# Uniform Attractors for Nonclassical Diffusion Equations with Memory

Authors: Yongqin Xie; Yanan Li; Ye Zeng
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We introduce a new method (or technique), asymptotic contractive method, to verify uniform asymptotic compactness of a family of processes. After that, the existence and the structure of a compact uniform attractor for the nonautonomous nonclassical diffusion equation with fading memory are proved under the following conditions: the nonlinearityf satisfies the polynomial growth of arbitrary order and the time-dependent forcing term g is only translation-bounded in Lloc2(R;L2(Ω)).


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## Body

## 1. Introduction

In this paper, we consider dynamical behavior of solutions for the following nonclassical diffusion equation with a fading memory term:(1)utt-Δutt-Δut-∫0∞ksΔut-sds+fut=gt,inΩ×τ,∞.The problem is supplemented with the boundary condition,(2)ux,t∂Ω=0,∀t∈R,and initial condition,(3)ux,t=uτx,t,t⩽τ,where Ω is a bounded smooth domain in Rn and g(t) is a given external time-dependent forcing.Concerning the memory kernelμ(s)=-k′(s), as in [1–6], we assume the following hypotheses:(4)μ∈C1R+∩L1R+,μs⩾0,μ′s⩽0∀s∈R+,and there is a constant δ>0, such that(5)μ′s+δμs⩽0∀s∈R+.From (4) and (5), we get(6)μ∞=lims→+∞⁡μs=0.We also denote(7)m0≔∫0∞μsds<∞.As Wang and Zhong [5], we introduce the past history of u, that is,(8)ηt=ηtx,s≔∫0sux,t-rdr,s∈R+,as a new variable of the system, which will be ruled by a supplementary equation. Denote (9)ηtt=∂∂tηt,ηst=∂∂sηt,and (1) transforms into the following system:(10)utt-Δutt-Δut-∫0∞μsΔηtsds+fut=gt,ηtt=-ηst+u,with initial-boundary conditions(11)ux,t∂Ω=0,ηtx,s∂Ω×R+=0,t≥τ,ux,τ=uτx,τ,x∈Ω,ητ=ητx,s=∫0suτx,τ-rdr,x,s∈Ω×R+.The past history uτ(τ-s) of the variable u satisfies the condition as follows: there exist positive constants R and ϱ≤δ (from (5)), such that(12)∫0∞e-ϱsuττ-s02ds≤R.The nonlinearityf∈C1 fulfills f(0)=0, along with arbitrary order polynomial growth restriction(13)γ1sp-β1≤fss≤γ2sp+β2,p≥2,and the dissipation condition(14)f′s≥-l,where γi,βi(i=1,2) and l are positive constants.This equation appears as an extension of the usual nonclassical diffusion equation in fluid mechanics, solid mechanics, and heat conduction theory (see [7–10]). Equation (1) with a one-time derivative appearing in the highest order term is called pseudoparabolic or Sobolev-Galpern equations [11–13]. Aifantis [7] proposed a general frame for establishing the equations. For certain classes of materials such as polymer and high-viscosity liquids, the diffusive process is nontrivially influenced by the past history of u, which is represented in (1) by the convolution term against a suitable memory kernel characterizing the diffusive species [14].Proving the existence of uniform attractors for (1) may be a hard task, mainly due to the fact that the nonlinearity satisfies arbitrary polynomial growth condition instead of critical, so Sobolev compact embedding theorems are no longer useful. The asymptotic compactness of solutions cannot be obtained by the usual method (used, e.g., in [2, 5, 6, 15, 16]).Whenμ is a Dirac measure at some fixed time instant or when it vanishes, (1) reduces to the usual nonclassical diffusion equation, which has been investigated extensively by many authors, especially about the asymptotic behavior of solutions; see, for example, [16–22] and the references therein. In [21], the author has proved the existence of global attractor in H01(Ω), when g(x)∈L2(Ω) under the assumptions that f satisfies critical exponent growth condition corresponding to N=3 and some additional condition for nonlinearity, which essentially requires that the nonlinearity is subcritical. Recently, the authors in [23] obtained the existence of a global attractor for g(x)∈H-1(Ω) only under critical nonlinearity, and when g(x,t)∈Lb2(R,L2(Ω)), they proved the asymptotic regularity and the existence of (nonautonomous) exponential attractor. The asymptotic behavior of solutions of this equation has received considerably less attention in the literature under the assumption that the nonlinearity satisfies arbitrary polynomial growth condition. In the ordinary case for some recent results on this equations the reader can refer to Sun et al. [24] and Anh and Toan [25, 26]. Hereafter in [23] the authors mention that these are some mistakes in the coauthor’s earlier paper [24]. In [25], they proved that the nonautonomous dynamical system generated by this class of solutions has a pullback attractor. In [27], they proved the existence of global attractor in H1(RN) with the initial data u0∈H2(RN) and g∈L2(RN).For convenience, hereafter let|u| be the modular (or absolute value) of u and let |·|p be the norm of Lp(Ω)(p≥1). Let ·0=|∇·|2 be the norm of V. Denote by H-1(Ω) the dual space of H01(Ω) and let ·H-1 be the norm of H-1(Ω).C denotes any positive constant which may be different from line to line even in the same line.LetLμ2(R+;V) be the Hilbert spaces of functions φ:R+→V, endowed with the inner product and norm, respectively. Consider(15)φ,ψμ,V=∫0∞μsφs,ψsVds,φμ,V2=∫0∞μsφs02ds.We also define the product spaceM1: (16)M1=V×Lμ2R+;V,endowed with the norm(17)zM12=u,ηtM12=12u22+u02+ηtμ,V2.For convenience, we first show the preliminary result as follows.Lemma 1 (see [5]).
LetI=[0,T], ∀T>0 and let the memory kernel μ(s) satisfy (4) and (5). Then for any ηt∈C(I;Lμ2(R+;V)) the following estimate (18)ηt,ηst≥δ2ηtμ,V2holds, where δ is from (5).For the time-dependent forcingg(x,t), we assume the following hypotheses: (19)g∈Lw,loc2R;L2Ω.Here Lw,loc2(R;L2(Ω)) is the space Lloc2(R;L2(Ω)) under the local weak convergence topology. Recall that a sequence {gn} converges to g as n→∞ in Lw,loc2(R;L2(Ω)) if and only if (20)limn→∞∫t1t2∫Ωvx,sgnx,s-gx,sds=0for all [t1,t2]⊂R and every v∈L2(t1,t2;L2(Ω)) holds.Letg0∈Lb2(R;L2(Ω)) (translation-bounded in Lw,loc2(R;L2(Ω))), and introduce a set of functions obtained by time-translation in g0: (21)Σ0=g0x,t+h:h∈R.We define the hull of g0, denoted as Σ=H(g0), as the closure of Σ0 with respect to the local weak convergence topology of Lloc2(R;L2(Ω)). If g∈Σ=H(g0), then g∈Lb2(R;L2(Ω)); that is, (22)gLb22=supt∈R⁡∫tt+1gs22ds<∞,where ·Lb2 means the norm of Lb2(R;L2(Ω)).Let{Uσ(t,τ)}, σ∈Σ, be a family of processes acting in a Banach space X with symbol space Σ; then for any σ∈Σ(23)Uσt,s∘Uσs,τ=Uσt,τ,for  anyt≥s≥τ,τ∈R,(24)Uστ,τ=Ididentity,for  anyτ∈R.The operators {H(s)}s≥0 are the translation semigroup on Σ; a family of processes {Uσ(t,τ)}, σ∈Σ, is called to satisfy the translation identity if(25)Uσt+s,s+τ=UHsσt,τ,for  anyσ∈Σ,t≥τ,τ∈R,s≥0,(26)HsΣ=Σ,for  anys≥0.We recall (e.g., see [28]) also that the kernel K of the process U(t,τ) consists of all bounded complete trajectories of the process; that is,(27)K=u·:utX≤Cu,ut=Ut,τuτ,∀t≥τ,τ∈Rand K(s)={u(s):u(·)∈K}⊂X is denoted by the kernel section at a time moment s∈R.It is well-known that the key point is to obtain certain asymptotic compactness for the solution operator in the study of the long time behavior, especially for attractors. The nonlinearity having an arbitrary polynomial growth brings a difficulty here even for the autonomous and without memory case; see, for example, [22, 24, 27]. The main contribution of this paper is to extend the method in [15, 22] to overcome the difficulty caused by a lack of Sobolev compact embedding theorems. The conception, asymptotic contractive function, and new a priori estimates for verifying uniform asymptotic compactness of the family of processes are devised. We also prove someweak continuity for the family of processes and then obtain the structure of the compact uniform attractors.The main results of this paper are given expression to in the following two theorems, which will be proved in Sections2 and 3, respectively.Theorem 2.
LetX be a Banach space and let {Uσ(t,τ)}, σ∈Σ, be a family of processes on X. Assume further that g0∈Lb2(R;L2(Ω)) and Σ is the hull of g0 in Lw,loc2(R;L2(Ω)). Then {Uσ(t,τ)}, σ∈Σ, has a uniform attractor in X provided that the following conditions hold true: (i)
{Uσ(t,τ)},σ∈Σ, has a bounded absorbing set B0 in X.(ii)
For anyε>0, there exist T=T(B0,ε)>0 and an asymptotic contractive function ψT on B0×B0 such that (28)Uσ1T,τx-Uσ2T,τyX≤ε+ψTx,y,σ1,σ2,∀x,y∈B0,σ1,σ2∈Σ,where ψT depend on T.Theorem 3 (uniform attractor).
LetΩ be a bounded domain in Rn with smooth boundary, and μ and f satisfy (4)–(6) and (13)-(14), respectively. Assume further that g0∈Lb2(R;L2(Ω)) and Σ is the hull of g0 in Lw,loc2(R;L2(Ω)). Then the family of processes {Uσ(t,τ)}, σ∈Σ, corresponding to (10)-(11), has a compact uniform (with respect to σ∈Σ) attractor A in M1. Moreover, this attractor can be decomposed as follows:(29)A=⋃σ∈ΣKσ0,where Kσ is the kernel of the process Uσ and Kσ(0) is the kernel section at time 0.This theorem gives the existence of the uniform attractor and its structure as the union of the kernel sections (see [29]) of the nonautonomous process.It is worth noting that Theorem3 is also interesting in the autonomous case (see, e.g., [5]) or in the unbounded domain case (see, e.g., [22, 23]). This is the basis for further considering the asymptotic behavior, such as, to construct the nonautonomous (pullback) exponential attractor [2, 25]. On the other hand, from Theorem 16, the existence of the uniform attractor is obtained directly. However, since the external forcing term g(x,t) is only assumed to betranslation-bounded, consequently the symbol space is only weak compact with respect to the localweak convergence topology. So, in order to obtain the structure equality (29), we need to verify someweak continuity for the family of processes, which is different from the usualstrong continuity. This may be the reason why some authors (e.g., see [1]) have to assume further that the external forcing term g(x,t) istranslation-compact.


## 2. Preliminaries

Definition 4 (see [15]).
LetX be a Banach space, B be a bounded subset of X and Σ be a symbol (or parameter) space. We call a function ϕ(·,·,·,·), defined on (X×X)×(Σ×Σ), to be a contractive function on B×B, if, for any sequence {xn}n=1∞⊂B, and any {σn}n=1∞⊂Σ, there is a subsequence {xnk}k=1∞⊂{xn}n=1∞ and {σnk}k=1∞⊂{σn}n=1∞ respectively, such that (30)limk→∞⁡liml→∞⁡ϕxnk,xnl,σnk,σnl=0.Definition 5.
LetX be a Banach space and let B be a bounded subset of X and let Σ be a symbol (or parameter) space. We call a function ψ(·,·,·,·), defined on (X×X)×(Σ×Σ), to be an asymptotic contractive function on B×B, if for any ε>0 and xi∈B, σi∈Σ(i=1,2), there is a contractive function ϕ on B×B such that (31)ψx1,x2,σ1,σ2≤ε+ϕx1,x2,σ1,σ2.We denote the set of all asymptotic contractive functions onB×B by E(B,Σ).In the following theorem, we present a new method (or technical) to verify the asymptotic compactness for the family of processes generated by evolutionary equations, which will be used in our later discussion.Theorem 6.
Let(X,·X) be a Banach space; {Uσ(t,τ)}, σ∈Σ, is a family of processes on X. Assume further that the following conditions hold true: (i)
{Uσ(t,τ)}, σ∈Σ, has a bounded absorbing set B0 in X.(ii)
For anyε>0 and x,y∈B0, σ1,σ2∈Σ, there exist T=T(B0,ε)>0 and ψT∈E(B0,Σ) such that (32)Uσ1T,τx-Uσ2T,τyX≤ε+ψTx,y,σ1,σ2.Then {Uσ(t,τ)}, σ∈Σ, is uniform (with respect to σ∈Σ) asymptotically compact in X, where ψT depend on T.Proof.
For anyτ∈R, we assume that B is any bounded subset of X, and {xn}n=1∞⊂B, σn∈Σ and tn≥τ satisfy tn→+∞ as n→∞. It is enough to show that, from the assumptions, there is N>0 such that Uσ(tN,τ)x∈B0 and tn>tN-τ for each n>N large enough and x∈B and furthermore that (33)Uσtn,τx=Uσtn,tN∘UσtN,τx=Uσtn,tNy=Uσ′tn′,τy,where y=Uσ(tN,τ)x∈B0, tn′=tn-tN+τ>0 and σ′=H(tN-τ)σ∈Σ. So we only need to consider the case as {xn}n=1∞⊂B0.
In the following, the existence of a Cauchy subsequence of{Uσn(tn,τ)xn}n=1∞ is proved by the diagonal method.
Takeεm>0 with εm→0 as m→∞.
At first, forε1, by the assumptions, there exist T1=T1(ε1) and ψT1∈E(B0,Σ) such that(34)Uσ1T1,τx-Uσ2T1,τyX≤ε1+ψT1x,y,σ1,σ2for  anyx,y∈B0,σ1,σ2∈Σ.
Sincetn→+∞, for some T1 fixed we assume that tn≫T1 is so large that Ugn(tn-T1+τ,τ)xn∈B0 for each n≥N and gn∈Σ.
Letyn=Ugn(tn-T1+τ,τ)xn; then from (23)–(26) we have(35)Ugntn,τxn-Ugmtm,τxmX=Ugntn,tn-T1+τ∘Ugntn-T1+τ,τxn-Ugmtm,tm-T1+τ∘Ugmtm-T1+τ,τxmX=UσnT1,τyn-UσmT1,τymX≤ε1+ψT1yn,ym,σn,σm,where σn=H(tn-T1)gn.
Due to the definition ofE(B0,Σ) and ψT1∈E(B0,Σ), we know that, for ε1 from (35), there is a contractive function ϕ on B0×B0 and {(yn,σn)}n=1∞ has a subsequence {(ynk(1),σnk(1))}k=1∞ such that one gets the following estimation:(36)ψT1ynk1,ynl1,σnk1,σnl1≤ε1+ϕT1ynk1,ynl1,σnk1,σnl1,(37)limk→∞⁡liml→∞⁡ϕT1ynk1,ynl1,σnk1,σnl1=0.
And similar to [15, 22, 30], we have(38)limk→∞supp∈N⁡Uσnk+p1tnk+p1,τxnk+p1-Uσnk1tnk1,τxnk1X≤limk→∞⁡supp∈N⁡limsupl→∞⁡Uσnk+p1tnk+p1xnk+p1-Uσnl1tnl1,τxnl1X+limsupk→∞⁡limsupl→∞⁡Uσnk1tnk1,τxnk1-Uσnl1tnl1,τxnl1X≤ε1+limk→∞⁡supp∈N⁡liml→∞⁡ψT1ynk+p1,ynl1,σnk+p1,σnl1+ε1+limk→∞⁡liml→∞⁡ψT1ynk1,ynl1,σnk1,σnl1≤4ε1+limk→∞⁡supp∈N⁡liml→∞⁡ϕT1ynk+p1,ynl1,σnk+p1,σnl1+limk→∞⁡liml→∞⁡ϕT1ynk1,ynl1,σnk1,σnl1
which, combined with (35) and (37), implies that(39)limk→∞supp∈N⁡Uσnk+p1tnk+p1,τxnk+p1-Uσnk1tnk1,τxnk1X≤5ε1.Therefore, there is K1 such that(40)Uσnk1tnk1,τxnk1-Uσnl1tnl1,τxnl1X≤6ε1∀k,l≥K1.By induction, we obtain that, for each m≥1, there is a subsequence {Uσnk(m+1)(tnk(m+1),τ)xnk(m+1)}k=1∞ of {Uσnk(m)(tnk(m),τ)xnk(m)}k=1∞ and certain Km+1 such that(41)Uσnkm+1tnkm+1,τxnkm+1-Uσnlm+1tnlm+1,τxnlm+1X≤6εm+1holds for all k,l≥Km+1. Now, we consider the diagonal subsequence {Uσnk(k)(tnk(k),τ)xnk(k)}k=1∞. Since, for each m∈N, {Uσnk(k)(tnk(k),τ)xnk(k)}k=1∞ is a subsequence of {Uσnk(m)(tnk(m),τ)xnk(m)}k=1∞, then(42)Uσnkktnkk,τxnkk-Uσnlltnll,τxnllX≤6εm∀k,l≥max⁡m,Km,which, combined with εm→0 as m→∞, implies that {Uσnk(k)(tnk(k),τ)xnk(k)}k=1∞ is a Cauchy sequence in X. This shows that {Uσn(tn,τ)xn}n=1∞ is precompact in X. Then the proof is complete.Proof of Theorem2.
From Theorem6, the family of processes {Uσ(t,τ)}, σ∈Σ, on X satisfy the following conditions:(i)
{Uσ(t,τ)}, σ∈Σ, has a bounded absorbing set B0 in X;(ii)
{Uσ(t,τ)}, σ∈Σ, is asymp(iii)
totically compact inX.Then {Uσ(t,τ)}, σ∈Σ, has a uniform attractor in X.Lemma 7 (see [31]).
LetX⋐H⊂Y be Banach spaces, with X reflexive. Suppose that un is a sequence that is uniformly bounded in L2(0,T;X) and dun/dt is uniformly bounded in Lp(0,T;Y), for some p>1. Then there is a subsequence of un that converges strongly in L2(0,T;H).


## 3. Uniform Attractors inH01(Ω)×Lμ2R+;H01(Ω)

### 3.1. A Priori Estimates

We start with the following general existence and uniqueness of solutions for the nonclassical diffusion equations with fading memory which can be obtained by theGalerkin approximation methods; here we formulate only the results.Lemma 8.
Assume thatg0∈Lb2(R;L2(Ω)) and g∈Σ and f satisfies (13)-(14). Then, for any initial data zτ∈M1 and any T>0, there exists a unique solution z=(u,ηt) for problem (10)-(11). Moreover, we have the following Lipschitz continuity: for any (zτi,gi)(zτi∈M1,gi∈Σ), denote by zi(i=1,2) the corresponding solutions of (10); then for all τ≤t≤T+τ(43)z1t-z2tM12≤Qzτ1M1,zτ2M1,Tzτ1-zτ2M12+g1-g2Lb22,where Q(·) is a monotonically increasing function.By Lemma8, we can define a process Ug(t,τ) in M1 as the following: (44)Ugt,τ:Rτ×M1⟶M1,zτ=uτ,ητ⟶zt=ut,ηt=Ugt,τzτ,and {Ug(t,τ)}, g∈Σ, is a family of processes on M1. See [5] for more details.Lemma 9.
Let (13)-(14) hold, and g∈Σ. Then there exists a positive ρ0, which depends only on gLb2, such that, for any bounded subset B⊂M1, there is T0=T0(BM1) such that(45)Ugt,τzτM12⩽ρ0,forallt-τ⩾T0andallzτ∈B.Proof.
Multiplying (10) by u and then integrating over Ω, it follows that(46)12ddtu22+u02+ηtμ,V2+u02+ηt,ηstμ,V+fu,u=g,u.Observe that(47)fu,u=∫Ωfuu≥γ1upp-β1mesΩ.
Using Lemma1, we have(48)ηt,ηstμ,V≥δ2ηtμ,V2.
Using the Hölder inequality, combining with (47) and (48), then (46) can be reformulated as follows:(49)12ddtu22+u02+ηtμ,V2+12u02+δ2ηtμ,V2+γ1upp≤β1mesΩ+12λ12g22.
According to Poincaré inequality, we get(50)12ddtu22+u02+ηtμ,V2+λ14u22+14u02+δ2ηtμ,V2+γ1upp≤C21+g22.
Letα=min⁡(λ1/2,1/2,δ); again, we have(51)ddtu22+u02+ηtμ,V2+αu22+u02+ηtμ,V2≤C1+g22.
Using theGronwall Lemma, it follows from (51) that(52)u22+u02+ηtμ,V2≤e-αt-τuτ22+uτ02+ητμ,V2+C1+gLb22.
We know thatgLb22 is bounded, and by (12) and (6) (53)ητμ,V2=∫0+∞μsητs02ds≤∫0+∞∫0sμsuττ-r02drds≤∫0+∞dr∫r+∞e-δsuττ-r02ds≤1δ∫0+∞e-δsuττ-s02ds≤δ-1R.So zτM12=|uτ|22+uτ02+ητμ,V2 is bounded.
For anyt-τ>T0=1/αln⁡zτM12/C(1+gLb22), we infer from (52) that(54)ut22+ut02+ηtμ,V2≤2C1+gLb22.
For anyt-τ>T0, we get(55)ut22+ut02+ηtμ,V2≤ρ0,where ρ0=2C(1+gLb22). Then we get (56)B0=z=u,ηt∈M1:zM12≤ρ0.
The proof is complete.Combining with (43), we know that, for any τ∈R, Ug maps the bounded set of M1 into a bounded set for all t⩾τ that is as follows.Corollary 10.
Let (13)-(14) hold, and g∈Σ. Then, for any bounded (in M1) subset B, there is MB=M(BM1,gLb2) such that(57)Ugt,τzτM12⩽MBforallt⩾τandallzτ∈B.Lemma 11.
Let (13)-(14) hold, and g∈Σ. Then, for any bounded (in M1) subset B, there exists a positive constant C=C(α,γ1,ρ0), such that (58)∫tt+1usppds<C1+gLb22,∫tt+1zsM12ds<C1+gLb22hold for any t-τ≥T0 (from Lemma 9).Proof.
Combining with (49), let α=min⁡(λ1/2,1/2,δ); it follows that(59)ddtu22+u02+ηtμ,V2+αu22+u02+ηtμ,V2+γ1upp≤2C1+g22.
For anyt-τ≥T0 (from Lemma 9) and integrating the inequality above from t to t+1, we can get(60)α∫tt+1zsM12ds+2γ1∫tt+1usppds≤ztM12+2C1+gLb22.
By Lemma9, we have(61)α∫tt+1zsM12ds+2γ1∫tt+1usppds≤ρ0+2C1+gLb22.Setting α0=min⁡(α,2γ1), then (62)∫tt+1zsM12ds≤1α0ρ0+2C1+gLb22,∫tt+1usppds≤1α0ρ0+2C1+gLb22.
LetC=Cα,γ1,ρ0=1/α0ρ0+2C; then we have that (63)∫tt+1zsM12ds≤C1+gLb22,∫tt+1usppds≤C1+gLb22hold for any t-τ≥T0.LetF(s)=∫0s‍f(υ)dυ; from assumption (3), there are positive constants β~1, β~2, γ~1, γ~2, such that(64)γ~1sp-β~1≤Fs≤γ~2sp+β~2.Lemma 12.
There is a positive constantρ1, for any zτ=(uτ,ητ)∈B0 (from Lemma 9); the following estimate (65)utpp≤ρ1holds for any t-τ≥1.Proof.
Multiplying (10) by ut and then integrating in Ω, we get(66)utt22+utt02+ddt12ut02+∫ΩFut=∫0+∞μsΔηts,uttds+gt,utt.By the Hölder inequality and the Young inequality, we have(67)∫0+∞μsΔηts,uttds≤∫0+∞μsηts0utt0ds≤m02∫0+∞μsηts02ds+12m0utt02∫0+∞μsds≤m02ηtμ,V2+12utt02≤m02ztM12+12utt02,here m0 is from (7). And(68)gt,utt≤12gt22+12utt22.
Let(69)Et=12ut02+∫ΩFut.
Then(70)12utt22+12utt02+ddtEt≤m02ztM12+12gt22.So we get the following inequality:(71)ddtEt≤m02ztM12+12gt22.Integrating the inequality above to t from s(t≤s≤t+1) to t+1, we have(72)Et+1≤Es+m02∫tt+1zsM12ds+12gLb22,and integrating (72) to s from t to t+1,(73)Et+1≤∫tt+1Esds+m02∫tt+1zsM12ds+12gLb22.
Besides, by the Hölder inequality, Young inequality, and (13), we can infer from (69) that(74)Es≤12us02+γ~2uspp+β~2mesΩ≤12zsM12+γ~2uspp+β~2mesΩ.
On the contrary,(75)Et+1≥12ut+102+γ~1ut+1pp-β~1mesΩ≥γ~1ut+1pp-β~1mesΩ.
Combining (73) and (74), we have(76)Et+1≤1+m02∫tt+1zsM12ds+γ~2∫tt+1usppds+12gLb22+β~2mesΩ.By Lemma 11, it follows from (76) that(77)Et+1≤C1+m02+γ~21+gLb22+12gLb22+β~2mesΩ,and by (75)–(77), we have(78)ut+1pp≤1γ~11+m02+γ~2C1+gLb22+12gLb22+β~1+β~2mesΩ.Let (79)ρ1=1γ~11+m02+γ~2C1+gLb22+12gLb22+β~1+β~2mesΩ.Then the proof is complete.Lemma 13 (bounded uniformly absorbing set).
Let (13)-(14) hold, and g∈Σ. Then there exist positive ρ0, ρ1, which depend only on gLb2, such that, for any bounded (in M1) subset B, there is T1=T1(BM1)=T0+1 such that(80)Ugt,τzτM12⩽ρ0,utpp≤ρ1hold for all t-τ⩾T1 and all zτ∈B.For brevity, in the sequel, letB0 be the bounded absorbing set obtained in Lemmas 9 and 12; that is,(81)B0=z=u,ηt∈M1:zM12≤ρ0,upp≤ρ1.Lemma 14.
There is a positive constantρ2, for any zτ=(uτ,ητ)∈B0 (from (81)); the following estimate (82)∫tt+1uts22+uts02ds≤ρ2,holds for any t-τ≥0.Proof.
By Lemma12, for any t-τ≥0, we integrate (70) to t from t to t+1; then we have(83)Et+1+∫tt+1uts22+uts02ds≤Et.
According to Lemma13, Corollary 10, and Lemma 12, we know that E(t) is bounded for any t-τ≥0, so there is a positive constant ρ2, and the conclusion is true.


### 3.2. Uniform Attractors

In the following, we will prove the existence of uniform attractors for system (10) with initial-boundary conditions (11) in M1 by using the method of asymptotic contractive function.Lemma 15.
Letf satisfy (13)-(14); zn(t)=(un(t),ηtn) is the sequence of solutions of (10) with initial data zτn=(uτn,ητn)∈B0 and gn∈Σ(n=1,2,…); then there is a subsequence unk of un(t) that converges strongly in L2(0,T;L2(Ω)) and(84)limk→∞⁡limj→∞⁡∫0T∫Ωgnkx,s-gnjx,sunkx,s-unjx,sds=0.Proof.
By Lemmas13 and 14, the sequence un is uniformly bounded in L2(0,T;H01(Ω)) and dun/dt is uniformly bounded in L2(0,T;H-1(Ω)). Since H01(Ω) is reflexive, is L2(0,T;H01(Ω)). Let X=H01(Ω), H=L2(Ω), and Y=H-1(Ω); then there is a subsequence unk(t) of {un(t)} that converges strongly in L2(0,T;L2(Ω)) by Lemma 7. One can write these as(85)limk→∞⁡limj→∞⁡∫0T∫Ωunkx,s-unjx,s2ds=0.Then by using Hölder inequality, we have (86)limk→∞⁡limj→∞⁡∫0T∫Ωgnkx,s-gnjx,sunkx,s-unjx,sds≤limk→∞⁡limj→∞⁡∫0T∫Ωgnkx,s-gnjx,s2ds1/2∫0T∫Ωunkx,s-unjx,s2ds1/2≤2TgLb2limk→∞⁡limj→∞⁡∫0T∫Ωunkx,s-unjx,s2ds1/2=0.Theorem 16.
LetΩ be a bounded domain in R3 with smooth boundary, and let μ and f satisfy (4)-(5) and (13)-(14), respectively. Assume further that g0∈Lb2(R;L2(Ω)) and Σ is the hull of g0 in Lw,loc2(R;L2(Ω)). {Uσ(t,τ)}, σ∈Σ, is the family of processes generated by the solutions of (10) with initial data zτ∈M1. Then {Uσ(t,τ)}, σ∈Σ, is uniformly (with respect to σ∈Σ) asymptotic compact in M1.Proof.
For anyzτi=(uτi,ηiτ)∈B0 and gi∈Σ, let zi(t)=(ui(t),ηit) be the corresponding solution with respect to initial data zτi(i=1,2) and gi∈Σ; that is, zi(t) is the solution of (10) with initial-boundary conditions (11).
For convenience, we denoteω(t)=u1(t)-u2(t) and ξt=η1t-η2t; then ω(t) satisfies the following equation:(87)ωtt-Δωtt-Δωt-∫0∞μsΔξtsds+fu1t-fu2t=g1t-g2tξtt=-ξst+ω,with initial-boundary conditions(88)ω∂Ω=0,ξtx,s∂Ω×R+=0,t≥τ,ωx,τ=ωτx=uτ1x-uτ2x,x∈Ω,ξτx,s=η1τx,s-η1τx,s,x,s∈Ω×R+.
Multiplying (87) by ω(t) and integrating in Ω, we obtain(89)12ddtω22+ω02+ξtμ,V2+ω02+ξts,ξstsμ,V+∫Ωfu1-fu2u1-u2=∫Ωg1t-g2tu1-u2.
By Lemma1 and assumption (14), we have (90)ξts,ξstsμ,V≥δ2ξtμ,V2,∫Ωfu1-fu2u1-u2≥-lω22.Using the Poincaré inequality, it follows that(91)12ddtω22+ω02+ξtμ,V2+λ12ω22+12ω02+δ2ξtμ,V2≤lω22+∫Ωg1t-g2tu1-u2.Let ϱ=min⁡(λ1,1,δ); then we get (92)ddtω22+ω02+ξtμ,V2+ϱω22+ω02+ξtμ,V2≤2lu1-u222+2∫Ωg1t-g2tu1-u2.
By theGronwall  Lemma, we have(93)ωt22+ωt02+ξtμ,V2≤e-ϱt-τωτ22+ωτ02+ξτμ,V2+2l∫τte-ϱt-su1s-u2s22ds+2∫τte-ϱt-s∫Ωg1s-g2su1s-u2sds.
For anyε>0, there exists T2=T2(ε)=max⁡(T1,1/ϱln⁡(ε/zτM12))≥T1, such that(94)ωt22+ωt02+ξtμ,V2≤ε+ψtholds for any t≥T2; here (95)ψt=ψtu1,u2,g1,g2=2l∫τte-ϱt-su1s-u2s22ds+2∫τte-ϱt-s∫Ωg1s-g2su1s-u2sds.Hereafter, we verify that there is T≥T2 such that ψT is an asymptotic contractive function on B0. In fact (96)ψt=2l∫τte-ϱt-su1s-u2s22ds+2∫τte-ϱt-s∫Ωg1s-g2su1s-u2sds≤2l∫τT1e-ϱt-su1s-u2s22ds+2∫τT1e-ϱt-s∫Ωg1s-g2su1s-u2sds+2l∫T1te-ϱt-su1s-u2s22ds+2∫T1te-ϱt-s∫Ωg1s-g2su1s-u2sds≤e-ϱt-T12l∫τT1u1s-u2s22ds+2∫τT1∫Ωg1s-g2su1s-u2sds+2l∫T1tu1s-u2s22ds+2∫T1t∫Ωg1s-g2su1s-u2sds≤2T1e-ϱt-T12l+1MB+gLb22+2l∫0t-T1u1s+T1-u2s+T122ds+2∫0t-T1∫Ωg1s+T1-g2s+T1u1s+T1-u2s+T1ds,where MB is from Corollary 10. Let(97)φt=2l∫0t-T1v1s-v2s22ds+2∫0t-T1∫Ωσ1s-σ2sv1s-v2sds,where vi(s)=ui(s+T1)∈B0 and σi(s)=gi(s+T1)∈Σ(i=1,2). For any ε>0, let t=T≥T1+1/ϱln⁡2T1((2l+1)MB+gLb22)/ε be fixed; then (98)ψTu1,u2,g1,g2≤ε+φTu1,u2,g1,g2.
Combining Lemma15 and Definition 5, we know that φT is the contractive function on B0×B0 and ψT is the asymptotic contractive function on B0×B0.
Then{Ug(t,τ)}t≥0, g∈Σ, is uniform (with respect to g∈Σ) asymptotic compact in M1 by Theorem 6.We also recall the following useful result, whose proof is simple and we omit it.Lemma 17 (see [15]).
LetX be a reflexive Banach space and xn⇀0 in X. Then, for each compact (in X∗) subset B⊂X∗, the uniform convergence holds: for any ε>0, there is Nε, depending only on ε, such that (99)f,xnX∗≤εforalln≥Nεandallf∈B.Proof of Theorem3.
Theorem2 implies that the family of processes {Ug(t,τ)}, g∈Σ, corresponding to (1), has a compact (in M1) uniform (with respect to g∈Σ) attracting set which is bounded in M1; consequently, as a direct application of the abstract theorem [29, Chapter IV, Theorem  3.1], we obtain the existence of a compact uniform (with respect to g∈Σ) attractor A, and A⊂M1.
We remark that the above existence does not require any continuity of the family of processes (e.g., see [29]). However, in order to obtain the explicit form of A, that is, equality (29), we need some continuity. Moreover, since the symbol space Σ now has onlyweak compactness, we need to verify the correspondingweak continuity.
Hence, in order to obtain (29), we need to verify the following: for any fixed τ∈R and t⩾τ, if znτ→zτ in M1 and gn⇀g with respect to the local weak convergence topology of Lloc2(R;L2(Ω)), then (100)Ugnt,τznτ⟶Ugt,τzτinM1.
In the following, we will verify that the continuity above holds onA×Σ, which obviously is a compact uniformly attracting set for the corresponding skew-product flow semigroup (generated by the family processes Ug(t,τ)zτ, g∈Σ, and the shift semigroup {T(t)}t⩾0). We will complete the proof of the week continuity.
Letzn(t)=(un,ηnt), let zn(t) be the solutions of (10) with initial data zτn∈A(n=1,2,…), and let the time-dependent forcing gn∈Σ, z(t) be the solutions of (10) with initial data zτ∈A and the time-dependent forcing g∈Σ; we assume the following hypotheses:(101)zτn⟶zτ,inM1,(102)gn⟶g,inLw,loc2R,L2Ω.Since (10) holds as an equality in Lq(τ,T;H-1(Ω)) (q is conjugate to p from (13)), this means that for any T(>τ) fixed(103)un⇀u,inL2τ,T;H01Ω,where u=Π1Ug(t,τ)zτ∈L2(τ,T;H01(Ω)), and Π1 is the projector from X×Y to X. Since A is bounded in M1, following Corollary 10, we know that there is M such that(104)supg∈Σ⁡supτ∈R⁡supt⩾τ⁡Ugt,τAM12⩽M<∞.Hence, (105)⋃g∈ΣΠ1Ugt,τzτ:t∈τ,T,zτ∈Ais  bounded  inL2τ,T;H01Ω,⋃g∈Σ∂tΠ1Ugt,τzτ:t∈τ,T,zτ∈Ais  bounded  inL2τ,T;H-1Ω.Then by the compactness lemma (Lemma 7) we know that(106)⋃g∈ΣΠ1Ugt,τzτ:t∈τ,T,zτ∈Ais  compact  inL2τ,T;L2Ω.It is (107)limn→∞⁡∫τT∫Ωuns-us2ds=0.Denote ω(t)=un(t)-u(t), ξt=ηnt-ηt, ωτ=unτ(t)-uτ, and zn(t)=(un(t),ηnt)=Ugn(t,τ)znτ-Ug(t,τ)zτ, znτ,zτ∈A, n=1,2,…. The proof of the following inequality is similar to (93): (108)ωt22+ωt02+ξtμ,V2≤e-ϱt-τωτM12+2l∫τtuns-us22ds+2∫τt∫Ωgns-gsuns-usds.
Due to (107) and (102), we only need to show that if gn→g in Lw,loc2(R;L2(Ω)), then(109)∫τtgnx,s-gx,s,wx,sds⟶0uniformly on L2(τ,t;L2(Ω))-compact set. But this is a direct application of Lemma 17. Therefore we complete the proof of thecontinuity on A×Σ.
Based on thecontinuity claim above, and by constructing a skew-product flow on A×Σ and applying [29, Chapter IV, Theorem  5.1], we obtain the structure equality (29). This completes the proof of Theorem 3.


## 3.1. A Priori Estimates

We start with the following general existence and uniqueness of solutions for the nonclassical diffusion equations with fading memory which can be obtained by theGalerkin approximation methods; here we formulate only the results.Lemma 8.
Assume thatg0∈Lb2(R;L2(Ω)) and g∈Σ and f satisfies (13)-(14). Then, for any initial data zτ∈M1 and any T>0, there exists a unique solution z=(u,ηt) for problem (10)-(11). Moreover, we have the following Lipschitz continuity: for any (zτi,gi)(zτi∈M1,gi∈Σ), denote by zi(i=1,2) the corresponding solutions of (10); then for all τ≤t≤T+τ(43)z1t-z2tM12≤Qzτ1M1,zτ2M1,Tzτ1-zτ2M12+g1-g2Lb22,where Q(·) is a monotonically increasing function.By Lemma8, we can define a process Ug(t,τ) in M1 as the following: (44)Ugt,τ:Rτ×M1⟶M1,zτ=uτ,ητ⟶zt=ut,ηt=Ugt,τzτ,and {Ug(t,τ)}, g∈Σ, is a family of processes on M1. See [5] for more details.Lemma 9.
Let (13)-(14) hold, and g∈Σ. Then there exists a positive ρ0, which depends only on gLb2, such that, for any bounded subset B⊂M1, there is T0=T0(BM1) such that(45)Ugt,τzτM12⩽ρ0,forallt-τ⩾T0andallzτ∈B.Proof.
Multiplying (10) by u and then integrating over Ω, it follows that(46)12ddtu22+u02+ηtμ,V2+u02+ηt,ηstμ,V+fu,u=g,u.Observe that(47)fu,u=∫Ωfuu≥γ1upp-β1mesΩ.
Using Lemma1, we have(48)ηt,ηstμ,V≥δ2ηtμ,V2.
Using the Hölder inequality, combining with (47) and (48), then (46) can be reformulated as follows:(49)12ddtu22+u02+ηtμ,V2+12u02+δ2ηtμ,V2+γ1upp≤β1mesΩ+12λ12g22.
According to Poincaré inequality, we get(50)12ddtu22+u02+ηtμ,V2+λ14u22+14u02+δ2ηtμ,V2+γ1upp≤C21+g22.
Letα=min⁡(λ1/2,1/2,δ); again, we have(51)ddtu22+u02+ηtμ,V2+αu22+u02+ηtμ,V2≤C1+g22.
Using theGronwall Lemma, it follows from (51) that(52)u22+u02+ηtμ,V2≤e-αt-τuτ22+uτ02+ητμ,V2+C1+gLb22.
We know thatgLb22 is bounded, and by (12) and (6) (53)ητμ,V2=∫0+∞μsητs02ds≤∫0+∞∫0sμsuττ-r02drds≤∫0+∞dr∫r+∞e-δsuττ-r02ds≤1δ∫0+∞e-δsuττ-s02ds≤δ-1R.So zτM12=|uτ|22+uτ02+ητμ,V2 is bounded.
For anyt-τ>T0=1/αln⁡zτM12/C(1+gLb22), we infer from (52) that(54)ut22+ut02+ηtμ,V2≤2C1+gLb22.
For anyt-τ>T0, we get(55)ut22+ut02+ηtμ,V2≤ρ0,where ρ0=2C(1+gLb22). Then we get (56)B0=z=u,ηt∈M1:zM12≤ρ0.
The proof is complete.Combining with (43), we know that, for any τ∈R, Ug maps the bounded set of M1 into a bounded set for all t⩾τ that is as follows.Corollary 10.
Let (13)-(14) hold, and g∈Σ. Then, for any bounded (in M1) subset B, there is MB=M(BM1,gLb2) such that(57)Ugt,τzτM12⩽MBforallt⩾τandallzτ∈B.Lemma 11.
Let (13)-(14) hold, and g∈Σ. Then, for any bounded (in M1) subset B, there exists a positive constant C=C(α,γ1,ρ0), such that (58)∫tt+1usppds<C1+gLb22,∫tt+1zsM12ds<C1+gLb22hold for any t-τ≥T0 (from Lemma 9).Proof.
Combining with (49), let α=min⁡(λ1/2,1/2,δ); it follows that(59)ddtu22+u02+ηtμ,V2+αu22+u02+ηtμ,V2+γ1upp≤2C1+g22.
For anyt-τ≥T0 (from Lemma 9) and integrating the inequality above from t to t+1, we can get(60)α∫tt+1zsM12ds+2γ1∫tt+1usppds≤ztM12+2C1+gLb22.
By Lemma9, we have(61)α∫tt+1zsM12ds+2γ1∫tt+1usppds≤ρ0+2C1+gLb22.Setting α0=min⁡(α,2γ1), then (62)∫tt+1zsM12ds≤1α0ρ0+2C1+gLb22,∫tt+1usppds≤1α0ρ0+2C1+gLb22.
LetC=Cα,γ1,ρ0=1/α0ρ0+2C; then we have that (63)∫tt+1zsM12ds≤C1+gLb22,∫tt+1usppds≤C1+gLb22hold for any t-τ≥T0.LetF(s)=∫0s‍f(υ)dυ; from assumption (3), there are positive constants β~1, β~2, γ~1, γ~2, such that(64)γ~1sp-β~1≤Fs≤γ~2sp+β~2.Lemma 12.
There is a positive constantρ1, for any zτ=(uτ,ητ)∈B0 (from Lemma 9); the following estimate (65)utpp≤ρ1holds for any t-τ≥1.Proof.
Multiplying (10) by ut and then integrating in Ω, we get(66)utt22+utt02+ddt12ut02+∫ΩFut=∫0+∞μsΔηts,uttds+gt,utt.By the Hölder inequality and the Young inequality, we have(67)∫0+∞μsΔηts,uttds≤∫0+∞μsηts0utt0ds≤m02∫0+∞μsηts02ds+12m0utt02∫0+∞μsds≤m02ηtμ,V2+12utt02≤m02ztM12+12utt02,here m0 is from (7). And(68)gt,utt≤12gt22+12utt22.
Let(69)Et=12ut02+∫ΩFut.
Then(70)12utt22+12utt02+ddtEt≤m02ztM12+12gt22.So we get the following inequality:(71)ddtEt≤m02ztM12+12gt22.Integrating the inequality above to t from s(t≤s≤t+1) to t+1, we have(72)Et+1≤Es+m02∫tt+1zsM12ds+12gLb22,and integrating (72) to s from t to t+1,(73)Et+1≤∫tt+1Esds+m02∫tt+1zsM12ds+12gLb22.
Besides, by the Hölder inequality, Young inequality, and (13), we can infer from (69) that(74)Es≤12us02+γ~2uspp+β~2mesΩ≤12zsM12+γ~2uspp+β~2mesΩ.
On the contrary,(75)Et+1≥12ut+102+γ~1ut+1pp-β~1mesΩ≥γ~1ut+1pp-β~1mesΩ.
Combining (73) and (74), we have(76)Et+1≤1+m02∫tt+1zsM12ds+γ~2∫tt+1usppds+12gLb22+β~2mesΩ.By Lemma 11, it follows from (76) that(77)Et+1≤C1+m02+γ~21+gLb22+12gLb22+β~2mesΩ,and by (75)–(77), we have(78)ut+1pp≤1γ~11+m02+γ~2C1+gLb22+12gLb22+β~1+β~2mesΩ.Let (79)ρ1=1γ~11+m02+γ~2C1+gLb22+12gLb22+β~1+β~2mesΩ.Then the proof is complete.Lemma 13 (bounded uniformly absorbing set).
Let (13)-(14) hold, and g∈Σ. Then there exist positive ρ0, ρ1, which depend only on gLb2, such that, for any bounded (in M1) subset B, there is T1=T1(BM1)=T0+1 such that(80)Ugt,τzτM12⩽ρ0,utpp≤ρ1hold for all t-τ⩾T1 and all zτ∈B.For brevity, in the sequel, letB0 be the bounded absorbing set obtained in Lemmas 9 and 12; that is,(81)B0=z=u,ηt∈M1:zM12≤ρ0,upp≤ρ1.Lemma 14.
There is a positive constantρ2, for any zτ=(uτ,ητ)∈B0 (from (81)); the following estimate (82)∫tt+1uts22+uts02ds≤ρ2,holds for any t-τ≥0.Proof.
By Lemma12, for any t-τ≥0, we integrate (70) to t from t to t+1; then we have(83)Et+1+∫tt+1uts22+uts02ds≤Et.
According to Lemma13, Corollary 10, and Lemma 12, we know that E(t) is bounded for any t-τ≥0, so there is a positive constant ρ2, and the conclusion is true.


## 3.2. Uniform Attractors

In the following, we will prove the existence of uniform attractors for system (10) with initial-boundary conditions (11) in M1 by using the method of asymptotic contractive function.Lemma 15.
Letf satisfy (13)-(14); zn(t)=(un(t),ηtn) is the sequence of solutions of (10) with initial data zτn=(uτn,ητn)∈B0 and gn∈Σ(n=1,2,…); then there is a subsequence unk of un(t) that converges strongly in L2(0,T;L2(Ω)) and(84)limk→∞⁡limj→∞⁡∫0T∫Ωgnkx,s-gnjx,sunkx,s-unjx,sds=0.Proof.
By Lemmas13 and 14, the sequence un is uniformly bounded in L2(0,T;H01(Ω)) and dun/dt is uniformly bounded in L2(0,T;H-1(Ω)). Since H01(Ω) is reflexive, is L2(0,T;H01(Ω)). Let X=H01(Ω), H=L2(Ω), and Y=H-1(Ω); then there is a subsequence unk(t) of {un(t)} that converges strongly in L2(0,T;L2(Ω)) by Lemma 7. One can write these as(85)limk→∞⁡limj→∞⁡∫0T∫Ωunkx,s-unjx,s2ds=0.Then by using Hölder inequality, we have (86)limk→∞⁡limj→∞⁡∫0T∫Ωgnkx,s-gnjx,sunkx,s-unjx,sds≤limk→∞⁡limj→∞⁡∫0T∫Ωgnkx,s-gnjx,s2ds1/2∫0T∫Ωunkx,s-unjx,s2ds1/2≤2TgLb2limk→∞⁡limj→∞⁡∫0T∫Ωunkx,s-unjx,s2ds1/2=0.Theorem 16.
LetΩ be a bounded domain in R3 with smooth boundary, and let μ and f satisfy (4)-(5) and (13)-(14), respectively. Assume further that g0∈Lb2(R;L2(Ω)) and Σ is the hull of g0 in Lw,loc2(R;L2(Ω)). {Uσ(t,τ)}, σ∈Σ, is the family of processes generated by the solutions of (10) with initial data zτ∈M1. Then {Uσ(t,τ)}, σ∈Σ, is uniformly (with respect to σ∈Σ) asymptotic compact in M1.Proof.
For anyzτi=(uτi,ηiτ)∈B0 and gi∈Σ, let zi(t)=(ui(t),ηit) be the corresponding solution with respect to initial data zτi(i=1,2) and gi∈Σ; that is, zi(t) is the solution of (10) with initial-boundary conditions (11).
For convenience, we denoteω(t)=u1(t)-u2(t) and ξt=η1t-η2t; then ω(t) satisfies the following equation:(87)ωtt-Δωtt-Δωt-∫0∞μsΔξtsds+fu1t-fu2t=g1t-g2tξtt=-ξst+ω,with initial-boundary conditions(88)ω∂Ω=0,ξtx,s∂Ω×R+=0,t≥τ,ωx,τ=ωτx=uτ1x-uτ2x,x∈Ω,ξτx,s=η1τx,s-η1τx,s,x,s∈Ω×R+.
Multiplying (87) by ω(t) and integrating in Ω, we obtain(89)12ddtω22+ω02+ξtμ,V2+ω02+ξts,ξstsμ,V+∫Ωfu1-fu2u1-u2=∫Ωg1t-g2tu1-u2.
By Lemma1 and assumption (14), we have (90)ξts,ξstsμ,V≥δ2ξtμ,V2,∫Ωfu1-fu2u1-u2≥-lω22.Using the Poincaré inequality, it follows that(91)12ddtω22+ω02+ξtμ,V2+λ12ω22+12ω02+δ2ξtμ,V2≤lω22+∫Ωg1t-g2tu1-u2.Let ϱ=min⁡(λ1,1,δ); then we get (92)ddtω22+ω02+ξtμ,V2+ϱω22+ω02+ξtμ,V2≤2lu1-u222+2∫Ωg1t-g2tu1-u2.
By theGronwall  Lemma, we have(93)ωt22+ωt02+ξtμ,V2≤e-ϱt-τωτ22+ωτ02+ξτμ,V2+2l∫τte-ϱt-su1s-u2s22ds+2∫τte-ϱt-s∫Ωg1s-g2su1s-u2sds.
For anyε>0, there exists T2=T2(ε)=max⁡(T1,1/ϱln⁡(ε/zτM12))≥T1, such that(94)ωt22+ωt02+ξtμ,V2≤ε+ψtholds for any t≥T2; here (95)ψt=ψtu1,u2,g1,g2=2l∫τte-ϱt-su1s-u2s22ds+2∫τte-ϱt-s∫Ωg1s-g2su1s-u2sds.Hereafter, we verify that there is T≥T2 such that ψT is an asymptotic contractive function on B0. In fact (96)ψt=2l∫τte-ϱt-su1s-u2s22ds+2∫τte-ϱt-s∫Ωg1s-g2su1s-u2sds≤2l∫τT1e-ϱt-su1s-u2s22ds+2∫τT1e-ϱt-s∫Ωg1s-g2su1s-u2sds+2l∫T1te-ϱt-su1s-u2s22ds+2∫T1te-ϱt-s∫Ωg1s-g2su1s-u2sds≤e-ϱt-T12l∫τT1u1s-u2s22ds+2∫τT1∫Ωg1s-g2su1s-u2sds+2l∫T1tu1s-u2s22ds+2∫T1t∫Ωg1s-g2su1s-u2sds≤2T1e-ϱt-T12l+1MB+gLb22+2l∫0t-T1u1s+T1-u2s+T122ds+2∫0t-T1∫Ωg1s+T1-g2s+T1u1s+T1-u2s+T1ds,where MB is from Corollary 10. Let(97)φt=2l∫0t-T1v1s-v2s22ds+2∫0t-T1∫Ωσ1s-σ2sv1s-v2sds,where vi(s)=ui(s+T1)∈B0 and σi(s)=gi(s+T1)∈Σ(i=1,2). For any ε>0, let t=T≥T1+1/ϱln⁡2T1((2l+1)MB+gLb22)/ε be fixed; then (98)ψTu1,u2,g1,g2≤ε+φTu1,u2,g1,g2.
Combining Lemma15 and Definition 5, we know that φT is the contractive function on B0×B0 and ψT is the asymptotic contractive function on B0×B0.
Then{Ug(t,τ)}t≥0, g∈Σ, is uniform (with respect to g∈Σ) asymptotic compact in M1 by Theorem 6.We also recall the following useful result, whose proof is simple and we omit it.Lemma 17 (see [15]).
LetX be a reflexive Banach space and xn⇀0 in X. Then, for each compact (in X∗) subset B⊂X∗, the uniform convergence holds: for any ε>0, there is Nε, depending only on ε, such that (99)f,xnX∗≤εforalln≥Nεandallf∈B.Proof of Theorem3.
Theorem2 implies that the family of processes {Ug(t,τ)}, g∈Σ, corresponding to (1), has a compact (in M1) uniform (with respect to g∈Σ) attracting set which is bounded in M1; consequently, as a direct application of the abstract theorem [29, Chapter IV, Theorem  3.1], we obtain the existence of a compact uniform (with respect to g∈Σ) attractor A, and A⊂M1.
We remark that the above existence does not require any continuity of the family of processes (e.g., see [29]). However, in order to obtain the explicit form of A, that is, equality (29), we need some continuity. Moreover, since the symbol space Σ now has onlyweak compactness, we need to verify the correspondingweak continuity.
Hence, in order to obtain (29), we need to verify the following: for any fixed τ∈R and t⩾τ, if znτ→zτ in M1 and gn⇀g with respect to the local weak convergence topology of Lloc2(R;L2(Ω)), then (100)Ugnt,τznτ⟶Ugt,τzτinM1.
In the following, we will verify that the continuity above holds onA×Σ, which obviously is a compact uniformly attracting set for the corresponding skew-product flow semigroup (generated by the family processes Ug(t,τ)zτ, g∈Σ, and the shift semigroup {T(t)}t⩾0). We will complete the proof of the week continuity.
Letzn(t)=(un,ηnt), let zn(t) be the solutions of (10) with initial data zτn∈A(n=1,2,…), and let the time-dependent forcing gn∈Σ, z(t) be the solutions of (10) with initial data zτ∈A and the time-dependent forcing g∈Σ; we assume the following hypotheses:(101)zτn⟶zτ,inM1,(102)gn⟶g,inLw,loc2R,L2Ω.Since (10) holds as an equality in Lq(τ,T;H-1(Ω)) (q is conjugate to p from (13)), this means that for any T(>τ) fixed(103)un⇀u,inL2τ,T;H01Ω,where u=Π1Ug(t,τ)zτ∈L2(τ,T;H01(Ω)), and Π1 is the projector from X×Y to X. Since A is bounded in M1, following Corollary 10, we know that there is M such that(104)supg∈Σ⁡supτ∈R⁡supt⩾τ⁡Ugt,τAM12⩽M<∞.Hence, (105)⋃g∈ΣΠ1Ugt,τzτ:t∈τ,T,zτ∈Ais  bounded  inL2τ,T;H01Ω,⋃g∈Σ∂tΠ1Ugt,τzτ:t∈τ,T,zτ∈Ais  bounded  inL2τ,T;H-1Ω.Then by the compactness lemma (Lemma 7) we know that(106)⋃g∈ΣΠ1Ugt,τzτ:t∈τ,T,zτ∈Ais  compact  inL2τ,T;L2Ω.It is (107)limn→∞⁡∫τT∫Ωuns-us2ds=0.Denote ω(t)=un(t)-u(t), ξt=ηnt-ηt, ωτ=unτ(t)-uτ, and zn(t)=(un(t),ηnt)=Ugn(t,τ)znτ-Ug(t,τ)zτ, znτ,zτ∈A, n=1,2,…. The proof of the following inequality is similar to (93): (108)ωt22+ωt02+ξtμ,V2≤e-ϱt-τωτM12+2l∫τtuns-us22ds+2∫τt∫Ωgns-gsuns-usds.
Due to (107) and (102), we only need to show that if gn→g in Lw,loc2(R;L2(Ω)), then(109)∫τtgnx,s-gx,s,wx,sds⟶0uniformly on L2(τ,t;L2(Ω))-compact set. But this is a direct application of Lemma 17. Therefore we complete the proof of thecontinuity on A×Σ.
Based on thecontinuity claim above, and by constructing a skew-product flow on A×Σ and applying [29, Chapter IV, Theorem  5.1], we obtain the structure equality (29). This completes the proof of Theorem 3.



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*Source: 5340489-2016-05-30.xml*