Non-compact global attractors for slowly non-dissipative equations seminars... · 2020-03-11 · Non-compact global attractors for slowly non-dissipative equations Juliana F. S. Pimentel

Non-compact global attractors for slowlynon-dissipative equations

Juliana F. S. Pimentel

Instituto Superior TécnicoUniversidade de Lisboa

June 2014

Joint work with C. Rocha

Juliana Pimentel (Universidade de Lisboa) Slowly non-dissipative systems June 2014 1 / 35

Outline

1 IntroductionDefinitions - dissipative and non-dissipative systemsMotivation - dissipative system and its global attractor

2 Problem statementSlowly non-dissipative systemMain objective

3 Preliminary resultsEquilibria at infinity

4 Main resultsNon-compact global attractorAssociated permutation

5 Strategy of the proof


Outline







Reaction-diffusion equation

1 Consider the scalar reaction-diffusion equation{ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,

(1)

where f : [0, π]× R2 → R is a C2 function.2 Equation (1) generates a local semiflow that is

I dissipative: global existence and ultimately boundedness for allsolutions;

I fast non-dissipative: at least one solution blows-up in finite-time; orI slowly non-dissipative: global existence for all solutions and there

exists a solution whose norm grows to infinity with time.




(1)








(1)








(1)








(1)






Dissipative system1 An example of sufficient conditions for (1) to be dissipative:

f (x ,u,0).u < 0, for |u| large enough,

and|f (x ,u,p)| ≤ c(1 + |p|γ),

with c > 0 and 0 ≤ γ < 2, uniformly for x and u in compact sets.2 There exists a global attractor A, i.e., a maximal compact invariant

set attracting each bounded set in the appropriate state space.3 Permutation σ associated to (1): assume hyperbolicity and let

E = {v1, ..., vn} be the set of equilibria of (1) with

v1(0) < v2(0) < ... < vn(0),

then σ is defined by

vσ(1)(π) < vσ(2)(π) < ... < vσ(n)(π).




and|f (x ,u,p)| ≤ c(1 + |p|γ),




v1(0) < v2(0) < ... < vn(0),


vσ(1)(π) < vσ(2)(π) < ... < vσ(n)(π).




and|f (x ,u,p)| ≤ c(1 + |p|γ),




v1(0) < v2(0) < ... < vn(0),


vσ(1)(π) < vσ(2)(π) < ... < vσ(n)(π).


Global attractor

Proposition (Henry(1981))

The global attractor A associated to the dissipative equation (1) iscomposed by the set of equilibria E and their heteroclinic connections.

Proposition (Fusco and Rocha(1991), Fiedler and Rocha(1996))The permutation σ ∈ S(n) determines which equilibria are connectedand which are not.


Outline







Slowly non-dissipative equation

Consider {ut = uxx + bu + g(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0.

(2)

Assumptions and notation:f (x ,u,ux) = bu + g(x ,u,ux)

b > 0 and g : [0, π]× R2 → R is a C2 functiong is bounded and g(x ,u,p) is globally Lipschitz in (u,p)X = L2([0, π]) with norm ‖ · ‖A = −∂xx − bIXα are the associated fractional power spaces

Xα := D((A + (b + 1)I)α).




(2)



Xα := D((A + (b + 1)I)α).




(2)



Xα := D((A + (b + 1)I)α).


Slowly non-dissipative systemIf b > 0 then equation (2) generates a slowly non-dissipative system.

1 Global existence for every initial condition.2 {ϕj}j∈N0 orthonormal basis in L2 and λj = j2 − b.3 Any solution u(t , x) is represented by u(t , x) =

∑∞j=0 uj(t)ϕj(x).

4 Apply the E0-projection to equation (2) and obtain

ddt

u0(t) = −λ0u0(t) + g0(t). (3)

5 The solution of (3) is given by

u0(t) = uh0(0)e

−λ0t +

∫ t

∞e−λ0(t−s)g0(s)ds.

6 Since λ0 = −b < 0, if take an initial condition u0 such thatuh

0(0) 6= 0 then u0(t)→∞ as t →∞.






ddt

u0(t) = −λ0u0(t) + g0(t). (3)


u0(t) = uh0(0)e

−λ0t +

∫ t



0(0) 6= 0 then u0(t)→∞ as t →∞.






ddt

u0(t) = −λ0u0(t) + g0(t). (3)


u0(t) = uh0(0)e

−λ0t +

∫ t



0(0) 6= 0 then u0(t)→∞ as t →∞.






ddt

u0(t) = −λ0u0(t) + g0(t). (3)


u0(t) = uh0(0)e

−λ0t +

∫ t



0(0) 6= 0 then u0(t)→∞ as t →∞.






ddt

u0(t) = −λ0u0(t) + g0(t). (3)


u0(t) = uh0(0)e

−λ0t +

∫ t



0(0) 6= 0 then u0(t)→∞ as t →∞.






ddt

u0(t) = −λ0u0(t) + g0(t). (3)


u0(t) = uh0(0)e

−λ0t +

∫ t



0(0) 6= 0 then u0(t)→∞ as t →∞.






ddt

u0(t) = −λ0u0(t) + g0(t). (3)


u0(t) = uh0(0)e

−λ0t +

∫ t



0(0) 6= 0 then u0(t)→∞ as t →∞.


Non-compact global attractor

1 Equation (2) possesses at least one solution blowing-up in infinitetime, i.e., a grow-up solution.

2 Non-compact global attractor : non-empty minimal set in the statespace Xα attracting all bounded sets of Xα.

3 Objective:I Obtain a decomposition for the non-compact global attractor.

F Ben-Gal (2010) - for g = g(u).I Describe the heteroclinic connections in terms of an associated

permutation.







permutation.







permutation.


Outline







Asymptotic behavior

There exists a Lyapunov functional for equation (2).Let u(t , ·) be a solution of equation (2). Then u(t , ·) eitherconverges to some (bounded) equilibrium as t goes to infinity oru(t , ·) is a grow-up solution.Obtain the limits of the unbounded solutions.


Asymptotic behavior



Asymptotic behavior



Grow-up solutions

Let {ϕj}j∈N be an orthonormal basis in L2([0, π]) comprised of theeigenfunctions of A = −∂xx − bI with Neumann boundaryconditions.If u(t , ·) is a grow-up solution then

u(t , ·)‖u(t , ·)‖

−→ ϕ±j (·) in L2,

with j ≤√

b and ϕ±j := ±ϕj .

The original orbit grows-up to infinity in the direction of ϕ±j .


Grow-up solutions


u(t , ·)‖u(t , ·)‖

−→ ϕ±j (·) in L2,

with j ≤√




Grow-up solutions


u(t , ·)‖u(t , ·)‖

−→ ϕ±j (·) in L2,

with j ≤√




Poincaré Projection

Figure: Projection of M = (u,1) ∈ Xα × {1}.

H = {(χ, z) ∈ Xα × R|〈χ, χ〉2α + z2 = 1, z ≥ 0}.Equilibrium points on He = {(χ,0) ∈ H}: ±Φj , for j ∈ N.

I Hell (2009).


Equilibria at infinity

Φ±j are equilibrium points on He.

We thus define objects Φ±,∞j at infinity as

P(Φ±,∞j ) = Φ±j , for j = 0,1, ..., [√

b],

and we refer to these as equilibria at infinity.


Equilibria at infinity

Φ±j are equilibrium points on He.

We thus define objects Φ±,∞j at infinity as

P(Φ±,∞j ) = Φ±j , for j = 0,1, ..., [√

b],

and we refer to these as equilibria at infinity.


Outline







Definitions - zero number and adjacency

1 Let u be a continuous function defined on [0, π]. We denote byz(u) the zero number of u, that is, the number of strict signchanges of u.

2 Let u, v ∈ Ef with z(v − u) = j and u(0) < v(0). We say that u andv are adjacent if there does not exist w ∈ Ef satisfying

u(0) < w(0) < v(0)

and z(v − w) = z(w − u) = j .


Definitions - zero number and adjacency

1 Let u be a continuous function defined on [0, π]. We denote byz(u) the zero number of u, that is, the number of strict signchanges of u.

2 Let u, v ∈ Ef with z(v − u) = j and u(0) < v(0). We say that u andv are adjacent if there does not exist w ∈ Ef satisfying

u(0) < w(0) < v(0)

and z(v − w) = z(w − u) = j .


The non-compact global attractor

Theorem

LetEc

f denote the set of (bounded) equilibria andE∞f denote the set of equilibria at infinity.

Then the non-compact global attractor Af of (2) is composed by theset of equilibria Ef = Ec

f ∪ E∞f and the heteroclinic connectionsbetween the equilibria,

Af = Ef ∪ {heteroclinic connections}.

Moreover, for any u, v ∈ Ef , there exists an orbit connecting them if,and only if, u and v are adjacent.


Permutation related to the non-dissipative equation

Theorem

There exists a permutation related to the non-dissipative equation (2)determining which equilibria in Ef are connected and which are not.


Outline







Permutation related to f

1 Generic assumption: all the equilibria are hyperbolic.2 Let Ec

f = {v1, v2, ..., vn} be the set of equilibria of equation (2),where

v1(0) < v2(0) < ... < vn(0).

We then define the permutation σf ∈ S(n) by

vσf (1)(π) < vσf (2)(π) < ... < vσf (n)(π).


Meander associated to σf

1 Consider the initial value problem

ux = v (4)vx = −f (x ,u,ux) = −bu − g(x ,u,ux)

u(0) = u0, v(0) = 0.

2 The set of solutions u = u(x ,u0), v = v(x ,u0) of (4) defines thetwo-dimensional manifold in [0, π]× R2

M = {(x ,u, v) ∈ [0, π]× R2|u = u(x ,u0), v = v(x ,u0) solve (4)}.

3 Let γf be the section curve of M at x = π

γf = {(x ,u(x ,u0), v(x ,u0))|u0 ∈ R, x = π}.


Example of a meander permutation

1 2 3 4 5 6 7 8 9

9 8 3 6 5 4 7 2 1

Figure: Meander related to σf = {9,8,3,6,5,4,7,2,1} ∈ S(9) considering thenonlinearity f (u) = 10u + 16 sin u.


Suspension

Definition

Let σ ∈ S(n) and k be a positive integer. We define the suspension σk

of the permutation σ as the permutation σk ∈ S(n + 2) which satisfies:(i) σk (j) = σ(j − 1) + 1, for j ∈ {2, ...,n + 1}; and(ii) if k is odd

σk (1) = 1 and σk (n + 2) = n + 2 ,

and if k is even

σk (1) = n + 2 and σk (n + 2) = 1 .


Example of suspension

Figure: k -th suspension of the meander γf for k = [√

10] + 1 = 4 andf (u) = 10u + 16 sin u.


Suspension of σf

1 The Morse indices of v1 and vn, i.e., the dimension of theircorresponding unstable manifolds, satisfy

i(v1) = i(vn) = [√

b] + 1 =: k .

2 Let σ1f ∈ S(n + 2k) be the k -th suspension of σf .


Sturm permutation method

1 σ1f is a Sturm permutation, i.e., there exists h realizing σ1

f .2 There exists a function h such that the dynamical system induced

by {ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0

(5)

I is dissipative andI the permutation σh defined by

wσh(1)(π) < wσh(2)(π) < ... < wσh(n+2k)(π),

where Eh = {w1, ...,wn+2k} is the set of equilibria of (5) ordered bytheir values at x = 0, satisfies

σh = σ1f .


Global attractor Ah

Proposition (Wolfrum (2002))Given any two equilibria u, v ∈ Eh, there exists an orbit connectingthem if, and only if, u and v are adjacent.


Morse indices and zero numbers given by σh

1 The Morse indices i(wm) are given in terms of σh by

i(wm) =m−1∑j=1

(−1)j+1 sign(σ−1h (j + 1)− σ−1

h (j)).

2 The zero numbers z(wl − wm) for 1 ≤ m < l ≤ n + 2k are given interms of σh by

z(wl − wm) =i(wm) +12[(−1)l sign(σ−1

h (l)− σ−1h (m))− 1]

+l−1∑

j=m+1

(−1)j sign(σ−1h (j)− σ−1

h (m)).

I Rocha (1985)I Fusco and Rocha (1991)I Rocha (1991)


Correspondence between the equilibria

1 Consider the set of equilibria for the dissipative equation

Eh = {w1, ...,wn+2k}.

2 Consider the set of equilibria for the non-dissipative equation

Ef = {Φ−,∞0 , ..., Φ−,∞k−1 , v1, ..., vn, Φ+,∞k−1 , ..., Φ

+,∞0 }.

3 We make the correspondence

Φ−,∞j ↔ wj+1 and Φ+,∞j ↔ wn+2k−j ,

for 0 ≤ j ≤ k − 1.4 Since h = f on B where B ⊂ Xα contains {v1, ..., vn},

wk+l = vl for l = 1, ...,n.


Morse indices for the dissipative equation

Lemma

The Morse indices of the equilibria satisfy(i) i(wk+l) = i(vl), for any 1 ≤ l ≤ n(ii) i(wj) = i(Φ−,∞j−1 ) = j − 1, for 1 ≤ j ≤ k

(iii) i(wn+2k−j) = i(Φ+,∞j ) = j , for 0 ≤ j ≤ k − 1

where i(Φ±,∞j ) := i(ϕ±j ) = j .


Zero numbers for the dissipative equation

LemmaFor any 1 ≤ l ≤ n, the following hold

z(wk+l − wj) = z(Φ−,∞j−1 − vl) = j − 1, for 1 ≤ j ≤ k

z(wn+2k−j − wk+l) = z(Φ+,∞j − vl) = j , for 0 ≤ j ≤ k − 1.

LemmaFor any 1 ≤ r < l ≤ n, the following holds

z(wk+l − wk+r ) = z(vl − vr ).


Heteroclinic connections in Af in terms of σh

1 σh determines the Morse indices and zero numbers of theequilibria in Eh.

2 The Morse indices and zero numbers are preserved by thecorrespondence.

3 The Sturm permutation σh explicitly determines the zero numbersand Morse indices of the equilibria in Ef .

4 The correspondence preserves the connections between theequilibria.

5 The permutation σh = σ1f determines, through the adjacency

notion, which equilibria in Ef = Ecf ∪ E∞f are connected and which

are not.









are not.









are not.









are not.









are not.


Cut-off functionConsider the non-dissipative equation{

ut = uxx + f (x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,

with f (x ,u,ux) = bu + g(x ,u,ux).We construct a cut-off function h such that

I h = f on B ⊂ Xα containing Ecf = {v1, ..., vn},

I h = cu, for c < 0, outside a larger set B, andI σ1

f is used to define h in the remaining portion of the domain.

Analyze the equation{ut = uxx + h(x ,u,ux), x ∈ [0, π]ux(t ,0) = ux(t , π) = 0,

using the theory for dissipative systems.















































Thank you for your attention.


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Non-compact global attractors for slowly non-dissipative equations seminars... · 2020-03-11 · Non-compact global attractors for slowly non-dissipative equations Juliana F. S. Pimentel