Pullback attractors for nonclassical diffusion equations

Pullback attractors for nonclassical diffusion equations Pullback attractors for nonclassical diffusion equations Pullback attractors for nonclassical diffusion equations Pullback attractors for nonclassical diffusion equations Pullback attractors for nonclassical diffusion equations Pullback attractors for nonclassical diffusion equations Pullback attractors for nonclassical diffusion equations Pullback attractors for nonclassical diffusion equations

MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

- TANG QUOC BAO

PULLBACK ATTRACTORS FOR NONCLASSICAL

2 Existence and upper semicontinuity of pullback attractors 132.1 Existence of pullback attractors 132.2 The upper semicontinuity of pullback attractors at ε = 0 30

Informat-Last, but not least, I am grateful to my family, my friends for theirencouragement, which helps me very much in completing the thesis.

Nonclassical diffusion equations arise as models to describe physical nonmena, such as non-Newtonian flows, soil mechanics, and heat conduc-tion (see e.g [1, 12, 18]) In the last few years, the existence and long-timebehavior of solutions to nonclassical difussion equations has attracted theattention of many mathematicians.

phe-Let us review some recent results on nonclassical diffusion equations.For autonomous case, that is the case g independent of time t, in [25], theauthor considered equation (0.1) in a bounded domain Ω with ε = 1, thetime-independent external force g ∈ L2(Ω) and the nonlinearity f satisfythe following conditions

lim sup

|s|→∞

f (s)

s < λ1, (0.2)where λ1 is the first eigenvalue of the operator −∆ in Ω with Dirichletcondition, and

|f0(s)| ≤ C(1 + |s|4), (0.3)and

|f (s)| ≤ C(1 + |s|γ), γ < 5 (0.4)

The growth condition (0.4) of f is usually called subcritical case Underconditions (0.2)-(0.4), the author proved the existence of a global attractor

in H01(Ω)

Also consider autonomous nonclassical diffusion equations in a boundeddomain, the authors in [22] assumed that the initial uτ ∈ H2(Ω) ∩ H01(Ω)and the nonlinearity f satisfies the subcritical growth condition Underthose assumptions, the authors proved that for each ε ∈ [0, 1], there exists

a global attractor Aε ⊂ H2(Ω) ∩ H01(Ω) for problem (0.1) Moreover, theyshowed that when ε → 0, Aε tends to A0 in the sense that

lim

ε→0distH1 (Ω)(Aε, A0) = 0 as ε → 0,where distH1 (Ω) is the Hausdorff semi-distance in H01(Ω)

For non-autonomous case, under a Sobolev growth condition of f , theauthors in [2] proved that there exists a pullback attractor Aε = {Aε(t) :

t ∈ R} ⊂ H01(Ω) of (0.1) for each ε ∈ [0, 1] and

lim

ε→0sup

t∈I

distL2 (Ω)(Aε(t), A0(t)) = 0 as ε → 0,

for any interval I ⊂ R

We refer the reader to [14, 25, 15, 24] for other results It is noticedthat all existing results are devoted in bounded domains The dynamic ofnonclassical diffusion equations in unbounded domains is not well under-stood In this thesis, we consider the existence and long-time behavior ofsolutions to problem (0.1) in the case of unbounded domains, the nonlinear-ity of polynomial type, and the unbounded external force g depending ontime t The main aims of this thesis are to prove the existence of pullbackattractors ˆAε = {Aε(t) : t ∈ R}, ε ∈ [0, 1], in H1(Rn) ∩ Lp(Rn) for problem(0.1) and to show the upper semicontinuity of ˆAε at ε = 0 The resultsobtained in this thesis has been accepted for publication in the journalCommunications on Pure and Applied Analysis [3]

The existence of a pullback attractor for problem (0.1) (on the entirespace Rn) in the case ε = 0 has been proved recently in [21]

In the case ε > 0, since equation (0.1) contains the term −ε∆ut, it

is different from the classical reaction-diffusion equation essentially For

example, the reaction-diffusion equation has some kind of ”regularity”, e.g.,although the initial datum only belongs to a weaker topology space, thesolution will belong to a stronger topology space with higher regularity.However, for problem (0.1) when ε > 0, because of −∆ut, if the initialdatum uτ belongs to H1(Rn) ∩ Lp(Rn), the solution u(t) with intial datumu(τ ) = uτ is always in H1(Rn) ∩ Lp(Rn) and has no higher regularity,which is similar to hyperbolic equations This brings some difficulty inestablishing the existence of pullback attractors for nonclassical diffusionequations On the other hand, notice that the domain Rn for (1.1) isunbounded, so Sobolev embeddings are no longer compact in this case Thisintroduces a major obstacle for examining the asymptotic compactness ofsolutions.

To try to overcome these difficulties, we combine the method of estimates [20] and the asymptotic a priori estimate method [11] to prove theasymptotic compactness of the corresponding process We first use thesemethods to prove the existence of an (H1(Rn) ∩ Lp(Rn), Lp(Rn))-pullbackattractor Then by verifying the condition (PDC) introduced in [7], weobtain the existence of a pullback attractor ˆAε in H1(Rn) ∩ Lp(Rn) Next,

tail-we study the continuous dependence on ε of solutions to problem (0.1)

as ε → 0 Hence using an abstract result derived recently by Carvalho

et al [5] and techniques similar to ones used in [2], we prove the uppersemicontinuity of pullback attractors ˆAε in L2(Rn) at ε = 0

The rest of the thesis is organized as follows In Chapter 1, we prove thewell-posedness of equation (0.1), that is, the existence and uniqueness ofsolutions In Chapter 2, the existence and upper semi-continuity of pullbackattractors are investigated In conclusion, we give some ways to extend theresults

Chapter 1

Existence and uniqueness of solutions

The aims of this Chapter is to prove the existence and uniqueness of lutions In Section 2.1, we give some hypothesis and a definition of weaksolutions The existence of weak solutions is investigated in Section 2.2

This thesis is concerned with the non-autonomous nonclassical diffusionequation

(

ut − ε∆ut − ∆u + f (x, u) + λu = g(t, x),u|t=τ = uτ, (1.1)where ε ∈ [0, 1] and λ > 0

To study problem (1.1), we assume the following conditions:

(H1) The initial datum uτ ∈ H1

(Rn), φ2 ∈ Lq

(Rn), q = p−1p , are nonnegativefunctions, α, β, γ are positive constants, |φ1(x)| ≤ C0 for all x ∈ Rn

For F (x, s) = R0sf (x, r)dr, we assume

γ1|u|p− φ3(x) ≤ F (x, u) ≤ γ2|u|p+ φ4(x),where γ1, γ2 > 0, φ3, φ4 ∈ L1

(Rn) are nonnegative functions Thisimplies there exist positive constants ζ1, ζ2, C3, C4 such that

ζ1kukpLp (R n ) − C3 ≤

Z

Rn

F (x, u)dx ≤ ζ2kukpLp (R n )+ C4 (1.5)(H3) The external force g ∈ Wloc1,2(R; L2(Rn)) satisfies

Z t

−∞

eσs

kg(s)k2L2 (R n )+ kg0(s)k2L2 (R n )



< +∞, ∀t ∈ R, (1.6)and

Definition 1.1 A function u(t, x) is called a weak solution of (1.1) on(τ, T ) iff

u ∈ L∞(τ, T ; H1(Rn)) ∩ Lp(τ, T ; Lp(Rn)), ∂u

∂t ∈ L2(τ, T ; H1(Rn)),u|t=τ = uτ a.e in Rn,

Z

Rn

gv,for all test functions v ∈ C0∞([0, T ] × Rn)

In this Section, we prove that under assumptions (H1) - (H3), the problem(1.1) has a unique weak solution Denote by k · k, (·, ·) the norm and scalar

product of L2(Rn) and C an arbitrary constant, which may be differentfrom line to line (and even in the same line).

In order to prove the existence of a weak solution, we consider the let problem in a bounded domain

a unique weak solution for any uτ ∈ H1

12

ddtZ

Ωrj

|∇urj|2+

Using (1.2) and the inequality | Ω

Ωrj

|∇urj|2

!+ 2Z

Ωrj

|∇urj(t)|2 + 2

Z t τ

Z

Ωrj

|∇urj(s)|2 + 2α

Z t τ

Z

Ωrj

|g(s, x)|2 + 2(T − τ )kφ1kL1

(R n ).(1.12)This inequality implies that

Ωrk of urj and denote them by ukj = Lkurj It is clear from (1.13) that {ukj}

is bounded in L∞(τ, T ; H1(Ωrk)) ∩ Lp(τ, T ; Lp(Ωrk)) It follows that thereexists a subsequence (denoted again by ukj) such that ukj = Lkurj → uk∞weakly in Lp(τ, T ; Lp(Ωrk)) and weakly-* in L∞(τ, T ; H1(Ωrk))

We now check that Lku∞ = uk∞ Indeed, let v ∈ C0∞([τ, T ] × Ωrk) Theweak-* convergence in L∞(τ, T ; H1(Ωrk)) gives

Z T τ

Z

Ωrk

Lkurjv →

Z T τ

Z

Ωrk

Lkurjv =

Z T τ

Z

Rn

urjv →

Z T τ

Z

Rn

u∞v =

Z T τ

Z

Ωrk

Lku∞v, (1.18)

so that uk∞ = Lku∞

We claim that Lku∞is a weak solution in [τ, T ]×Ωrk Let v ∈ C0∞([τ, T ]×

Ωrk) Since Ωrk ⊂ Ωrj, it follows that v ∈ C0∞([τ, T ] × Ωrj), and using thefact that urj is a weak solution in Ωrj we have

=

Z T τ

Z

Ωrj

(∂turjv + ε∇∂turj∇v+ ∇urj∇v + f (x, urj)v + urjv − g(t, x)v) = 0

2

+ 14Z

Ωrk

|g(t, x)|2

(1.20)

Thus, we deduce that

{∂tLkurj} is bounded in L2(τ, T ; L2(Ωrk)) (1.21)Combining (1.15) (with Lkurj in place of urj, uk∞ in place of u∞) and (1.21),using the Aubin-Lions Lemma in [9] we get

Lkurj → uk∞ strongly in L2(τ, T ; L2(Ωrk)), up to a subsequence,and therefore

Lkurj → uk∞ a.e in [τ, T ] × Ωrk.Since f is a continuous function, we have

{f (Lkurj)} is bounded in Lq(τ, T ; Lq(Ωrk)) (1.23)From (1.22), (1.23) and Lemma 1.3 in [9, Chapter 1], we have

f (x, Lkurj) → f (x, uk∞) in Lq(τ, T ; Lq(Ωrk)) (1.24)Passing to the limit in (1.19) and using (1.24), we get that Lku∞ is aweak solution in [τ, T ] × Ωrk Hence we get that u∞ is a weak solution ofproblem (1.1) Indeed, for any v ∈ C0∞([τ, T ] × Rn), there exists rk suchthat v ∈ C0∞([τ, T ] × Ωrk), using Lku∞ solving (1.1) in [τ, T ] × Ωrk we canconclude that u∞ is a weak solution of (1.1) in [τ, T ] × Rn

It remains to prove the uniqueness of solution Let u and v be twosolutions of problem (1.1) Denote w = u − v, we have

wt − ε∆wt − ∆w + f (x, u) − f (x, v) + λw = 0 (1.25)

Taking the inner product of (1.25) with w in L2(Rn) we get

d

dt kwk2 + εk∇wk2
≤ 2γ(kwk2 + εk∇wk2),thus

kw(t)k2 + εk∇w(t)k2 ≤ e2γ(t−τ )(kw(τ )k2 + εk∇w(τ )k2)

by Gronwall’s lemma This implies the uniqueness (if u(τ ) = v(τ )) and thecontinuous dependence of the solution

Chapter 2

Existence and upper semicontinuity

of pullback attractors

We first recall some basic concepts related to pullback attractors for namical systems

dy-Let X, Y be two Banach spaces with the norms k · kX and k · kY tively, and X ⊂ Y Denote by B(X) the set of all bounded subsets of X.For A, B ⊂ X, the Hausdorff semi-distance between A and B is defined by

respec-distX(A, B) = sup

x∈A

inf

y∈Bkx − ykX.Let {U (t, τ ) : t ≥ τ, τ ∈ R} be a process in X, i.e., U (t, τ ) : X → Y suchthat U (τ, τ ) = Id and U (t, s)U (s, τ ) = U (t, τ ) for all t ≥ s ≥ τ, τ ∈ R.Definition 2.1 The process {U (t, τ )} is said to be (X, Y ) - pullback asymp-totically compact if for any t ∈ R, any sequence τn → −∞, and any boundedsequence {xn}, the sequence {U (t, τn)xn} is relatively compact in Y

Definition 2.2 A process {U (t, τ )} is called satisfying Condition (PDC)

in Y if for any fixed t ∈ R, D ∈ B(X), and any η > 0, there exist τ0 =

τ0(D, η, t) ≤ t and a finite dimensional subspace Y1 of Y such that:

(i) P (Sτ ≤τ

0U (t, τ )D) is bounded in Y ,(ii) k(IdY − P )ykY ≤ η, ∀y ∈ S

τ ≤τ 0U (t, τ )D,

where P : Y → Y1 is a bounded projector, IdY is the identity.

Lemma 2.1 [7] If a process {U (t, τ )} satisfies Condition (PDC) in Y ,then {U (t, τ )} is (X, Y ) - pullback asymptotically compact

Definition 2.3 The family ˆA = {A(t) : t ∈ R} ⊂ B(X) is said to be an(X, Y ) - pullback attractor for {U (t, τ )} if

(1) A(t) is compact in Y , for all t ∈ R,

Theorem 2.2 [7] Let {U (t, τ )} be a process satisfying the following ditions:

con-(i) {U (t, τ )} is norm-to-weak continuous on Y , i.e., U (t, τ )xn * U (t, τ )x

in Y , as xn → x in X, for all t ≥ τ, τ ∈ R;

(ii) there exists a family of (X, Y ) - pullback absorbing sets B = {B(t) : t ∈R} ⊂ Y , i.e., for any t ∈ R, any D ∈ B(X), there is τ0 = τ0(D, t) ≤ tsuch that

[

τ ≤τ 0

U (t, τ )D ⊂ B(t);

(iii) {U (t, τ )} is (X, Y ) - pullback asymptotically compact

Then {U (t, τ )} has a unique (X, Y ) - pullback attractor ˆA = {A(t) : t ∈ R},and

Remark 2.1 In fact, the results of Lemma 2.1 and Theorem 2.2 wereproved in [7] for the case of usual pullback attractors (i.e the case X = Y )instead of bi-spaces pullback attractors But they can be obtained almostdirectly from [7] with some obvious changes, so we omit the proof here.The following result, which is about the upper semicontinuity of pullbackattractors was proved [5].

Definition 2.4 [5] Let {Uε(t, τ ) : ε ∈ [0, 1]} be a family of evolutionprocesses in a Banach space X with corresponding pullback attractors ˆAε ={Aε(t) : ε ∈ [0, 1]} For any bounded interval I ⊂ R, we say that { ˆAε}ε∈[0,1]

is upper semicontinuous in Y at ε = 0 for t ∈ I if

t ∈ I if for each t ∈ R, for each compact subset K of X and each T > 0,the following conditions hold:

Aε(t) is compact in Y for each t ∈ R

Now, we want to establish the existence of pullback attractors for tion (1.1) Thanks to Theorem 1.1, problem (1.1) defines a process

equa-U (t, τ ) : H1(Rn) ∩ Lp(Rn) → H1(Rn) ∩ Lp(Rn),with U (t, τ )uτ is the unique weak solution of (1.1) subject to uτ as initialdatum at time τ

Proposition 2.4 Under assumptions (1.2)-(1.6), for any t ∈ R, any D ∈B(H1(Rn) ∩ Lp(Rn)), there exists τ0(t, D) such that

ku(t)k2 + k∇u(t)k2 + ku(t)kpLp (R n ) ≤ C

for any τ ≤ τ0, any uτ ∈ D, where u(t) = U (t, τ )uτ This implies thatthe process {U (t, τ )} corresponding to (1.1) has a family of (H1(Rn) ∩

Lp(Rn), H1(Rn) ∩ Lp(Rn)) - pullback absorbing sets B = {B(t) : t ∈ R}.Proof Taking the inner product of (1.1) with u in L2(Rn), we have

d

dt kuk2 + εk∇uk2
+ σ(kuk2

+ εk∇uk2)+ C

kuk2 + k∇uk2 + kukpLp (R n )

(R n )



≤ Ceσtkg(t)k2 + Ceσt

(2.3)Integrating (2.3) from τ to s, s ∈ [τ, t − 1], we get

eσs(ku(s)k2+εk∇u(s)k2) ≤ eστ(kuτk2+εk∇uτk2)+C

and using (2.4), in particular we get

Z s+1

s

eσt

kuk2 + k∇uk2 + kukpLp

(R n )



≤ Ceσs(ku(s)k2 + εk∇u(s)k2) + C

Z s+1 s

Z s+1

s

eσt

λkuk2 + k∇uk2 + 2

eσt

kuk2 + k∇uk2 + kukpLp

ddt

λkuk2 + k∇uk2 + 2

d

dte

σt

λkuk2 + k∇uk2 + 2

Z

Rn

F (u)

+ eσtkg(t)k2

(2.9)

From (2.6) and (2.9) and using uniform Gronwall’s inequality, we get

Because uτ ∈ D is bounded in H1

(Rn) ∩ Lp(Rn),lim sup

τ →−∞

eστ(kuτk2 + εk∇uτk2) = 0

Then, we get τ0 ≤ t such that,

kuk2 + k∇uk2 + kukpLp (R n ) ≤ C

Remark 2.2 The family of (H1(Rn) ∩ Lp(Rn), H1(Rn) ∩ Lp(Rn)) - pullbackabsorbing sets B = {B(t)}t∈R of {U (t, τ )} obtained in Proposition 2.4 isalso (H1(Rn) ∩ Lp(Rn), L2(Rn), (H1(Rn) ∩ Lp(Rn), Lp(Rn)) and (H1(Rn) ∩

Lp(Rn), H1(Rn)) - pullback absorbing sets of the process {U (t, τ )}

Lemma 2.5 [26] Let X, Y be two Banach spaces, X∗, Y∗ be respectivelytheir dual spaces Suppose that X is dense in Y , the injection i : X → Y

is continuous and its adjoint i∗ : Y∗ → X∗ is dense, and {U (t, τ )} is acontinuous or weak continuous process on Y Then {U (t, τ )} is norm-to-weak continuous on X if and only if for t ≥ τ , τ ∈ R, U (t, τ ) maps acompact set of X to be a bounded set of X

By the above lemma and the fact that {U (t, τ )} is continuous in H1(Rn)and L2(Rn), we deduce that {U (t, τ )} is norm-to-weak in Lp(Rn).

Lemma 2.6 For any t ∈ R, any D ∈ B(H1(Rn) ∩ Lp(Rn)), there exists

for any τ ≤ τ0, any uτ ∈ D, where ut(s) = dtd (U (t, τ )uτ) |t=s

Proof By differentiating equation (1.1) with respect to t, we have

utt− ε∆utt− ∆ut + fu0(x, u)ut + λut = g0(t) (2.13)Taking the inner product of (2.13) with ut in L2(Rn) and using (1.3), weget

d

dt kutk2 + εk∇utk2
+ 2k∇utk2 + 2λkutk2 ≤ 2γkutk2 + (g0(t), ut) (2.14)

By Young’s inequality we can obtain

Z

Rn

F (u)

+ Ceσtkg(t)k2

From (2.15) and (2.17), using uniform Gronwall’s lemma, we get τ0 ≤ tsuch that

Next, we estimate the tail of solutions

Lemma 2.7 For any η > 0, any t ∈ R, and any D ∈ B H1(Rn) ∩ Lp(Rn)
,there exist K0 > 0 and τ0 ≤ t such that

Z

|x|≥K

(|u|2 + ε|∇u|2) ≤ η, for all K ≥ K0, τ ≤ τ0, uτ ∈ D (2.19)

Proof Let θ : R+ → R+ be a smooth function satisfying 0 ≤ θ(s) ≤ 1 for

(2.20)First, we estimate some terms on the left hand side of (2.20) By (1.2), wehave

|x|≥k

φ1(x)

(2.21)