# Global Attractors of the Extensible Plate Equations with Nonlinear Damping and Memory

Authors: Xiaobin Yao; Qiaozhen Ma
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We prove in this paper the existence of a global attractor for the plate equations of Kirchhoff type with nonlinear damping and memory using the contraction function method.


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## Body

## 1. Introduction

Let us consider the long-time behavior for the following Kirchhoff plate equations with fading memory and nonlinear damping:(1)utt+gut+αΔ2u-∫0∞μsΔ2ut-sds+λu+p-β∫Ω∇u2dxΔu+fu=hx,u=∂u∂ν=0on∂Ω×R,ux,τ=u0x,τ,utx,τ=∂tu0x,τ,x,τ∈Ω×-∞,0,where Ω⊂RN is a bounded domain with smooth boundary ∂Ω;  α,  λ, and β are positive constants, p∈R,  h∈L2(Ω); ν is the unit outer normal on ∂Ω; u0:Ω×(-∞,0]→R is the prescribed past history of u.Problem (1) arises from the isothermal viscoelastic theory; it describes a fourth-order viscoelastic plate with a lower order perturbation and also models transversal vibrations of a thin extensible elastic plate in a history space, which is established based on the framework of elastic vibration by Woinowsky-Krieger [1] and Berger [2], and can be seen as an elastoplastic flow equation with some kind of memory effects (see [3, 4] for details). The convolution term means that the stress at any instant t depends on the whole history of strains which the material has undergone and produced a weak damping mechanism (see [5, 6]).In the case whereμ(s)≡0,  p=β=0, (1) becomes the normal plate equation which has been treated in many papers such as [7–14]. For instance, the authors investigated the existence of the compact attractor for the plate equation on both the bounded domain [8, 10, 13] and the unbounded domain in [7, 11, 12], respectively. Yue and Zhong [9] proved the existence of global attractors to the plate equations when the nonlinear function satisfies the critical exponent in a locally uniform space. In [14], the authors studied the existence of the random attractor for the stochastic strongly damping plate equations with additive noise and critical nonlinearity.The case ofp=β≡0 in problem (1) has been studied by several authors (see [2, 5, 15–23] and references therein). For instance, Wu [15] scrutinized the existence of global attractors for the nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws when g(ut)=-Δut. Recently, Conti and Geredeli [5] paid attention to the existence of a smooth global attractor for the nonlinear viscoelastic equations with memory, in which they required the nonlinear damping g(ut) to be the polynomial growth. Shen and Ma [21] studied the existence of the random attractor for plate equations with memory and additive white noise. On the other hand, the asymptotic behavior of solutions for the extensible plate equations without memory affection was studied by several authors in [24–27].We focus on the existence of the extensible plate equations with nonlinear damping and history memory in the present paper. To prove the existence of a compact global attractor, the key goal is to establish the compact property of the semigroup associated with the dynamical system. Regarding problem (1), we need to overcome the following difficulties. One difficulty is caused by the critical nonlinearity and nonlinear damping. In order to overcome these obstacles, we apply the contraction function method into our problem. Another difficulty is brought about by the memory kernel, because there is no compact embedding in the history space; besides, we cannot use the finite rank method. We solve this term by introducing a new variable and defining an extending phase space (see [20] for details). In addition, the terms (p-β∫Ω|∇u|2dx)Δu(p∈R) make the estimates more complex, so we have to deal with them through accurate computation. Our main result is Theorem 12.As in [18], we define (2)η=ηtx,s=ux,t-ux,t-s,x,s∈Ω×R+,t≥0.By assuming thatμ∈L1(R+) and taking α=1+∫∞0μ(s)ds, the original memory term can be rewritten as (3)∫0∞μsΔ2ut-sds=∫0∞μsdsΔ2u-∫0∞μsΔ2ηtsds, and then problem (1) can be transformed into the following system: (4)utt+gut+Δ2u+∫0∞μsΔ2ηtsds+λu+p-β∫Ω∇u2dxΔu+fu=hxinΩ×R+,(5)ηt=-ηs+ut,inΩ×R+×R+.Here, (5) is obtained by differentiating (2). Initial-boundary value conditions are given as follows: (6)u=∂u∂ν=0on∂Ω×R+,η=∂η∂ν=0,on∂Ω×R+×R+,ux,0=u0x,utx,0=u1x,ηtx,0=0,η0x,s=η0x,s,where (7)u0x=u0x,0,x∈Ω,u1x=∂tu0x,tt=0,x∈Ω,η0x,s=u0x,0-u0x,-s,x,s∈Ω×R+.This paper is organized as follows: in Sections2 and 3, we make some preparations for our consideration; in Section 4, we will show the existence of bounded absorbing set and compact global attractors.


## 2. Assumptions

The following conditions are necessary for our main result.Concerning the nonlinear termf∈C1(R), there exists a constant k0>0 such that (8)f′s≤k01+sρ,where (9)0<ρ≤4N-4ifN≥5,ρ>0if1≤N≤4.Condition (9) implies that H02(Ω)↪L2(ρ+1)(Ω). Also, we say that ρ=4/N-4 is a critical exponent for the growth of f(u) when N≥5. In addition, we assume that (10)liminfs→∞fss>-λ1,where λ1>0 is the principal eigenvalue of Δ2 in H02(Ω).The nonlinear damping functiong∈C1(R) satisfies (11)g0=0,g′s≥l>0,(12)gs≤k11+sq,with 1≤q<∞ if N≤4 and 1≤q≤N+4/N-4 if N≥5.With respect to the memory kernelμ, we assume that (13)∫0∞μsds=μ0>0,μ∈C1R+∩L1R+,(14)μ′s≤0≤μs,∀s∈R+,and that there exists a constant k2>0 such that (15)μ′s+k2μs≤0,∀s∈R+.Now, we consider the Hilbert spaces that will be used in our analysis. Let(16)H=V0=L2Ω,V=V1=H02Ω, equipped with the respective inner products and norms, (17)u,vV=Δu,Δv,uV=Δu2, where (·,·) is L2-inner product and we use ·p to denote Lp-norms.We define the following weightedL2-space: (18)Lμ2R+;V=η:R+⟶V∣∫0∞μsηsV2ds<∞, which is a Hilbert space endowed with inner product and norm (19)u,vμ,V=∫0∞μrur,vrVdr,uμ,V2=u,uμ,V=∫0∞μrurV2dr.Finally, we introduce the phase space(20)H=V×H×Lμ2R+;V, equipped with the norm (21)u,v,ξH2=Δu22+v22+ξμ,V2.In order to obtain the global attractor of problems (4)–(6), we need the following theorem. Under our hypotheses, we can derive an existence result by standard Faedo-Galerkin method (see [12, 28, 29]). For arguments involving the memory term, we follow Giorgi et al. [19, 20].Theorem 1.
Let (8)–(15) hold. If h∈L2(Ω), then the following results hold. (i)
Provided that the initial data(u0,u1,η0)∈H, then problems (4)–(6) have a weak solution (22)u,ut,η∈C0,T;H,∀T>0,satisfying(23)u∈L∞0,T;V,ut∈L∞0,T;H,η∈L∞0,T;Lμ2R+;V.(ii)
Assume thatzi=ui,uti,ηi are weak solutions of problems (4)–(6) corresponding to initial data zi(0)=u0i,u1i,η0i,  i=1,2. Then, (24)z1t-z2tH≤ectz10-z20H,t∈0,T,for some constantc>0.Remark 2.
Let(u(t),ut(t),ηt) be the unique weak solution of problems (4)–(6); by Theorem 1, we can define a semigroup S(t):H→H as follows: (25)Stu0,u1,η0=ut,utt,ηt,t≥0. And S(t) is continuous on H.


## 3. Some Abstract Results

In this section, we will recall some basic theories of infinite dimensional dynamical systems; we refer to [30] for more details.Definition 3.
The global attractorA is the maximal compact invariant set (26)StA=A∀t≥0and the minimal set that attracts all bounded sets: (27)limt→∞distHStB,A=0,for any bounded set B⊂H, where distH is the Hausdorff semidistance in H.Definition 4.
A semigroup is dissipative if it possesses a bounded absorbing set; that is, for any bounded setB⊂H, there exists t0(B) such that (28)StB⊂B,∀t≥t0B.Definition 5.
(
H
,
S
(
t
)
) is asymptotically smooth if, for any bounded positively invariant set B⊂H, there exists a compact set K⊂B- such that (29)limt→∞distHStB,K=0.Theorem 6.
A dissipative dynamical system(H,S(t)) has a compact global attractor if and only if it is asymptotically smooth.Theorem 7.
For any bounded positively invariant setB⊂H, S(t) is called asymptotically smooth in H, if, for any ε>0 and any sequence {zn} in B, there exists T=T(ε,B) such that (30)STx-STyH≤ε+ϕTx,y,∀x,y∈B, where ϕT:B×B→R satisfies (31)liminfn→∞liminfm→∞ϕTzn,zm=0.


## 4. Existence of Attractors

In this section, we will use the abstract results presented in Section3 to prove our main result.Lemma 8.
Under assumptions (8)–(15), the semigroup {S(t)}t≥0 corresponding to problems (4)–(6) has a bounded absorbing set in H.Proof.
Taking the scalar product inH of (4) with ut, we infer that (32)ddt12ut22+12Δu22+λ2u22+∫ΩFu-hudx+η,utμ,V+∫Ωgututdx+p-β∇u22Δu,ut=0,where F(u)=∫0uf(τ)dτ.
Thanks to (5), (13), and (15) and Hölder inequality, we get (33)η,utμ,V=η,ηt+ηsμ,V=12ddtημ,V2+∫0∞μsηs,ηssVds=12ddtημ,V2+12∫0∞μsdηsV2=12ddtημ,V2-12∫0∞μ′sηsV2ds≥12ddtημ,V2+k22∫0∞μsηsV2ds=12ddtημ,V2+k22ημ,V2,(34)p-β∇u22Δu,ut=12ddtβ2β∇u22-p2β2.We conclude from (32)–(34) that (35)ddtEt+k22ημ,V2+∫Ωgututdx≤0,where (36)Et=12ut22+12Δu22+λ2u22+∫ΩFu-hudx+12ημ,V2+12β2β∇u22-p2β2. Since (37)gut=gut-g0=g′θutut,0<θ<1, from (11), we obtain (38)∫Ωgututdx≥lut2. Thus, (35) implies (39)Et≤E0,∀t≥0. It follows from (10) that there exist λ′  (λ1>λ′>0) and C0>0, such that (40)fu,u>-λ′u22-C0meas⁡Ω,∫ΩFudx>-λ′2u22-C0meas⁡Ω.Using (40) and Poincaré and Young inequalities, we end up with (41)E0≥Et=12ut22+12Δu22+λ2u22+∫ΩFu-hudx+12ημ,V2+12β2β∇u22-p2β2≥12ut22+12Δu22+λ2u22-λ′2u22+12ημ,V2-C0meas⁡Ω-1λh2-λ4u22≥12ut22+CΔu22+12ημ,V2-C1meas⁡Ω+h2≥-C1meas⁡Ω+h2,where C=1/2-λ′/2λ1,  C1=max⁡{C0,1/λ}.
By (35) and (41), we deduce that (42)∫0t∫Ωgututdxds≤-∫0tddtEsds=E0-Et≤E0+C1meas⁡Ω+h2,∀t≥0.Now, we set v=ut+δu and rewrite the equation of (4) as follows: (43)vt-δut+gut+Δ2u+∫0∞μsΔ2ηtsds+λu+p-β∇u22Δu+fu=hxinΩ×R+.
We formally take the scalar product inH of (43) with v; after a computation, we find (44)ddt12v22+12Δu22+λ2u22-δ22u22+∫ΩFu-hudx+gut-δut,ut+δgut,u+δΔu22+p-β∇u22Δu,v+η,utμ,V+δη,uμ,V+λδu22+δfu,u-δh,u=0,where (45)p-β∇u22Δu,v=p-β∇u22Δu,ut+δu=p-β∇u22Δu,ut+p-β∇u22Δu,δu≥12ddtβ2β∇u22-p2β2+δβ2β∇u22-p2β2-δp22β. Denote (46)Eδt=12v22+12Δu22+λ2u22-δ22u22+∫ΩFu-hudx+12ημ,V2+12β2β∇u22-p2β2,(47)Ht=gut-δut,ut+δgut,u+δΔu22+k22ημ,V2+δβ2β∇u22-p2β2-δp22β+δη,uμ,V+λδu22+δfu,u-δh,u.Therefore, together with (33) and (44), we get(48)ddtEδt+Ht≤0.From (40), for any δ sufficiently small, using Hölder and Young inequalities, we get (49)Eδt≥12v22+CΔu22+12ημ,V2-C1meas⁡Ω+h2.
Thanks to (11), we get (50)gut-δut,ut≥l-δut2.Using the Young inequality, we obtain (51)δη,uμ,V≥-k24ημ,V2-μ0δ2k2Δu22.From (11) and (12), we deduce that (52)gsq+1/q=gs1/q·gs≤C21+sgs≤C2,s≤1,2C2gss,s≥1,where C2 is a constant which is independent of s.
In line with (52), the Hölder and Young inequalities, and the Sobolev embedding H02↪Lq+1, similar to the progress of [31], we conclude (53)∫Ωgutudx≤∫Ωut≤1gutudx+∫Ωut≥1gutudx≤∫Ωut≤1C2udx+∫Ωut≥1gutq+1/qdxq/q+1∫Ωut≥1uq+1dx1/q+1≤∫Ωut≤1C2udx+2C2∫Ωut≥1gututdxq/q+1∫Ωut≥1uq+1dx1/q+1≤C24ε∫Ωdx+C2εu2+Cε∫Ωut≥1gututdx·uq+1q-1/q+εuq+12≤C24εmeasΩ+C2εu2+CsCεΔu2q-1/q∫Ωgututdx+εCsΔu22,where ε is a small positive constant and Cs is an embedding constant.
Therefore, we chooseε and δ small enough, such that (54)λ2-C2ε>0,2C-μ0δk2-εCs>C4,l-δ>l4.Together with (40), (50), (51), and (53), we have (55)Ht≥l-δut2+δΔu22+k24ημ,V2-μ0δ2k2Δu22+δβ2β∇u22-p2β2-δp22β+λδu22-δλ′u22-δC0measΩ-δλ2u2-δ2λh2-δC24εmeasΩ+C2εu2+CsCεΔu2q-1/q∫Ωgututdx+εCsΔu22≥l-δut2-Cδ′measΩ+h2+δλ2-C2εu2+δ2C-μ0δk2-εCsΔu2+k24ημ,V2-δCsCεΔu2q-1/q∫Ωgututdx-δp22β≥l4ut2-Cδ′measΩ+h2+δC4Δu2+k24ημ,V2-CE0∫Ωgututdx-δp22β,where CE(0) is a constant which depends on δ,Cs,Cε, and E(0), while Cδ′ is a constant which depends on δ,C0,λ,C2, and ε.
Notice that(56)ut2+Δu2+ημ,V2=ut+δu-δu2+Δu2+ημ,V2=v-δu2+Δu2+ημ,V2≤2v2+2δ2u2+Δu2+ημ,V2≤2v2+2δ2λ1+1Δu2+ημ,V2≤c0v2+Δu2+ημ,V2, where c0=max{2,1+2δ2λ1-1}.
Integrating (48) from 0 to t, combining with (42), (49), and (55), we arrive at (57)ut2+Δu2+ημ,V2-4c0CC1measΩ+h2+Eδ0+CE0E0+C1measΩ+h2≤-∫0tδ′c0Cuts2+Δus2+ηsτμ,V2-4c0CCδ′measΩ+h2-4c0C·δp22βds, where δ′=min⁡{l,δC,k2}.
Therefore, for anyr>4Cδ′βmeasΩ+h2+2δp2/δ′β, there exists t0>0, such that (58)utt02+Δut02+ηt0sμ,V2≤r. Denote (59)B0=u0,v0,η0∈H:Δu02+v02+η0μ,V2≤r; by the argument above, we know that B0 is a bounded absorbing set.
Define(60)B=⋃t≥0StB0; then B is also a bounded absorbing set. This shows that the semigroup {S(t)}t≥0 corresponding to problems (4)–(6) has a bounded absorbing set in H.Remark 9.
If(u,ut,η) is a solution of (4)–(6) with initial data (u0,u1,η0) in a bounded set B, then we have (61)ut,utt,ηtH≤CB,∀t≥0,where CB>0 is a constant depending on B.In order to prove that the dynamical system(H,S(t)) is asymptotically smooth, we need the following lemma.Lemma 10.
Assume that assumptions (8)–(15) hold and g∈L2(Ω); B⊂H is a bounded set; let z1=(u,ut,η) and z2=(v,vt,ξ) be two solutions of problems (4)–(6), such that z1(0)=(u0,u1,η0) and z2(0)=(v0,v1,ξ0) are in B. Then,(62)z1t-z2tH2≤C∗e-α2t+C∗∫0te-α2t-swts22+ws2ρ+12ds,for any t≥0, where C∗>0 and α2>0 are constants.Proof.
For convenience, we denotew=u-v and ζ=η-ξ. Then, (w,ζ) satisfy the following equations: (63)wtt+gut-gvt+Δ2w+∫0∞μsΔ2ζtsds+λw+p-β∇u22Δu-p-β∇v22Δv+fu-fv=0,(64)ζt=-ζs+wt,and initial datum (65)w0=u0-v0,wt0=u1-v1,ζ0=η0-ξ0. Taking the scalar product in H of (63) with φ=wt+σw, we obtain (66)ddt12φ22+12Δw22+g′vt+θut-vt-σwt,φ+σΔw22+p-β∇u22Δu,φ-p-β∇v22Δv,φ+λw,φ+ζ,wtμ,V+σζ,wμ,V+fu-fv,φ=0,where 0<θ<1.
Noting the similar estimate used in Lemma8, we obtain (67)g′vt+θut-vt-σwt,φ≥l-σwt,φ≥l-σφ22-σl-σw,φ,ζ,wtμ,V≥12ddtζμ,V2+k22ζμ,V2,σζ,wμ,V≥-k24ζμ,V2-μ0σ2k2Δw22.Using the Poincaré, Hölder, and Young inequalities and taking σ small enough, such that (68)l2-σ>l4,1-μ0σk2-σl2λ1>12, then we obtain (69)l-σφ22-σl-σw,φ+σ-μ0σ2k2Δw22≥l-σφ22-σ2l2λ1Δw22+l2φ22+σ1-μ0σk2Δw22≥l2-σφ22+σ1-μ0σk2-σl2λ1Δw22≥l4φ22+σ2Δw22.Combining (69) with (66), we get (70)ddt12φ22+12Δw22+12ζμ,V2+σ2Δw22+l4φ22+k24ζμ,V2+p-β∇u22Δu,φ-p-β∇v22Δv,φ≤-λw,φ-fu-fv,φ;besides, (71)-λw,φ≤λ2σw22+σ4φ22≤λ2c1σw2ρ+12+σ4φ22,where c1>0 is an embedding constant for L2(ρ+1)(Ω)↪L2(Ω).
By virtue of the generalized Hölder inequality withρ/2(ρ+1)+1/2(ρ+1)+1/2=1, (8), (61), and Young inequality, we have (72)-∫Ωfut-fvtφtdx≤k0∫Ω1+utρ+vtρwtφtdx≤k0Ωρ/2ρ+1+u2ρ+1ρ+v2ρ+1ρw2ρ+1φ2≤CBw2ρ+1φ2≤CB2σw2ρ+12+σ4φ22. Now, we estimate ((p-β∇u22)Δu,φ)-((p-β∇v22)Δv,φ).
Set(73)p-β∇u22Δu,φ-p-β∇v22Δv,φ=∫Ωp-β∇u22Δu-p-β∇v22Δvφtdx=I1+I2,where (74)I1=∫ΩpΔu-pΔvφtdx,I2=∫Ω-β∇u22Δu+β∇v22Δvφtdx. Taking advantage of the Hölder and Young inequalities, (61), it follows that (75)I1=∫ΩpΔu-pΔvφtdx≤∫ΩpΔwφtdx≤σ2p2Δw22+14σ2φ22≤σ2p2Δw22+14σ2wt+σw22≤σ2p2Δw22+12σ2wt22+12w22≤σ2p2Δw22+12σ2wt22+c12w2ρ+12,I2=∫Ωβ∇u22Δu-β∇v22Δvφtdx≤∫Ωβ∇u22Δu-β∇u22Δv+β∇u22Δv-β∇v22Δvφtdx≤β∫Ω∇u22Δwφtdx+β∫Ω∇u22+∇v22Δvφtdx≤CBβ2∫ΩΔwφtdx+CB2β∫ΩΔvφtdx≤CBβ2Δw2φ2+CB2βΔv2φ2≤CBβσ2Δw22+14σ2φ22≤CBβσ2Δw22+CBβ2σ2wt22+CBβc12w2ρ+12, where we have used the fact that Δv2=Δu-Δw2≤Δu2+Δw2≤CB/2Δw2.
Plugging the above two inequalities into (73), we obtain (76)p-β∇u22Δu,φ-p-β∇v22Δv,φ≥-σ2p2+CBβσ2Δw22-12σ2+CBβ2σ2wt22-c12+CBβc12w2ρ+12.Combining with (71), (73), and (76), we deduce from (70) (77)ddt12φ22+12Δw22+12ζμ,V2+σ2-σ2p2-CBβσ2Δw22+l4-σ2φ22+k24ζμ,V2≤12σ2+CBβ2σ2wt22+λ2c1σ+CB2σ+c12+CBβc12w2ρ+12. We can choose σ so small such that (78)σ2-σ2p2-CBβσ2=σ12-σp2-CBβσ>0,l4-σ2>0. Then, we set (79)EWt=φ22+Δw22+ζμ,V2,ddtEWt+α2EWt≤C3wt22+w2ρ+12,where (80)α2=min⁡σ-2σ2p2-2CBβσ2,l2-σ,k22,C3=max⁡1σ2+CBβσ2,2λ2c1σ+2CB2σ+c1+CBβc1.By the Gronwall lemma, we get (81)EWt≤EW0e-α2t+C3∫0te-α2t-swts22+ws2ρ+12ds.Hence, (82)z1t-z2tH2=Δw22+wt22+ζμ,V2=Δw22+φ-σw22+ζμ,V2≤Δw22+2φ22+2σ2w22+ζμ,V2≤1+2σ2λ1Δw22+2φ22+ζμ,V2≤C4φ22+Δw22+ζμ,V2, where C4=max{1+2σ2/λ1,2}.
Namely,(83)z1t-z2tH2≤C4EWt≤C4EW0e-α2t+C3C4∫0te-α2t-swts22+ws2ρ+12ds. It is not difficult to know that (62) holds, where C∗=max⁡{C4EW(0),C3C4}.Now, we shall show that the dynamical system(H,S(t)) is asymptotically smooth.Lemma 11.
Assume that assumptions (8)–(15) hold and g∈L2(Ω). Then, the dynamical system (H,S(t)) is asymptotically smooth.Proof.
Given a bounded positively invariant setB of H, for z1,z2∈B, denote S(t)z1=(u(t),ut(t),ηt) and S(t)z2=(v(t),vt(t),ξt) as the solutions of (4)–(6). Then, given ε>0, choosing T>0, such that C∗e-α2T/2<ε, we can infer from (62) that (84)STz1-STz2H≤ε+CB∫0Tus-vs2ρ+12+uts-vts22ds1/2,where CB is a constant which depends only on the size of B.
Now, we estimate the right hand side of (84). Condition (9) implies that 2<2(ρ+1)<∞ if 1≤N≤4 and 2<2(ρ+1)≤2N/N-4 if N≥5. Taking θ=N/4(1-1/ρ+1) and applying the Galiardo-Nirenberg interpolation inequality, we have (85)ut-vt2ρ+1≤CθΔut-vt2θut-vt21-θ≤CBut-vt21-θ. Since u(t)2 and v(t)2 are uniformly bounded, there exists a constant CB>0 such that (86)ut-vt2ρ+12≤CBut-vt221-θ.Using (86), we can rewrite (84) as (87)STz1-STz2H≤ε+ΦTz1,z2, with (88)ΦTz1,z2=CB∫0TCBus-vs221-θ+uts-vts22ds1/2. We claim that ΦT satisfies (31). In fact, let zn∈B with zn=(u0n,u1n,η0n) be given. Denote S(t)zn=unt,utnt,ηn,t. It can be shown that B is invariant by the property of St, and hence (un(t),utn(t),ηn,t) are uniformly bounded in H.
Particularly,(89)un,utnis bounded inC0,T,V×H,T>0. Then, by the fact V↪H and the Aubins lemma, it is not hard to show that there exists a subsequence {unk} that converges strongly in C([0,T],H); therefore, (90)limk→∞liml→∞∫0TCBunks-unls221-θds=0. Without loss of generality (at most by passing subsequences), we assume that (91)utn⟶utweakly star inL∞0,T;L2Ω; following similar argument given in [32, Lemma 4.4] and [31], we get (92)limk→∞liml→∞∫0TCBunks-unls221-θ+utnks-utnls22ds=0; then, the asymptotic smoothness property of (H,S(t)) follows from Theorem 7.Finally, we obtain our main results.Theorem 12.
Assume that assumptions (8)–(15) hold; then, the dynamical system (H,S(t)) corresponding to system (4)–(6) has a compact global attractor A⊂H.Proof.
We note that Lemmas8 and 11 imply that (H,S(t)) is a dissipative dynamical system which is asymptotically smooth. By Theorem 6, it has compact global attractor A in the phase space H.



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*Source: 4896161-2017-04-04.xml*