# Pullback Attractors for a Class of Semilinear Second-Order Nonautonomous Evolution Equations with Hereditary Characteristics

Authors: Fang-hong Zhang; Xiao-hua Chen
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we investigate the long-time behavior for the nonautonomous semilinear second-order evolution equation∂2u/∂t2−Δu−Δ∂u/∂t−Δ∂2u/∂t2=ft,ux,t−ρt+gt,x,inτ,∞×Ω with some hereditary characteristics, where Ω is an open-bounded domain of ℝNN≥3 with smooth boundary ∂Ω. Firstly, we establish the existence of solutions for the second-order nonautonomous evolution equation by the standard Faedo–Galerkin method, but without the uniqueness of solutions. Then by proving the pullback asymptotic compactness for the multivalued process Ut,τ on CH01Ω,H01Ω, we obtain the existence of pullback attractors in the Banach spaces CH01Ω,H01Ω for the multivalued process generated by a class of second-order nonautonomous evolution equations with hereditary characteristics and ill-posedness.


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## Body

## 1. Introduction

The study of nonlinear dynamics is a fascinating question which is at the very heart of understanding of many important problems of the natural sciences. The long-time behavior of PDEs can be described in the terms of attractors of the corresponding semigroups, such as Babin and Vishik [1], Chepyzhov and Vishik [2], Chueshov and Lasiecka [3], Hale [4], Ladyzhenskaya [5], or Temam [6], and the references therein. The study of pullback attractor for infinite dimensional dynamical systems has attracted much attention and has made fast progress in recent decades [7–13].In this paper, we consider the following nonautonomous semilinear second-order evolution equation with delays:(1)∂2u∂t2−Δu−Δ∂u∂t−νΔ∂2u∂t2=ft,ux,t−ρt+gt,x,inτ,∞×Ω,ut,x=ϕt−τ,x,t∈τ−h,τ,x∈Ω,∂ut,x∂t=∂ϕt−τ,x∂t,t∈τ−h,τ,x∈Ω,ut,x=0,onτ,∞×∂Ω,where Ω is an open-bounded domain of ℝNN≥3 with smooth boundary ∂Ω, τ is the initial time, and ϕ is the initial data on the interval τ−h,τ with h>0.The nonlinearityf⋅ and the external force gt,x satisfy the following conditions, respectively.InA1, there exist positive constants α1 and α2 such that the functions f∈Cℝ×ℝN;ℝ and ρ∈Cℝ;0,h satisfy(2)ft,v2≤α12+α22v2,∀t∈ℝ,v∈ℝN,(3)ρ′t≤ρ∗<1,∀t∈ℝ.InA2, the external force gt,x belongs to the space Lloc2ℝ,L2Ω such that(4)∫−∞τeλrgr,⋅L2Ω2dr<∞,∀τ∈ℝ,λ>0.Whenν=0 and without variable delays, Equation (1) becomes the usual strongly damped wave equation:(5)∂2u∂t2−Δu−Δ∂u∂t=fu+gt,x.Its asymptotic behavior has been studied extensively in terms of attractors [1,14–20] . The long-time behavior for the strongly damped wave equation with delays has been investigated in Refs. [7,12].For each fixedν>0 and without variable delays, Equation (1) becomes(6)∂2u∂t2−Δu−Δ∂u∂t−νΔ∂2u∂t2=fu+gt,x.It is a special form of the so-called improved Boussinesq equation (see [21–24]) with damped term −Δut, which was used to describe ion-sound waves in plasma by Makhankov [22,25] and also known to represent other sorts of “propagation problems” of, for example, lengthways waves in nonlinear elastic rods and ion-sonic waves of space transformations by a weak nonlinear effect [21]. Carvalho and Cholewa [26] presented systematic results including the existence-uniqueness and long-time behavior of Equation (6) by using the semigroup approach. The long-time behavior of, especially the global attractor, exponential attractors has been extensively studied by several authors [26–29] . For the nonautonomous semilinear second-order evolution (6) with the memory term, we get(7)utt−Δu−Δut−∫0∞kεsΔutt−sds−νΔutt+fu=gx,t.Zhang et al. in Ref. [30] constructed the existence of robust family of exponential attractors while the nonlinearity is critical and the time-dependent external forcing term is assumed to be only translation-bounded.Indeed, for Equation (6), in all above results, we require the solution operator given as follows:(8)St:u0↦ut.To be well-defined and continuous in a proper phase space. However, for many interesting problems, the well-posedness of the solution operatorSt is not known or does not hold true [11–13, 31–33] .To the best of our knowledge, the long-time dynamics of Equation (1) with hereditary characteristics has not been considered by predecessors. There are some barriers encountered. On the one hand, Equation (1) contains the term −Δutt, and it is essentially different from the usual wave equation in Refs. [1, 7, 12, 14–20]. For example, the wave equation has some smoothing effect; for example, although the initial data only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. However, for Equation (1), if the initial data ϕ∈CH01Ω,H01Ω, then the solution ut⋅ is always in CH01Ω,H01Ω and has no higher regularity because of −Δutt, and it will cause some difficulties [26–29]. On the other hand, suppose that A1−A2 hold true, then g∈Lloc2ℝ;H and ϕ∈CH01Ω,H01Ω, then the uniqueness of the weak solutions for Equation (1) are lost; that is, we need to overcome some difficulties brought by ill-posedness. In addition, the delay term also causes some difficulties to obtain the pullback attractors.This paper is organized as follows. In Section1, we have expounded on research progress regarding our research problem and have given some assumptions. In Section 2, we introduce some notations and functions spaces, and we recall some useful results on nonautonomous multivalued dynamical systems and pullback attractors. In Section 3, we prove the existence of solutions for Equation (1) in ϕ∈CH01Ω,H01Ω. The existence of pullback attractor for the multivalued process Ut,τ corresponding to Equation (1) is proved in Section 4.


## 2. Preliminaries

Next, we iterate some definitions and abstract results concerning the multivalued dynamical systems and the pullback attractor, which is necessary to obtain our main results [7–13,34] .LetX be a complete metric space with metric dX⋅,⋅, and let PX be the class of nonempty subsets of X. Denoted by HXsemi⋅,⋅ the Hausdorff semidistance between two nonempty subsets of a complete metric space X can be defined as(9)HXsemiA,B=supa∈Ainfb∈BdXa,b.Definition 1.
A family of mappingsUt,τ:X⟶PX and t≥τ, τ∈ℝ is called to be a multivalued process if(10)Uτ,τx=x,∀τ∈ℝ,x∈X,(11)Ut,rUr,τx=Ut,τxfor allτ≤r≤t,x∈X.
LetD be a nonempty class of parametrized sets D=Dtt∈ℝ⊂PX.Definition 2.
A collectionD of some families of nonempty closed subsets of X is said to be inclusion-closed if for each D=Dtt∈ℝ∈D, we have(12)D˜t:D˜tis a non empty subset ofDt,∀t∈ℝ.
which also belongs toD.Definition 3.
LetUt,τ be a multivalued process on X, then we get those as follows:(1)
Q=Qtt∈ℝ∈D is called a D-pullback absorbing set for Ut,τ if for any ℬ=Btt∈ℝ∈D and each t∈ℝ, there exists a t0=t0ℬ,t∈ℝ+ such that(13)Ut,t−sBt−s⊂Qt,∀s≥t0(2)
Ut,τ is said to be D-pullback asymptotically upper-semicompact in X with respect to ℬ if for each fixed t∈ℝ, any sequence yn∈Ut,t−snxn has a convergent subsequence in X whenever sn⟶+∞n⟶∞ and xn∈Bt−sn with ℬ=Btt∈ℝ∈DTheorem 1.
A family of nonempty compact subsetsA=Att∈ℝ∈D of X is called to be a D-pullback attractor for the multivalued process Ut,τ if(1)
A=Att∈ℝ is an invariant, i.e.,(14)Ut,τAτ=At,∀t≥τ,τ∈ℝ.(2)
A attracts every member of D, i.e., for every ℬ=Btt∈ℝ∈D and any fixed t∈ℝ, we get(15)lims⟶+∞HXsemiUt,t−sB,At=0.Theorem 2.
LetUt,τ be a multivalued process on Banach space X, and let Q=Qtt∈ℝ be a D− pullback absorbing set for Ut,τ in D. Suppose that U can be written as(16)Ut,τ=U1t,τ+U2t,τ,∀t≥τ,
and for any fixedt∈ℝ, then we get those as follows:(1)
lims⟶∞U1t,t−sQt−sX=0(2)
For any fixeds>0, every sequence yn∈U2t,t−sQt−s is a Cauchy sequence in X
ThenUt,τ is D-pullback asymptotically upper-semicompact in X.Theorem 3.
LetD be an inclusion-closed collection of some families of nonempty closed subsets of X and Ut,τ be a multivalued process on X. Also, U has a closed values and let Ut,τx be upper semicontinuous in x for fixed t≥τ,τ∈ℝ. Suppose that Ut,τ is D-pullback asymptotical upper-semicompact in X, then Ut,τ has a D-pullback absorbing set Q=Qtt∈ℝ∈D, and Qt is closed for every t∈ℝ. Then, the D-pullback attractor A=Att∈ℝ is unique for each t∈ℝ and is given by(17)At=∩T∈ℝ+∪s≥TUt,t−sQt−s¯.LetH=L2Ω and V=H01Ω, which are Hilbert spaces for the usual inner products and associated norms. Let Au=−Δu for any u∈DA, where DA=u∈V:Au∈H=H01Ω∩H2Ω. Note that DA is also a Hilbert space for the norm uDA=Au, u∈DA.LetX be a Banach space with norm ⋅X. Let h>0 be a given positive number, which will denote the delay time, and let CX denote the Banach space C0−h,0;X with the sup-norm, then we get(18)ψCX≔=sups∈−h,0ψsX,forψ∈CX.We can denote byCX,X the Banach spaces CX∩C1−h,0;X with the norm ⋅CX,X that is defined by(19)ψCX,X2≔=ψCX2+ψ′CX2,forψ∈CX,X.Givenτ∈ℝ,T>τ and u:τ−h,T⟶X, for each t∈τ,T, we can denote as(20)ut:−h,0⟶X.Denote the function defined on by(21)uts=ut+s,s∈−h,0.Without the loss of generality, we assume thatν=1 in the following discussion.


## 3. Existence of Solutions

In this section, we want to prove the existence of solutions which can be obtained by the standard Faedo–Galerkin methods (see [1,6,35]), and the multivalued evolution processes corresponding to Equation (1) will be constructed. We only give the sketch of proof, and the details similar to the proof of Theorem 4 in Ref. [2], Sec. XV.3 and the arguments in [6] Sec. IV. 4.4.Theorem 4.
Suppose thatA1−A2 holds true, g∈Lloc2ℝ;H and ϕ∈CV,V, then there exist solutions ut of Equation (1) such that(22)u∈C1τ−h,T;V,∂u∂t∈C1τ−h;V,∀T>τ.Proof.
(Sketch)
LetAu=−Δu for any u∈DA, where DA=u∈V:Au∈H=H01Ω∩H2Ω. Since A is self-adjoint, positive operator and has a compact inverse, and there exists a complete set of eigenvectors ωii=1∞ in H, and the corresponding eigenvalues λii=1∞ satisfy(23)Aωi=λiωi,0<λ1≤λ2≤…≤λi≤…⟶+∞,i⟶+∞.
SettingVm=spanω1,ω2,…,ωm and Pm is the orthogonal projection onto Hm, then we get(24)Pmu=∑i=1mu,ωiωi,u∈H.
We consider the approximate solutions of Equation (1) in the form(25)umt=∑j=1mαjmtwj.
Thenumt satisfies(26)∂2um∂t2−Δum−Δ∂um∂t−νΔ∂2um∂t2=ft,umx,t−ρt+Pmgt,x,umt,x=Pmϕt−τ,x,t∈τ−h,τ,∂umt,x∂t=∂Pmϕt−τ,x∂t,t∈τ−h,τ.
Letvm=um′+ηum0<η<1, and we write Equation (26) as(27)ddtvm−ηvm+η2um−1−η+η2Δum−Δddtvm−1−ηΔvm=ft,umt−ρt+Pmgt.
Multiplying Equation (27) by vm in L2Ω, we infer that(28)12ddtvm2+∇vm2+η2um2+1−η+η2∇um2−ηvm2+η3um2+η1−η+η2∇um2+1−η∇vm2=ft,umt−ρt,vm+gt,vm.
Noting (2), using Young’s inequality, we get that(29)ft,umt−ρt,vm≤η2vm2+α222ηumt−ρt2+α12Ω2η,and(30)gt,vm≤η2vm2+12ηgt2.
By the Poincáre inequalityλ1u2≤∇u2, we get that(31)12ddtvm2+∇vm2+η2um2+1−η+η2∇um2+λ11−η2−2ηvm2+1−η2∇vm2+η3um2+η1−η+η2∇um2≤α222ηumt−ρt2+12ηgt2+α12Ω2η.
Choosingη=min1/3,λ1/λ1+6, we infer that(32)12ddtvm2+∇vm2+η2um2+1−η+η2∇um2+ηvm2+∇vm2+η2um2+1−η+η2∇um2≤α222ηumt−ρt2+12ηgt2+α12Ω2η.
Note that for anyη>0, we have(33)vm2+∇vm2+η2um2+1−η+η2∇um2∼vm2+∇vm2+um2+∇um2∼∇vm2+∇um2.
Now integrating (32) from τ to t, we get(34)∇vmt2+∇umt2+η∫τt∇vms2+∇ums2ds≤∇vmτ2+∇umτ2+α22η∫τtums−ρs2ds+1η∫τtgs2ds+α12Ωηt−τ.
In view ofρs∈0,h and the fact(35)11−ρ′s≤11−ρ∗,for all s∈ℝ, setting r=s−ρs, we arrive at(36)α22η∫τtums−ρs2ds≤α22η1−ρ∗∫τ−hτums2ds+∫τtums2ds≤α22ηλ11−ρ∗ϕV,V2+α22ηλ11−ρ∗∫τt∇vms2+∇ums2ds.
Thus, we obtain that(37)∇vmt2+∇umt2≤∇vmτ2+∇umτ2+α22ηλ11−ρ∗ϕV,V2+1η∫τtgs2ds+α12Ωηt−τ+α22ηλ11−ρ∗∫τt∇vms2+∇ums2ds.
By the integral form of Gronwall lemma, we infer that(38)∇vmt2+∇umt2≤CeCt−τϕV,V2+t−τ+∫τtgs2ds.
Then,(39)um,um′is a bounded set ofL∞τ−h,T;V×Vasm⟶∞.Thus, we can extract a subsequence, still denoted as m, such that(40)um⟶uinL∞τ−h,T;Vweak−star,asm⟶∞,and(41)um′⟶uinL∞τ−h,T;Vweak−star,asm⟶∞.
Furthermore,(42)um⟶uinL2Ω×τ−h,Tstrongly,
and(43)um⟶ufor almost everyt,x∈τ−h,T×Ω.
Note thatf∈Cℝ×ℝN;ℝ, then(44)fum⟶fuinL2τ−h,T;Vweakly.
We then pass the limit in Equation (26), and we can find that u is a solution of Equation (1) such that(45)u∈L∞τ−h,T;Vandu′∈L∞τ−h,T;V.
The continuity properties(46)u∈C1τ−h,T;V,∂u∂t∈C1τ−h,T;V,∀T>τ,can be established with the methods indicated in Section II.3 and II.4 in the research by Temam [6] (e.g., Theorem 3.1 and 3.2).
This completes the proof.Remark 1.
According to Theorem4, we can define a family of multivalued mappings Ut,τ on CV,V as(47)Ut,τ:CV,V⟶CV,V,corresponding to Equation (1) by(48)Ut,τϕ=ut⋅;τ,ϕ|u⋅are solutions of Equation1withϕ∈CV,V.
Then,Ut,τ is multivalued process on CV,V.


## 4. Pullback Attractors inCV,V

We denote byℜ the set of all functions r:ℝ⟶0,+∞ such that(49)limt⟶−∞eδ/3tr2t=0,where δ>0 is defined in (50) and is denoted by DCV,V in the class of all families D=Dtt∈ℝ⊂PCV,V such that Dt⊂N¯0,rDt, for some rD∈ℜ, where PCV,V denotes the family of all nonempty subsets of CV,V, and N¯0,rDt denotes the closed ball in CV,V centered at zero with radius rDt.Lemma 1.
(Existence ofD-pullback absorbing set) Suppose that A1−A2 holds true, then g∈Lloc2ℝ;H and ϕ∈CV,V, and there exists a constant δ satisfying(50)0<δ∗<δ<δ∗<1,where δ∗ satisfies(51)δ∗4=9α22eδ∗h/351−ρ∗.(52)δ∗=min34,λ1λ1+3.
λ1 is the positive constant in the Poincáre inequality. Then the multivalued process Ut,τ possesses a DCV,V-pullback absorbing set QCV,V in DCV,V.Proof.
Letv=u′+δu, and we write Equation (1) as(53)ddtv−δv+δ2u−1−δ+δ2Δu−Δddtv−1−δΔv=ft,ut−ρt+gt.
Multiplying Equation (1) by v in L2Ω, we infer that(54)12ddtv2+∇v2+δ2u2+1−δ+δ2∇u2−δv2+δ3u2+δ1−δ+δ2∇u2+1−δ∇v2=ft,ut−ρt,v+gt,v.
Noting (2), using Young’s inequality, for ϵ1,ϵ2>0, we infer that(55)ft,ut−ρt,v≤ε12v2+α222ε1ut−ρt2+α12Ω2ε1,and(56)gt,v≤ε22v2+12ε2gt2.
Applying the Poincáre inequality, we get that(57)12ddtv2+∇v2+δ2u2+1−δ+δ2∇u2+λ11−δ2−δ−ε12−ε22v2+1−δ2∇v2+δ3u2+δ1−δ+δ2∇u2≤α222ε1ut−ρt2+α12Ω2ε1+12ε2gt2.
Letδ′>0 be determined later, then we infer that(58)ddteδ′tv2+∇v2+δ2u2+1−δ+δ2∇u2=δ′eδ′tv2+∇v2+δ2u2+1−δ+δ2∇u2+eδ′tddtv2+∇v2+δ2u2+1−δ+δ2∇u2≤2δ+ϵ1+ϵ2+δ′−λ11−δeδ′tv2+δ+δ′−1eδ′t∇v2+δ2δ′−2δ3eδ′tu2+δ′−2δ1−δ+δ2eδ′t∇u2+α22ϵ1eδ′tut−ρt2+α12Ωϵ1eδ′t+1ϵ2eδ′tgt2.
Now integrating (58) from τ to t, we get(59)eδ′tv2+∇v2+δ2u2+1−δ+δ2∇u2≤eδ′τvτ2+∇vτ2+δ2uτ2+1−δ+δ2∇uτ2+2δ+ϵ1+ϵ2+δ′−λ11−δ∫τteδ′svs2ds+δ+δ′−1∫τteδ′s∇vs2ds+δ2δ′−2δ3∫τteδ′sus2ds+δ′−2δ1−δ+δ2∫τteδ′s∇us2ds+α22ϵ1∫τteδ′sus−ρs2ds+α12Ωϵ1∫τteδ′sds+1ϵ2∫τteδ′sgs2ds.
Note thatρs∈0,h and the fact(60)11−ρ′s≤11−ρ∗,for all s∈ℝ.
Settingr=s−ρs, we arrive at(61)α22ε1∫τteδ′sus−ρs2ds≤α22eδ′hε11−ρ∗∫τ−hτeδ′rur2dr+∫τteδ′rur2dr≤α22eδ′h+τhϕV,V2ε1λ11−ρ∗+α22eδ′hε11−ρ∗∫τteδ′rur2dr.
Thus, we obtain(62)eδ′tv2+∇v2+δ2u2+1−δ+δ2∇u2≤eδ′τvτ2+∇vτ2+δ2uτ2+1−δ+δ2∇uτ2+2δ+ϵ1+ϵ2+δ′−λ11−δ∫τteδ′svs2ds+δ+δ′−1∫τteδ′s∇vs2ds+δ2δ′−2δ3+α22eδ′hϵ11−ρ∗∫τteδ′sus2ds+δ′−2δ1−δ+δ2∫τteδ′s∇us2ds+α12Ωϵ1δ′eδ′t+1ϵ2∫τteδ′sgs2ds+α22eδ′h+τhϕV,V2ϵ1λ11−ρ∗.
Choosingε1=ε2=δ′=δ/3, and noting that(63)δ∗=min34,λ1λ1+3,δ<δ∗<1,we get(64)2δ+ε1+ε2+δ′−λ11−δ<0,δ+δ′−1<0.
Note thatδ∗ satisfies δ∗4=9α22eδ∗h/3/51−ρ∗ if 0<δ∗<δ, then we have(65)δ∗4>9α22eδ∗h/351−ρ∗.
Then we have(66)δ2δ′−2δ3+α22eδ′hε11−ρ∗<0,δ′−2δ1−δ+δ2<0.
Therefore,(67)eδ/3tv2+∇v2+δ2u2+1−δ+δ2∇u2≤eδ/3tvτ2+∇vτ2+δ2uτ2+1−δ+δ2∇uτ2+9α12Ωδ2eδ/3t+δ3∫τteδ/3sgs2ds+3α22eδ/3h+τhϕV,V2δλ11−ρ∗.
Settingt+θ instead of t, where θ∈−h,0, and multiplying by e−δ/3t+θ, we get(68)vt+θ2+∇vt+θ2+δ2ut+θ2+1−δ+δ2∇ut+θ2≤e−δ/3t+θeδ/3τvτ2+∇vτ2+δ2uτ2+1−δ+δ2∇uτ2+9α12Ωδ2+δ3e−δ/3t+θ∫τt+θeδ/3sgs2ds+3α22e−δ/3teδ/32h+τhϕV,V2δλ11−ρ∗.
Note thatv=u′+δu, and by (68), we infer that(69)utCV,V2=utCV2+ut′CV2=sups∈−h,0∇ut+θ2+sups∈−h,0∇u′t+θ2≤sups∈−h,0∇ut+θ2+2sups∈−h,0∇vt+θ2+2δ2∇ut+θ2≤1+2δ2sups∈−h,0∇ut+θ2+2sups∈−h,0∇vt+θ2≤Cδe−δ/3teδ/3τ+hvτ2+∇vτ2+δ2uτ2+1−δ+δ2∇uτ2+Cδ9α12Ωδ2+Cδδ3e−δ/3teδ/3h∫τt+θeδ/3sgs2ds+3Cδα22e−δ/3teδ/32h+τhϕV,V2δλ11−ρ∗≤C1e−δ/3t+C2+C3e−δ/3t∫−∞teδ/3sgs2ds,where(70)C1=Cδeδ/3τ+hvτ2+∇vτ2+δ2uτ2+1−δ+δ2∇uτ2+3Cδα22eδ/32h+τhϕV,V2δλ11−ρ∗,(71)C2=Cδ9α12Ωδ2,
and(72)C3=Cδδ3eδ/3h.
Now, we denote byRt the nonnegative number given for each t∈ℝ by(73)R2t=2C2+C3e−δ/3t∫−∞teδ/3sgs2ds,
and consider the family of closed bounded ballsQCV,V=Qtt∈ℝ in CV,V that is defined by(74)Qt=φ∈CV,V:φCV,V≤Rt.
Clearly,QCV,V∈DCV,V, and moreover, according to (49) and (69), the family of QCV,V is DCV,V− pullback absorbing for the multivalued process Ut,τ on CV,V.
This completes the proof.Lemma 2.
Under the assumptions of Lemma1, the multivalued process Ut,τ on CV,V is DCV,V− pullback asymptotically upper-semicompact.Proof.
Note that for anyT≥t−s with s≥0, we get(75)UT,t−sϕ=uT⋅;t−s,ϕ|u⋅is a solution of Equation1withϕ∈Qt−s,where QCV,V=Qtt∈ℝ is DCV,V− pullback absorbing for the multivalued process Ut,τ on CV,V. Now, we decompose Equation (1) as follows:(76)uT,x=wT,x+yT,x,where wT,x and yT,x satisfy the following equations, respectively:(77)∂2w∂T2−Δw−Δ∂w∂T−Δ∂2w∂T2=0,T≥t−s,wT,x=ϕT−t+s,x,t−s−h≤T≤t−s,∂wT,x∂T=∂ϕT−t+s,x∂T,t−s−h≤T≤t−s,w|∂Ω=0,T≥t−s,(78)∂2y∂T2−Δy−Δ∂y∂T−Δ∂2y∂T2=fT,ux,T−ρT+gT,x,T≥t−s,yT,x=0,t−s−h≤T≤t−s,∂yt,x∂T=0,t−s−h≤T≤t−s,y|∂Ω=0,T≥t−s.
For Equation (77), similar to the proof of Theorem 4, we can easily obtain(79)wtCV,V2=wtCV2+wt′CV2≤Ce−δ/3sϕCV,V2.
That is, we get(80)lims⟶∞U1t,t−sQt−sCV,V=0.
Now, from Theorem2, we only need to show that for any fixed s>0, every sequence yn∈U2t,t−sQt−s is a Cauchy sequence in CV,V.
We investigate two solutions ofuT1 and uT2 for Equation (1) corresponding to the initial data ϕ1 and ϕ2, respectively. Let zT=y1T−y2T, then zT satisfies(81)∂2z∂T2−Δz−Δ∂z∂T−Δ∂2z∂T2=fT,u1x,T−ρT−fT,u2x,T−ρT,T≥t−s,zT,x=0,t−s−h≤T≤t−s,∂zt,x∂T=0,t−s−h≤T≤t−s,z|∂Ω=0,T≥t−s.
Multiplying Equation (81) by z′ in L2Ω, we infer(82)12ddT∇z′T2+z′T2+∇z2+∇z′T2=fT,u1x,T−ρT−fT,u2x,T−ρT,z′T.
Integrating fromt−s to t+θ (where θ∈−h,0), we have(83)∇z′t+θ2+z′t+θ2+∇zt+θ2≤2∫t−st+θfT,u1x,T−ρT−fT,u2x,T−ρT,z′TdT≤2∫t−st+θ∫ΩfT,u1x,T−ρT−fT,u2x,T−ρT‖z′TdxdT≤2z′TL2Ω×t−s,tfT,u1x,T−ρT−fT,u2x,T−ρTL2Ω×t−s,t.
LetunT∈UT,t−sϕn be with ϕn∈Qt−s. According to (69), without of generality, we can assume that(84)un⟶uweakly star inL∞t−s−h,t;V,
and(85)un′⟶uweakly star inL∞t−s−h,t;V.
Then, we infer that(86)un⟶uinL∞t−s−h,t;V,
and(87)unT,x⟶uT,xn⟶∞fora.e.T,x∈t−s−h,t×Ω.
Note that whenf∈Cℝ×ℝN;ℝ, by (87), we have(88)fT,unT,x⟶fT,uT,xn⟶∞fora.e.T,x∈t−s−h,t×Ω.
Applying the Lebesgue dominated convergence theorem, we infer that(89)limn⟶∞limm⟶∞fT,unx,T−ρT−fT,umx,T−ρTL2Ω×t−s,t=0.
It follows from (83)and 89 that(90)ynt−ymtCV,V2=ynt−ymtCV2+ynt′−ymt′CV2=sups∈−h,0∇ynt+θ−∇ymt+θ2+sups∈−h,0∇yn′t+θ−∇ym′t+θ2≤2yn′T−ym′TL2Ω×t−s,t×fT,unx,T−ρT−fT,umx,T−ρTL2Ω×t−s,t.
Combining (89) with (90), we get (91)limn⟶∞limm⟶∞ynt−ymtCV,V2=0.
This completes the proof.Theorem 5.
(Existence ofDCV,V pullback attractors) Under the assumptions of Lemma 3.2, the multivalued process Ut,τ on CV,V possesses an unique pullback attractor ACV,Vtt∈ℝ in DCV,V.Proof.
By Lemma2, we know that the multivalued process Ut,τ corresponding to Equation (1) is DCV,V pullback asymptotically upper-semicompact on CV,V. Furthermore, according to Lemma 1, the multivalued process Ut,τ possesses a DCV,V pullback absorbing set QCV,V in DCV,V. In order to obtain the existence of DCV,V pullback attractors, we only need to show the negative invariance of ACV,Vtt∈ℝ, where(92)ACV,Vt=ωtQCV,V=∩T∈ℝ+∪s≥TUt,t−sQt−s¯,∀t∈ℝ,and QCV,V=Qtt∈ℝ is the DCV,V pullback absorbing set of Ut,τ in CV,V.
Lety∈ACV,Vt. Then there exist sequences sn∈ℝ+, sn⟶+∞n⟶∞, xn∈Qt−sn, and yn∈Ut,t−snxn such that(93)yn⟶y∈CV,Vasn⟶∞.
On the other hand, forn that is large enough, we get(94)yn∈Ut,t−snxn=Ut,τUτ,t−snxn.
Lemma2 implies that the multivalued process Ut,τ corresponding to Equation (1) is DCV,V pullback asymptotically upper-semicompact on CV,V, then there is a subsequence of x˜n∈Uτ,t−snxn=Ut,τ−τ+sn−txn, which we still relabel as x˜n such that yn∈Ut,τx˜n and(95)x˜n⟶x∈CV,Vasn⟶∞.
Clearly,x∈ACV,Vτ.
By slightly modifying the proof of the existence of solutions Theorem4, we can see that(96)yn⋅⇀u⋅+t,τ,xinL2−h,0;V∩H1−h,0;V,where u⋅ is a solution in Theorem 4. This together with 9495, we can deduce that(97)y∈Ut,τx⊂Ut,τACV,Vτ.
This completes the proof.Remark 2.
In this article, we showed the existence of pullback attractors in the Banach spacesCH01Ω,H01Ω for the multivalued process generated by a class of second-order nonautonomous evolution equations with hereditary characteristics. Furthermore, it would be interesting to consider the long-time behavior for the following nonautonomous semilinear second-order evolution equation with the delay term and critical exponent as(98)∂2u∂t2−Δu−Δ∂u∂t−Δ∂2u∂t2=hu+ft,ux,t−ρt+gt,x,inτ,∞×Ω,where the nonlinearity h⋅ fulfills the critical exponential growth condition, and the nonlinearity f⋅ and the external force gt,x satisfy A1 and A2, respectively. Maybe we can extend the results presented here in this case and can possibly address them in forthcoming papers.



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*Source: 8264550-2022-10-12.xml*