

H′

(t) ≤ −η1

ˆΩ

(divv)2 dx − η2

ˆΩ

|∇v|2 dx − η3

ˆΩ

|∇θt|2 dx

−η4

ˆΩ

|∇θ|2 dx − εˆ

Ω

|vt|2 dx

≤ −η5

[ˆΩ

(divv)2 dx +

ˆΩ

|∇v|2 dx +

ˆΩ

|θt|2 dx +

ˆΩ

|∇θ|2 dx +

ˆΩ

|vt|2 dx

]for η1 − η5 are strictly positive constants. Therefore

H′

(t) ≤ −µE (t) ≤ −ζH (t) . (3.142)

3.4 Boundary stabilization for a multi-dimensional nonlinear system99

A simple integration of (3.42) then leads to

H (t) ≤ H (0) e−ζt.

By using the fact that E and H are equivalent, the assertion of the theorem is established.Lemma 3.4.1 There exist two positive constants C1and C2 such that

C1E (t) ≤ H (t) ≤ C2E (t) .

Proof: We have from (3.120)

|H (t) − E (t)| = |εF (t)|

=

∣∣∣∣∣εˆΩ

[M (x) .vt + (n − 1)v.vt] dx + cˆ

Ω

θθtdx + β

ˆΩ

θdivvdx∣∣∣∣∣ .

Let

ρ1 (t) =

∣∣∣∣∣ˆΩ

[M (x) .vt + (n − 1)v.vt] dx∣∣∣∣∣

=

∣∣∣∣∣∣∣n∑

i=1

ˆΩ

[Mi (x) v

′

i + (n − 1)vvt]

dx

∣∣∣∣∣∣∣≤ (n − 1)

ˆΩ

v2dx + nˆ

Ω

v2t dx + R2

ˆΩ

|∇v|2 dx

and

ρ2 (t) =

∣∣∣∣∣c ˆΩ

θθtdx + β

ˆΩ

θdivvdx∣∣∣∣∣

≤c2

ˆΩ

θ2t dx +

c + β

2

ˆΩ

θ2dx +β

2

ˆΩ

(divv)2 dx.

By using Poincare’s inequality we obtain

|εF (t)| ≤ εCE (t)

where C is positive constant. It then flows that

(1 − εC) E (t) ≤ H (t) ≤ (1 + εC) E (t) .

By choosing ε small enough we arrive at

C1E (t) ≤ H (t) ≤ C2E (t) .

100 Thermoelasticity of type III

Chapter 4

Appendix

In this appendix we give two important results on the global nonexistence results for special classof nonlinear wave equations with nonlinear damping and source terms as well as for the nonlin-early damped multidimensional Boussinesq equation. The main tool, used in both cases, followscarefully the techniques of Georgiev and Todorova [19]. More precisely, in section 4.1 we considerthe nonlinear wave equation

utt − ∆ut − div(|∇u|α−2 ∇u

)− div

(|∇ut|

β−2 ∇ut)

+ a |ut|m−2 ut = b |u|p−2 u,

where a, b > 0. The above problem is associated with initial and Dirichlet-boundary conditions. Weprove, under suitable conditions on α, β,m, p, that any weak solution with negative initial energyblows up in finite time. This improves a result by Yang [85], who requires that the initial energy besufficiently negative and relates the blow-up time to the size of Ω.

In section 4.2 we consider a multi-dimensional nonlinear initial-boundary value problem re-lated to the Boussinesq equation and prove a global nonexistence result. This work improves anearlier one by Gmira and Guedda [20].

102 Appendix

4.1 Global non-existence of solutions of a class of wave equations withnonlinear damping and source terms

4.1.1 Introduction

In this section we are concerned with the following initial boundary value problem .utt − ∆ut − div

(|∇u|α−2 ∇u

)− div

(|∇ut|

β−2 ∇ut)

+ a |ut|m−2 ut = b |u|p−2 u, x ∈ Ω, t > 0

u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ Ω

u (x, t) = 0, x ∈ ∂Ω, t > 0(4.1)

where a, b > 0, α, β, m, p > 2, and Ω is a bounded domain of Rn (n ≥ 1), with a smooth boundary∂Ω.

Equation (4.1) appears in the models of nonlinear viscoelasticity ( See [2] and [85]). It alsocan be considered as a system governing the longitudinal motion of a viscoelastic configurationobeying a nonlinear Voight model ( See [85] and [34]).

In the absence of viscosity and strong damping, equation (4.1) becomes

utt − div(|∇u|α−2 ∇u

)+ a |ut|

m−2 ut = b |u|p−2 u, x ∈ Ω, t > 0. (4.2)

For b = 0, it is well known that the damping term assures global existence and decay of the solutionenergy for arbitrary initial data (see [37], [3]).

For a = 0, the source term causes finite time blow up of solutions with negative initial energyif p > maxα,m (see [3] and [31]).

The interaction between the damping and the source terms was first considered by Levine [41]and [45] in the linear damping case (α = m = 2). He showed that solutions with negative initialenergy blow up in finite time. Georgiev and Todorova [19] extended Levine’s result to the nonlineardamping case (m > 2). In their work, the authors considered (4.2) with α = 2 and introduced amethod different than the one known as the concavity method. They determined suitable relationsbetween m and p, for which there is global existence or alternatively finite time blow up. Precisely;they showed that solutions with negative energy continue to exist globally ’in time’ if m ≥ p andblow up in finite time if p > m and the initial energy is sufficiently negative. This result was latergeneralized to an abstract setting and to unbounded domains by Levine and Serrin [43] and Levineand Park [42]. In these papers, the authors showed that no solution with negative energy can beextended to [0, ∞) if p > m and proved several noncontinuation theorems. This generalizationallowed them also to apply their result to quasilinear situations (α > 2), of which the problem in[19] is a particular case. Vitillaro [84] combined the arguments in [19] and [43] and extended theseresults to situations where the damping is nonlinear and the solution has positive initial energy.Similar results have also been established by Todorova [80, 81] for different Cauchy problems.

4.1 Global non-existence of solutions of a class of wave equations with nonlinear dampingand source terms 103

In [85], Yang studied (4.1) and proved a blow up result under the condition p > maxα,m,α > β, and the initial energy is sufficiently negative (see condition ii, theorem 2.1 of [85]). In factthis condition made it clear that there exists a certain relation between the blow-up time and |Ω|( see remark 2 of [85]). We should note here that (4.1) corresponds to equation (5) of [85] butthe same conclusions hold for equation (1) of the same paper, under suitable conditions, stated intheorem 2.3 of [85].

In this work we show that any solution of (4.1), with negative initial energy, blows up in finitetime if p > maxα,m, α > β. Therefore our result improves the one of [85]. Our technique ofproof follows closely the argument of [51] with the modifications needed for our problem.

4.1.2 Blow up

In order to state and prove our result, we introduce the following function space

Z = L∞([0,T ); W1,α

0 (Ω))∩W1,∞

([0,T ); L2(Ω)

)∩W1,β

([0,T ); W1,β

0 (Ω))∩W1,m (

[0,T ); Lm(Ω)),

for T > 0 and the energy functional

E (t) =12

ˆΩ

u2t dx +

1α

ˆΩ

‖∇u‖αdx −bp

ˆΩ

|u|p dx. (4.3)

Theorem 4.1.1 Assume that α, β,m, p ≥ 2 such that β < α, and maxm, α < p < rα, where rα isthe Sobolev critical exponent of W1,α

0 (Ω). Assume further that

E (0) < 0. (4.4)

Then the solution of (4.1) blows up in finite time.Remark 4.1.1 We remind that rα = nα/(n − α) if n > α, rα > α if n = α, and rα = ∞ if n < αProof. We suppose that the solution exists for all time and we reach to a contradiction. For thispurpose we multiply equation (4.1) by ut and integrate over Ω to get

E′

(t) = −

ˆΩ

|∇ut|2 dx −

ˆΩ

|∇ut|β dx − a

ˆΩ

|ut|m dx ≤ 0 (4.5)

for any regular solution. This remains valid for u ∈ Z by density argument. Hence E (t) ≤ E (0) ,∀t ≥ 0.

By setting H (t) = −E (t), we get

0 < H (0) ≤ H (t) ≤bp

ˆΩ

|u|p dx, ∀ t ≥ 0. (4.6)

We then defineL (t) = H1−σ (t) + ε

ˆΩ

uutdx (4.7)

104 Appendix

for ε small to be chosen later and

0 < σ ≤ min(α − 2

p,

α − β

p(β − 1),

p − mp(m − 1)

,α − 2

2α

). (4.8)

Our goal is to show that L (t) satisfies a differential inequality of the form

L′

(t) ≥ ξLq (t) , q > 1.

This, of course, will lead to a blow up in finite time.

By taking a derivative of (4.7) we obtain

L′

(t) = (1 − σ) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx + ε

ˆΩ

uuttdx. (4.9)

By using equation (4.1), the estimate (4.9) gives

L′

(t) = (1 − σ) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx

−ε

ˆΩ

∇u∇utdx − εˆ

Ω

|∇u|α dx

−ε

ˆΩ

|∇ut|β−2 ∇ut∇udx (4.10)

−aεˆ

Ω

|ut|m−2 utudx + bε

ˆΩ

|u|p dx.

We then exploit Young’s inequality to get

ˆΩ

|ut|m−2 utudx ≤

δm

m

ˆΩ

|u|m dx +

m − 1m

δ−m/(m−1)ˆ

Ω

|ut|m dx. (4.11)

ˆΩ

∇u∇utdx ≤1

4µ

ˆΩ

|∇u|2 dx + µ

ˆΩ

|∇ut|2 dx. (4.12)

ˆΩ

|∇ut|β−1 ∇udx ≤

λβ

β − 1

ˆΩ

|∇u|β dx (4.13)

+β − 1β

λ−β/(β−1)ˆ

Ω

|∇ut|β dx.

4.1 Global non-existence of solutions of a class of wave equations with nonlinear dampingand source terms 105

A substitution of (4.11) - (4.13) in (4.10) yields

L′

(t) ≥ (1 − σ) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx

−ε

4µ

ˆΩ

|∇u|2 dx − µεˆ

Ω

|∇ut|2 dx

−ε

ˆΩ

|∇u|α dx − ελβ

β

ˆΩ

|∇u|β dx

−εβ − 1β

λ−β/(β−1)ˆ

Ω

|∇ut|β dx (4.14)

+bεˆ

Ω

|u|p dx − aεδm

m

ˆΩ

|u|m dx

−aεm − 1

mδ−m/(m−1)

ˆΩ

|ut|m dx.

Therefore by choosing δ, µ, λ so that

δ−m/(m−1) = M1H−σ (t) ,µ = M2H−σ (t) ,λ−β/(β−1) = M3H−σ (t) ,

for M1, M2, and M3 to be specified later, and using (4.14) we arrive at

L′

(t) ≥ (1 − σ) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx

−ε

4M2Hσ (t)

ˆΩ

|∇u|2 dx − εˆ

Ω

|∇u|α dx

−εM−(β−1)

3

βHσ(β−1) (t)

ˆΩ

|∇u|β dx (4.15)

−aεm

M−(m−1)1 Hσ(m−1) (t)

ˆΩ

|u|m dx + bεˆ

Ω

|u|p dx

−

[εM2

ˆΩ

|∇ut|2 dx +

β − 1β

M3

ˆΩ

|∇ut|β dx

+am − 1

mM1

ˆΩ

|ut|m dx

]H−σ (t)

106 Appendix

If M = M2 + (β − 1)M3/β + (m − 1)M1/m then (4.15) takes the form

L′

(t) ≥ ((1 − σ) − εM) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx

−ε

4M2Hσ (t)

ˆΩ

|∇u|2 dx − εˆ

Ω

|∇u|α dx

−εM−(β−1)

3

βHσ(β−1) (t)

ˆΩ

|∇u|β dx (4.16)

−aεm

M−(m−1)1 Hσ(m−1) (t)

ˆΩ

|u|m dx + bεˆ

Ω

|u|p dx.

We then use the embedding Lp (Ω) → Lm (Ω) and (4.6) to get

Hσ(m−1) (t)ˆ

Ω

|u|m dx ≤(

bp

)σ(m−1) (ˆΩ

|u|p dx)m+σp(m−1)

p

. (4.17)

We also exploit the inequalityˆ

Ω

|∇u|2 dx ≤ C(ˆ

Ω

|∇u|α dx)2/α

,

the embedding W1,α0 (Ω) → Lp (Ω) , and (4.4) to obtain

Hσ (t)ˆ

Ω

|∇u|2 dx ≤ C(

bp

)σ (ˆΩ

|∇u|α dx) pσ+2

α

. (4.18)

Since α > β we have ˆΩ

|∇u|β dx ≤ C(ˆ

Ω

|∇u|α dx) βα

;

consequently

Hσ(β−1) (t)ˆ

Ω

|∇u|β dx ≤ C(

bp

)σ(β−1) (ˆΩ

|∇u|α dx) pσ(β−1)+β

α

, (4.19)

where C is a constant depending on Ω only. By using (4.8) and

zν ≤ z + 1 ≤(1 +

1a

)(z + a) , ∀z ≥ 0, 0 < ν ≤ 1, a ≥ 0, (4.20)

we have the following(ˆΩ

|u|p dx)m+σp(m−1)

p

≤

(ˆΩ

|∇u|α dx)m+σp(m−1)

α

≤ d(ˆ

Ω

|∇u|α dx + H (0)), (4.21)

≤ d(ˆ

Ω

|∇u|α dx + H (t))∀ t ≥ 0.

4.1 Global non-existence of solutions of a class of wave equations with nonlinear dampingand source terms 107

(ˆΩ

|∇u|α dx) pσ+2

α

≤ d(ˆ

Ω

|∇u|α dx + H (t)), ∀ t ≥ 0. (4.22)

(ˆΩ

|∇u|α dx) pσ(β−1)+β

α

≤ d(ˆ

Ω

|∇u|α dx + H (t)), ∀ t ≥ 0. (4.23)

where d = 1 + 1/H (0) . Inserting the estimates (4.17) - (4.19) and (4.21) - (4.23) into (4.16) we get

L′

(t) ≥ ((1 − σ) − εM) H−σ (t) H′

(t)

+kH (t) +

(ε +

k2

) ˆΩ

u2t dx

−εC2

M2

(ˆΩ

|∇u|α dx + H (t))− ε

ˆΩ

|∇u|α dx

−εC3

Mβ−13

(ˆΩ

|∇u|α dx + H (t))

+kα

ˆΩ

|∇u|α dx (4.24)

−εC1

Mm−11

(ˆΩ

|∇u|α dx + H (t))

+ b(ε −

kp

)ˆΩ

|u|p dx,

for some constant k and

C1 =aCdm

(bp

)σ(m−1)

,C2 =Cd4

(bp

)σ,C3 =

Cdβ

(bp

)σ(β−1)

.

Putting k = εp, we arrive at

L′

(t) ≥ ((1 − σ) − εM) H−σ (t) H′

(t) + ε

(p + 2

2

)ˆΩ

u2t dx

+ε

p −C2

M2−

C3

Mβ−13

−C1

Mm−11

H (t) (4.25)

+ε

pα−

C2

M2−

C3

Mβ−13

−C1

Mm−11

− 1

ˆΩ

|∇u|α dx.

At this point, we choose M1, M2, M3 large enough so that

L′

(t) ≥ ((1 − σ) − εM) H−σ (t) H′

(t)

+γε

[H (t) +

ˆΩ

u2t dx +

ˆΩ

|∇u|α dx], (4.26)

where γ is a positive constant (this is possible since p > α). By choosing ε < (1 − σ) /M so that

L (0) = H1−σ (0) + ε

ˆΩ

u0u1dx > 0

108 Appendix

we obtainL (t) ≥ L (0) > 0, ∀ t ≥ 0.

and

L′

(t) ≥ γε[H (t) +

ˆΩ

u2t dx +

ˆΩ

|∇u|α dx]. (4.27)

Next, it is clear that

L1

1−σ (t) ≤ 21

1−σ

H (t) + ε1

1−σ

(ˆΩ

utudx) 1

1−σ .

By the Cauchy-Schwarz inequality and the embedding of the Lp (Ω) spaces we have∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ ≤ (ˆ

Ω

u2dx)1/2 (ˆ

Ω

u2t dx

)1/2

≤ C(ˆ

Ω

|u|α dx)1/α (ˆ

Ω

u2t dx

)1/2

,

which implies ∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 1

1−σ

≤ C(ˆ

Ω

|u|α dx) 1

(1−σ)α(ˆ

Ω

u2t dx

) 12(1−σ)

.

Also Young’s inequality gives∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 1

1−σ

≤ C

(ˆΩ

|u|α dx) µ

(1−σ)α

+

(ˆΩ

u2t dx

) θ2(1−σ)

for 1/µ + 1/θ = 1. We take θ = 2 (1 − σ) , (hence µ =

2(1−σ)(1−2σ) ) to get

∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 1

1−σ

≤ C

(ˆΩ

|u|α dx) 2

(1−2σ)α

+

ˆΩ

u2t dx

.By Poincare’s inequality, we obtain∣∣∣∣∣ˆ

Ω

utudx∣∣∣∣∣ 1

1−σ

≤ C

(ˆΩ

|∇u|α dx) 2

(1−2σ)α

+

ˆΩ

u2t dx

.By using (4.8) and (4.20) we deduce(ˆ

Ω

|∇u|α dx) 2

(1−2σ)α

≤ (1 +1

H (0))(ˆ

Ω

|∇u|α dx + H (t)).

Therefore, ∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 1

1−σ

≤ C[H (t) +

ˆΩ

‖∇u‖αdx +

ˆΩ

u2t dx

], ∀t ≥ 0.

4.1 Global non-existence of solutions of a class of wave equations with nonlinear dampingand source terms 109

consequently

L1

1−σ (t) ≤ Γ

[H (t) +

ˆΩ

|∇u|α dx +

ˆΩ

u2t dx

](4.28)

where Γ is positive constant. A combination of (4.27) and (4.28) , thus, yields

L′

(t) ≥ ξL1

1−σ (t) , ∀t ≥ 0. (4.29)

Integration of (4.29) over (0, t) gives

Lσ

1−σ (t) ≥1

L−σ

1−σ (0) − ξσ(1−σ) t

;

hence L (t) blow up in time

T ∗ ≤1 − σ

ξσLσ

1−σ (0). (4.30)

Remark 4.1.2 The time estimate (4.30) shows that the larger L(0) is the quicker the blow up takesplace.Remark 4.1.3 In (4.6) we only require that H(0) > 0, Unlike Yang [85], where it is required thatH(0) > A, a constant depending on the size of Ω. See condition ii), theorem 2.1 of [85].Remark 4.1.4 If we consider

utt − ∆ut − div (σ(∇u)∇u) − div (β(∇u)∇ut) + f (ut) = g(u), x ∈ Ω, t > 0

with the initial and boundary conditions of (4.1) we can establish a similar blow up result under thegrowth conditions of theorem 2.3 of [85] on f , g, σ and β.

110 Appendix

4.2 Global nonexistence result for the nonlinearly damped multi-dimensionalBoussinesq equation

4.2.1 Introduction

The Boussinesq equation

utt + αuxxxx − uxx = β(u2

)xx, x ∈ R, t > 0 (4.31)

where α, β > 0,was first derived by Boussinesq [5] in 1872 and since then so many mathematicianshave studied it and used it to model real world problems such as the propagation of long waves onshallow water and oscillations of nonlinear elastic beams. Varlamov [82] considered the dampedequation of the form

utt − 2butxx + αuxxxx − uxx = β(u2

)xx, x ∈ (0, π), t > 0, (4.32)

for small initial data and constructed, for the case α > b2, the solution in the form of Fourierseries. He also showed that, on [0,T ),T < ∞, the solution of (4.32) is obtained by letting b goto zero. In 2001, Varlamov [83] improved his earlier result by considering the three-dimensionalversion of (4.32) in the unit ball and used the eigenfunctions of the Laplace operator to constructsolutions. He examined the problem, for homogeneous boundary conditions and small initial data,and obtained global mild solutions in appropriate Sobolev spaces. He also addressed the issueof the uniqueness and the long-time behavior of the solution. Lai and Wu [38] considered thefollowing more generalized equation

utt − auttxx − 2butxx + cuxxxx − uxx = −p2u + β(u2

)xx, x ∈ R, t > 0, (4.33)

where a, b, c > 0, p , 0 and β is a real number. They used the Fourier transform and the perturba-tion theory to establish the well-posedness of global solutions to small initial data for the Cauchyproblem. The same techniques have been applied by Lai et al [75] to establish a global existenceand an exponential decay results for an initial-boundary value problem related to (4.33).

For the nonexistence, we mention the result of Levine and Sleeman [44], in which the authorsconsidered an initial boundary value problem related to the equation

utt = 3uxxxx + uxx − 12(u2

)xx

(4.34)

and showed that, under appropriate conditions for the initial data, no positive weak or classical solu-tion can exist for all time. Recently Bayrack and Can [4] studied the behavior of a one-dimensionalriser vibrating due to effects of waves and current involving linear dissipation. Precisely, they

4.2 Global nonexistence result for the nonlinearly damped multi-dimensional Boussinesqequation

111

looked into the following problem

utt + αut + 2βuxxxx − 2 [(ax + b) ux]x +β3

(u3

x

)xxx

−[(ax + b) u3

x

]x− β

(u2

xxux)

x= f (u) , (x, t) ∈ (0, 1) × (0,T ) ,

u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ (0, 1) ,

u (0, t) = u (1, t) = 0, uxx (0, t) = uxx (1, t) = 0, t ∈ (0,T ) .

(4.35)

and proved that, under suitable conditions on f and the initial data, all solutions of (4.35) blowup in finite time in the L2 space. To establish their result, the authors used the standard concavitymethod due to [41]. Gmira and Guedda [20] extended the result of [4] to the multi-dimensionalversion of the problem (4.35). So they considered

utt + ρ (x) ut + β∆2u − div (g (x)∇u) + Γ4(|∇u|2 ∆u

)−div

(h (x) |∇u|p−2 ∇u

)− Γdiv

((∆u)2 ∇u

)= f (u)

(4.36)

and established a nonexistence result, under suitable conditions on u0, u1, f , by using the ”mod-ified” concavity method introduced in [31]. The use of the latter method by Gmira and Gueddaallowed them to remove the condition of cooperative initial data

(´Ω

u0u1dx > 0)

imposed byBayrack and Can [4]. However, some conditions can be further weakened.

In this section we are concerned with the following nonlinearly damped problem

utt + ρ (x) |ut|m−2 ut + β∆2u − div (g (x)∇u) + Γ∆

(|∇u|2 ∆u

)−div

(h (x) |∇u|p−2 ∇u

)− Γdiv

((∆u)2 ∇u

)= |u|l−2 u, x ∈ Ω, t > 0,

u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ Ω,

u (x, t) =∂u∂η

(x, t) = 0, x ∈ ∂Ω, t > 0,

(4.37)

where Ω ⊂ Rn, n ≥ 1, is a bounded domain with sufficiently smooth boundary, η is the unit outernormal on ∂Ω, ρ ≥ 0, is a smooth bounded function given on Ω, g, h ∈ C1

(Ω,Rn

), p, l, m ≥

1, and β and Γ are nonnegative constants. In addition to allowing the damping to be nonlinear, weestablish a blow up result under weaker conditions than those required in [20], on the initial dataas well as the constants p, l, and m. To achieve our goal we exploit the method of Georgiev andTodorova [19] (see also [51]) . This work is divided into three subsections. In subsection two westate and demonstrate our main result. In subsection three, the linear damping (m = 2) case istreated.

112 Appendix

4.2.2 Blow up in the nonlinear damping case

In order to state and prove our result, we introduce the energy functional

E (t) =12

ˆΩ

u2t dx +

β

2

ˆΩ

(∆u)2 dx +12

ˆΩ

g |∇u|2 dx (4.38)

+Γ

2

ˆΩ

(∆u)2 |∇u|2 dx +1p

ˆΩ

h |∇u|p −1l

ˆΩ

|u|l dx

Theorem 4.2.1. Assume that m, p ≥ 1 and l > max 4,m, p. Assume further that

E (0) < 0. (4.39)

Then any classical solution of (4.37) blows up in finite time.Remark 4.2.1. In [2], the authors require that l > 2(4 + γ) > p, γ > 0, which is obviously strongerthan our requirements on both l and p. ( See (2.4) of [20]). Moreover in [20], l may depend on||ρ||∞ since γ does ( See (2.6) of [20] again).Remark 4.2.2. The result can be established for weak solution by means of density.Proof. A multiplication of equation (4.37) by ut and integration over Ω yields

E′

(t) = −

ˆΩ

ρ(x) |ut|m dx ≤ 0, (4.40)

By setting H (t) = −E (t), we get from (4.38) and (4.39),

0 < H (0) ≤ H (t) ≤1l

ˆΩ

|u|l dx, ∀ t ≥ 0. (4.41)

We then define

L (t) = H1−σ (t) + ε

ˆΩ

uutdx (4.42)

for ε small to be chosen later and

0 < σ ≤ min(

l − ml(m − 1)

,l − 2

2l

). (4.43)

By taking a derivative of (4.42) we obtain

L′

(t) = (1 − σ) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx + ε

ˆΩ

uuttdx (4.44)

4.2 Global nonexistence result for the nonlinearly damped multi-dimensional Boussinesqequation

113

By using (4.37), equation (4.44) becomes

L′

(t) = (1 − σ) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx (4.45)

−βε

ˆΩ

(∆u)2 dx − εˆ

Ω

ρ |ut|m−2 utudx

−ε

ˆΩ

g |∇u|2 dx − 2Γε

ˆΩ

|∇u|2 (∆u)2 dx

−ε

ˆΩ

h |∇u|p dx + ε

ˆΩ

|u|l dx.

We then exploit Young’s inequality to getˆΩ

ρ |ut|m−1 udx ≤

λm

m

ˆΩ

|u|m dx +m − 1

mλ−m/(m−1)

ˆΩ

∣∣∣ρ1/(m−1)ut∣∣∣m dx

≤λm

m

ˆΩ

|u|m dx + bm − 1

mλ−m/(m−1)

ˆΩ

ρ |ut|m dx.

where b = ‖ρ‖1/(m−1)∞ . This yields, by substitution in (4.45),

L′

(t) ≥ (1 − σ) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx

−βε

ˆΩ

(∆u)2 dx − ελm

m

ˆΩ

|u|m dx

−εbm − 1

mλ−m/(m−1)

ˆΩ

ρ |ut|m dx (4.46)

−ε

ˆΩ

g |∇u|2 dx − 2Γε

ˆΩ

|∇u|2 (∆u)2 dx

−ε

ˆΩ

h |∇u|p dx + ε

ˆΩ

|u|l dx.

Therefore, choosing λ so thatλ−m/(m−1) = MH−σ (t) ,

for large M to be specified later, and substituting in (4.46), we arrive at

L′

(t) ≥ (1 − σ) H−σ (t) H′

(t) + ε

ˆΩ

u2t dx

−βε

ˆΩ

(∆u)2 dx − εM−(m−1)

mHσ(m−1) (t)

ˆΩ

|u|m dx

−εbm − 1

mMH−σ (t) H

′

(t) − εˆ

Ω

g |∇u|2 dx

−2Γε

ˆΩ

|∇u|2 (∆u)2 dx − εˆ

Ω

h |∇u|p dx + ε

ˆΩ

|u|l dx.

114 Appendix

That is

L′

(t) ≥[(1 − σ) − εb

m − 1m

M]

H−σ (t) H′

(t) + ε

ˆΩ

u2t dx

−βε

ˆΩ

(∆u)2 dx − εM−(m−1)

mHσ(m−1) (t)

ˆΩ

|u|m dx

−ε

ˆΩ

g |∇u|2 dx − 2Γε

ˆΩ

|∇u|2 (∆u)2 dx (4.47)

−ε

ˆΩ

h |∇u|p dx + ε

ˆΩ

|u|l dx.

We then use the embedding Ll (Ω) → Lm (Ω) to get

ˆΩ

|u|m dx ≤ C(ˆ

Ω

|u|l dx)m/l

,

where C is a positive constant depending on Ω only. So we have, from (4.41) ,

Hσ(m−1) (t)ˆ

Ω

|u|m dx ≤Cl

(ˆΩ

|u|l dx)σ(m−1)+(m/l)

.

By using (4.43) and the inequality

zν ≤ z + 1 ≤(1 +

1a

)(z + a) , ∀z > 0, 0 < ν ≤ 1, a ≥ 0, (4.48)

we have the following(ˆΩ

|u|l dx)σ(m−1)+(m/l)

≤ d(ˆ

Ω

|u|l dx + H (0))

(4.49)

≤ d(ˆ

Ω

|u|l dx + H (t)), ∀ t ≥ 0,

where d = 1 + 1/H (0) . Inserting the estimate (4.49) into (4.47) we get

L′

(t) ≥[(1 − σ) − εb

m − 1m

M]

H−σ (t) H′

(t) + ε

ˆΩ

u2t dx

−βε

ˆΩ

(∆u)2 dx − εCdM−(m−1)

lm

(ˆΩ

|u|l dx + H (t))

−ε

ˆΩ

g |∇u|2 dx − 2Γε

ˆΩ

|∇u|2 (∆u)2 dx

−ε

ˆΩ

h |∇u|p dx + ε

ˆΩ

|u|l dx.

4.2 Global nonexistence result for the nonlinearly damped multi-dimensional Boussinesqequation

115

By using (4.38) and H (t) = −E (t), we can write, for some positive constant K,

L′

(t) ≥[(1 − σ) − εb

m − 1m

M]

H−σ (t) H′

(t) +

(K2

+ ε)ˆ

Ω

u2t dx (4.50)

+β(K

2− ε

)ˆΩ

(∆u)2 dx +

(K − εCd

M−(m−1)

lm

)H (t)

+

(ε −

Kl− εCd

M−(m−1)

lm

)ˆΩ

|u|l dx +

(K2− ε

)ˆΩ

g |∇u|2 dx

+Γ

(K2− 2ε

) ˆΩ

|∇u|2 (∆u)2 dx +

(Kp− ε

)ˆΩ

h |∇u|p dx

At this point we choose K = rε, for r = maxp, 4; hence (4.50) becomes

L′

(t) ≥[(1 − σ) − εb

m − 1m

M]

H−σ (t) H′

(t) + ε( r2

+ 1) ˆ

Ω

u2t dx

+βε( r2− 1

) ˆΩ

(∆u)2 dx + ε

(r −Cd

M−(m−1)

lm

)H (t) (4.51)

+ε

(1 −

rl−Cd

M−(m−1)

lm

)ˆΩ

|u|l dx + ε( r2− 1

) ˆΩ

g |∇u|2 dx

+Γε( r2− 2

)ˆΩ

|∇u|2 (∆u)2 dx + ε

(rp− 1

)ˆΩ

h |∇u|p dx

≥

[(1 − σ) − εb

m − 1m

M]

H−σ (t) H′

(t) + ε( r2

+ 1) ˆ

Ω

u2t dx

+ε

(r −Cd

M−(m−1)

lm

)H (t) + ε

(1 −

rl−Cd

M−(m−1)

lm

) ˆΩ

|u|l dx

We then choose M large enough so that

a1 = r −CdM−(m−1)

lm> 0, a2 = 1 −

rl−Cd

M−(m−1)

lm> 0.

Therefore (4.51) yields

L′

(t) ≥((1 − σ) − εb

m − 1m

M)

H−σ (t) H′

(t) (4.52)

+γε

[H (t) +

ˆΩ

u2t dx +

ˆΩ

|u|l dx],

whereγ = maxa1, a2,

r2

+ 1.

116 Appendix

Once M is fixed (hence γ), we choose ε so small that

(1 − σ) − εbm − 1

mM ≥ 0

and

L (0) = H1−σ (0) + ε

ˆΩ

u0u1dx > 0.

Therefore we have, from (4.52),

L′

(t) ≥ γε[H (t) +

ˆΩ

u2t dx +

ˆΩ

|u|l dx]. (4.53)

andL (t) ≥ L (0) > 0, ∀ t ≥ 0.

Next, it is clear that

L1

1−σ (t) ≤ 21

1−σ

H (t) + ε1

1−σ

(ˆΩ

utudx) 1

1−σ .

By the Cauchy-Schwarz inequality and the embedding of the Lp (Ω) spaces we have∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ ≤ (ˆ

Ω

u2dx)1/2 (ˆ

Ω

u2t dx

)1/2

≤ C(ˆ

Ω

|u|l dx)1/l (ˆ

Ω

u2t dx

)1/2

,

which implies ∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 1

1−σ

≤ C(ˆ

Ω

|u|l dx) 1

(1−σ)l(ˆ

Ω

u2t dx

) 12(1−σ)

.

Also Young’s inequality gives∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 1

1−σ

≤ C

(ˆΩ

|u|l dx) µ

(1−σ)l

+

(ˆΩ

u2t dx

) θ2(1−σ)

for 1/µ + 1/θ = 1. We take θ = 2 (1 − σ) , (hence µ =

2(1−σ)(1−2σ) ) to get

∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 1

1−σ

≤ C

(ˆΩ

|u|l dx) 2

(1−2σ)l

+

ˆΩ

u2t dx

.Again by using (4.43) and (4.48) we deduce, as in (4.49),

4.2 Global nonexistence result for the nonlinearly damped multi-dimensional Boussinesqequation

117

(ˆΩ

|u|l dx) 2

(1−2σ)l

≤ d(ˆ

Ω

|u|l dx + H (t)).

Therefore, ∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 1

1−σ

≤ C[H (t) +

ˆΩ

|u|l dx +

ˆΩ

u2t dx

], ∀t ≥ 0;

consequently

L1

1−σ (t) ≤ C1

[H (t) +

ˆΩ

|u|l dx +

ˆΩ

u2t dx

](4.54)

where C1 is positive constant. A combination of (4.53) and (4.54) , thus, yields

L′

(t) ≥ ξL1

1−σ (t) , ∀t ≥ 0. (4.55)

where ξ = γε/C1. Integration of (4.55) over (0, t) gives

Lσ

1−σ (t) ≥1

L−σ

1−σ (0) − ξσ(1−σ) t

;

hence L (t) blow up in a time

T ∗ ≤1 − σ

ξσLσ

1−σ (0). (4.56)

This completes the proof.

4.2.3 Blow up in the linear damping case

The next result improves the one given in Remark 3.3 of [20]. In fact we will show that theblow up for solutions of (4.37), when the damping is linear (m = 2), takes places if

´Ω

u0u1dx >−1

2

´Ωρu2

0dx instead of´

Ωu0u1dx > 0.

Theorem 4.2.2. Assume that p ≥ 1 and l > max 4, p. Assume further that

E (0) ≤ 0,ˆ

Ω

u0u1dx > −12

ˆΩ

ρu20dx. (4.57)

Then the solution of (4.37), for m = 2, blows up in finite time.

Proof. Let

L (t) =

ˆΩ

utudx +12

ˆΩ

ρ (x) u2dx. (4.58)

118 Appendix

By taking a derivative of (4.58) and using (4.37) we obtain

L′

(t) =

ˆΩ

u2t dx − β

ˆΩ

(∆u)2 dx

−

ˆΩ

g |∇u|2 dx − 2Γ

ˆΩ

|∇u|2 (∆u)2 dx

−

ˆΩ

h |∇u|p dx +

ˆΩ

|u|l dx. (4.59)

For some positive constant K, (4.59) takes the form

L′

(t) =

(K2

+ 1) ˆ

Ω

u2t dx + β

(K2− 1

) ˆΩ

(∆u)2 dx + KH (t)

+

(K2− 1

) ˆΩ

g |∇u|2 dx + Γ

(K2− 2

) ˆΩ

|∇u|2 (∆u)2 dx

+

(Kp− 1

) ˆΩ

h |∇u|p dx +

(1 −

Kl

) ˆΩ

|u|l dx. (4.60)

We choose K so that l > K > max 4, p, then we have from (4.60),

L′

(t) ≥(1 −

Kl

) [ˆΩ

(∆u)2 dx +

ˆΩ

u2t dx +

ˆΩ

|u|l dx + H (t)]. (4.61)

ThereforeL (t) ≥ L (0) =

ˆΩ

u0u1dx +12

ˆΩ

ρu20dx > 0, ∀ t ≥ 0.

Next, it is clear that

L2l

l+2 (t) ≤ C

(ˆ

Ω

utudx) 2l

l+2

+

(ˆΩ

ρu2dx) 2l

l+2 (4.62)

By the Cauchy-Schwarz inequality we have∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ ≤ (ˆ

Ω

u2dx)1/2 (ˆ

Ω

u2t dx

)1/2

≤ C(ˆ

Ω

|u|l dx)1/l (ˆ

Ω

u2t dx

)1/2

,

which implies ∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 2l

l+2

≤ C(ˆ

Ω

|u|l dx) 2

l+2(ˆ

Ω

u2t dx

) ll+2

.

With the help of Young’s inequality, we get∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 2l

l+2

≤ C

(ˆΩ

|u|l dx) 2µ

l+2

+

(ˆΩ

u2t dx

) θll+2

,

4.2 Global nonexistence result for the nonlinearly damped multi-dimensional Boussinesqequation

119

for 1/µ + 1/θ = 1.Choosing θ = l+2l , (hence µ = l+2

2 ), we obtain∣∣∣∣∣ˆΩ

utudx∣∣∣∣∣ 2l

l+2

≤ C[ˆ

Ω

|u|l dx +

ˆΩ

u2t dx

](4.63)

Similarly we also haveˆ

Ω

ρu2dx ≤(ˆ

Ω

(ρu)2 dx)1/2 (ˆ

Ω

u2dx)1/2

,

which gives (ˆΩ

ρu2dx) 2l

l+2

≤ C(ˆ

Ω

(ρu)l dx) 2

l+2(ˆ

Ω

u2dx) l

l+2

≤ C

(ˆΩ

|ρu|l dx) 2µ

l+2

+

(ˆΩ

u2dx) θl

l+2

With the same choice of θ and µ as in above and the use of the boundary conditions, we easilydeduce (ˆ

Ω

ρu2dx) 2l

l+2

≤ C[ˆ

Ω

|ρu|l dx +

ˆΩ

u2dx]

≤ C[‖ρ‖l∞

ˆΩ

|u|l dx +

ˆΩ

u2dx]

≤ C1

[ˆΩ

|u|l dx +

ˆΩ

u2dx]

≤ C2

[ˆΩ

|u|l dx +

ˆΩ

(∆u)2 dx]

(4.64)

Combining (4.63) and (4.64) , we have

L2l

l+2 (t) ≤ C[ˆ

Ω

(∆u)2 dx +

ˆΩ

u2t dx +

ˆΩ

|u|l dx]

(4.65)

≤ C[ˆ

Ω

(∆u)2 dx +

ˆΩ

u2t dx +

ˆΩ

|u|l dx + H (t)].

A combination of (4.61) and (4.65) leads to

L′

(t) ≥1C

(1 −

Kl

)L

2ll+2 (t) . (4.66)

A simple integration of (4.66) yields

L(l+2)/(l−2)(t) ≥1

L−(l+2)/(l−2)(0) − at, (4.67)

120 Appendix

where a = (1/C)[2l/(l − 2)][1 − (K/l)]. Therefore (4.67) shows that L blows up in finite time.Remark 4.2.3. The above result remains valid if |u|l−2u if replaced by f (u) provided that

u f (u) − KF(u) ≥ δ|u|l, δ > 0 F(u) =

ˆ u

0f (s)ds

Again this is a weaker requirement than (2.4) of [20].Remark 4.2.4. We do not require that u1 , 0 as in Theorem 2.1 of [20].

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