# Attractors for a Class of Abstract Evolution Equations with Fading Memory

Authors: Xuan Wang; Fenxia Duan; Didi Hu
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

In this paper, we study the dynamics of an abstract evolution equation with fading memory with a critical growing nonlinearity. By use of some new methods and asymptotic estimate techniques, we first verify the asymptotic compact of solution semigroup and then prove the existence of global attractors in weak topological space and strong topological space, while the forcing term only belongs toH-1(Ω) or L2(Ω), respectively. The results are new and appear to be optimal.


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## Body

## 1. Introduction

In this paper, we consider the asymptotic behaviors of solutions for the following abstract evolution equation with fading memory in a bounded domainΩ⊂Rn:(1)utt+αut+k0Aθu+∫0∞k′sAθut-sds+fu=gx,Ω×R+,ux,t=0,x∈∂Ω,t∈R,ux,t=u0x,t,x∈Ω,t⩽0,where θ∈[2n/n+2,n/2).About the forcing termg, we consider two cases: the weak solutions case g(x)∈H-1(Ω) and the strong solutions case g(x)∈L2(Ω). For the nonlinearity, we assume that f∈C1(R) with f(0)=0 and satisfies the following conditions.Dissipation condition:(2)limsups→∞⁡fss>-λ1θ,where λ1 is the first eigenvalue of A=-Δ in H01(Ω).Growth condition:(3)f′s⩽C1+sp,∀s∈R,p⩽4n-2,n⩾3,p⩾0is arbitrary,n=1,2.Since the nonlinearities critical exponent is 4/n-4 in the strong topological space, we need f to satisfy the following condition instead of (3):(4)f′s⩽C1+sp,∀s∈R,p⩽4n-4,n⩾5,p⩾0is arbitrary,n=3,4,(5)fss⩾0,∀s∈R.And there exists a constant l>0, such that(6)f′s⩽l,∀s∈R,and assumption (6) about f will be used to prove the existence of absorbing set in the strong topological space and that it has nothing to do with the testing of asymptotic compactness.There are many functions satisfying the above assumption (6). For example, when f(u)=u-u3, f(u) satisfies this condition. Without loss of generality, we also assume f(0)=0.The effects of fading memory in this equation are shown through the linear time convolution of the functionAθu(·) and memory kernel k(·). As that in [1], we also assume that the memory kernel k(·)∈C2(R+),k(s)⩾0,k′(s)⩽0, for every s∈R+. Besides, we also presume that the function μ(s)=-k′(s) satisfies the following hypotheses:(7)μ∈C1R+∩L1R+,μs⩾0,μ′s⩽0,∀s∈R+;(8)∫0∞μsds=k0>0;(9)μ′s+δμs⩽0,∀s⩾0,where δ is a positive constant. Obviously, it follows that the kernel μ(s) decays to zero with exponential rate. This behavior displays the fading memory of the far history in model (1).Ifθ=1 and A=-Δ, (1) represents a semilinear hyperbolic equation in viscoelasticity (cf. [2, 3]), and this model is originally derived from the theory of isothermal viscoelasticity and describes a process of energy dissipation of a homogeneous and isotropic viscoelastic solid (cf. [4]). Furthermore, if k′(s)≡0, (1) reduces to semilinear wave equation, where g denotes some displacement-dependant body's force density (cf. [5]); and, under these conditions, if f(u)=sin⁡u, (1) can be transformed into a model of Sine-Gordon equation (cf. [6]); if f(u)=|u|γu, (1) can be turned into the relative quantum mechanical equation (cf. [6]). If k′(s)≡0, θ=2, and A=-Δ, it expresses a floating beam equation; u denotes bent movement crosswise of beam (cf. [7, 8]).For the semilinear hyperbolic equation in viscoelasticity, the asymptotic behaviors of the solution have been studied in [1, 9–13]. In [1, 9, 10], the authors investigated the asymptotic behaviors of solutions when nonlinearity is subcritical growth. In [11], Sun et al. considered large-time behaviors of the solutions and gained the existence of global attractors for nonautonomous strongly damped wave-type evolution equation with the critical nonlinearity and linear memory. In [12], Cavalcanti et al. studied the long-time dynamics of a semilinear wave equation with degenerate viscoelasticity defined in a bounded domain Ω of R3, with Dirichlet boundary condition and nonlinear forcing term with critical growth. In [13], Zhou and Zhao proved the existence of random attractors for the continuous random dynamical systems generated by stochastic damped nonautonomous wave equations with linear memory and additive white noise when the nonlinearity has a critically growing exponent and studied the upper semicontinuity of random attractors.For the floating beam equation, the authors considered the dynamics and obtained the existence of a global attractor for the deterministic floating beam in [8] and deeply studied the existence of a compact random attractor for the random dynamical system generated by a model for nonlinear oscillations in a floating beam equation with strong damping and white noise in [14].The above equations are all special cases of an abstract evolution equation with fading memory that we will study in this paper. Until now, we find that no one else has studied the long-time behavior of the solutions about the problem with weak damping and critical nonlinearity, which just causes our strong research motivation and interest. Based on this, we will study the asymptotic behaviors and regularity of solutions for this problem in this paper.As we know, if we want to prove the existence of global attractors, the key point is to obtain the compactness of the semigroup in some sense. However, there exist some essential difficulties to test the continuity and compactness (asymptotic compactness) of the semigroup of solutions for (1). First, due to the memory terms, we cannot use (I-Pm)u or (I-Pm)Aθu as test function to verify the asymptotic compactness of the solution semigroup; second, the critical nonlinearity leads to the fact that some obstacles are hard to avoid in the process of energy estimation. Furthermore, since Aθ is an abstract operator, it brings about more substantive barriers in the process of proof. Despite all this, we still overcome the above bottlenecks in the process of estimation and proof by applying semigroup theory and decomposition technology. Finally, we test the continuity and compactness (asymptotic compactness) of the solution semigroup of solutions and obtain the existence of global attractors of (1) in the weak topological space Vθ×L2(Ω)×Lμ2(R+,Vθ) and the strong topological space V2θ×Vθ×Lμ2(R+,V2θ).This paper is organized as follows. Some preliminaries, including some notations that we will use, the assumption on nonlinearity, and some general abstract results about dynamical system are presented in Section2. Then, the proof of our main result about compactness testing and the existence of global attractor for the dynamical system generated by the solution of (1) are given in Sections 3 and 4. The main results are Theorems 11 and 18.


## 2. Preliminaries

LetH=L2(Ω) and b(u,v) be a bilinear continuous form on H which is symmetric and coercive. With this form, we associate the linear operator A in H by setting (Au,v)=b(u,v),∀u,v∈H, and A can be considered as a self-adjoint unbounded operator in H with domain D(A)⊂H.We assume that{λj}j∈N, {ωj}j∈N are eigenvalue and eigenvectors of A, so {ωj}j∈N can form a group of orthonormal bases of H; then (10)Aωj=λjωj,∀j∈N,0<λ1⩽λ2⩽⋯⩽λj,λj⟶∞,asj⟶∞.Using this group of bases we can easily define the powersAθ of A (see [6]), θ∈[2n/n+2,n/2), with domain D(Aθ)⊂H. we can consider a family of Hilbert spaces(11)DAθ=u∈H,∑j=1∞λj2θu,ωj2<∞,θ∈2nn+2,n2,with the standard inner product and norms, respectively,(12)·,·DAθ=Aθ·,Aθ·,·DAθ2=Aθ·2;here 〈·,·〉 and · mean L2(Ω) inner product and norm, respectively. Then, we know that(13)DAθ1↪DAθ2,for anyθ1>θ2,DAθ/2↪L2n/n-2θΩ,∀θ∈0,n2.The following interpolation inequality also holds: given s>r>q, for any ϵ>0, there exists a positive constant Cϵ=Cϵ(s,r,q) such that(14)Ar/2u⩽ϵAs/2u+CϵAq/2u,for anyu∈DAs/2.For convenience, we set, fors∈[2n/n+2,n/2),(15)Vs=DAs/2,with norm·Vs=·DAs/2;then, V0=H=L2(Ω), Vθ=D(Aθ/2), and V2θ=D(Aθ).As in [1, 9], we denote(16)ηtx,s=ux,t-ux,t-s.Setμ(s)=-k′(s) and use the assumption k(∞)=1; (1) can be transformed into the following system:(17)utt+αut+Aθu+∫0∞μsAθηtsds+fu=gx,ηtt=-ηst+ut.The associated initial-boundary conditions are(18)ux,t=0,ηtx,s=0,x,s∈∂Ω×R+,t⩾0,ux,t=u0x,t,∂tux,t=∂tu0x,t,x,t∈Ω×-∞,0,η0x,s=u0x,0-u0x,-s,x,s∈Ω×R+,where u(·) satisfies the condition as follows: there exist two positive constants R and ϱ=min⁡{δ/2,λ1/2}; here λ1 is the first eigenvalue of -Δ, such that(19)∫0∞e-ϱs∇u-s2ds⩽R;here · means L2(Ω) norm.In view of (7), we let Lμ2(R+;Vθ) be a family of Hilbert spaces of Vθ-valued functions on R+ and Lμ2(R+;V2θ) be a family of Hilbert spaces of V2θ-valued functions on R+, endowed with the following inner product and norm, respectively: (20)φ,ψμ,Vθ=∫0∞μs∫ΩAθ/2φAθ/2ψdxds,φμ,Vθ2=∫0∞μs∫ΩAθ/2φ2dxds,φ,ψμ,V2θ=∫0∞μs∫ΩAθφAθψdxds,φμ,V2θ2=∫0∞μs∫ΩAθφ2dxds.So, we introduce the family of Hilbert spaces:(21)H1=Vθ×H×Lμ2R+;Vθ;H2=V2θ×Vθ×Lμ2R+;V2θand endow this family of spaces with norm, respectively, (22)zH1=u,ut,ηtH1=12uVθ2+ut2+ηtμ,Vθ21/2;zH2=u,ut,ηtH2=12uV2θ2+utVθ2+ηtμ,V2θ21/2.In order to estimate conveniently, using assumptions (7)–(9), we can obtain the preliminary result as follows (cf. [9, 15]).Lemma 1.
SetI=[0,T],∀T>0. If memory kernel μ(s) satisfies (7)–(9), then, for any ηt∈C1(I;Lμ2(R+;Vr)), 0<r<2θ, θ∈[2n/n+2,n/2), there exists a constant δ>0, such that (23)ηt,ηstμ,Vr⩾δ2ηtμ,Vr2.The following results will be used to prove the compactness of the memory term and the existence of strong global attractors.Lemma 2 (see [9, 15, 16]).
Assume thatμ∈C1(R+)∩L1(R+) is a nonnegative function and satisfies the following: if there exists s0∈R+, such that μ(s0)=0, then μ(s)=0 for all s⩾s0 holds. Moreover, let B0, B1, B2 be Banach space, where B0, B1 are reflexive and satisfy(24)B0↪B1↪B2,where the embedding B0↪B1 is compact. Let C⊂Lμ2(R+;B1) and satisfy(i)
C in Lμ2(R+;B0)∩Hμ1(R+;B2);(ii)
supη∈Cη(s)B12⩽h(s),∀s∈R+,h(s)∈Lμ1(R+).Then C is relatively compact in Lμ2(R+;B1).Lemma 3 (see [6]).
LetH be a complete metric space, {S(t)}t⩾0 is a C0 semigroup in H, and {S(t)}t⩾0 has a bounded absorbing set B0 in H. If, for every t⩾0, the operator S(t) allows the decomposition S(t)=S1(t)+S2(t) and satisfies that(i)
the semigroup{S1(t)}t⩾0 is uniformly compact, as t is increasing sufficiently;(ii)
the operatorS2(t):H→H is continuous and, for any bounded set B⊂H, as t→∞, (25)rBt=supφ∈BS2tφH⟶0,then ω-limit set of absorbing set B0 is global attractors of {S(t)}t⩾0.


## 3. Global Attractors inH1

### 3.1. The Existence and Uniqueness of Weak Solution

At first, we define the concept of weak solution for dynamical systems (17)-(18) (cf. [15] about the corresponding definition of solutions for wave equations).Definition 4.
DefineI=[0,T],∀T>0. Let g∈H-1(Ω) and z0∈H2. A ternary form z=(u,ut,ηt) which fulfils(26)u∈CI;Vθ,ut∈L2I;Vθ,ηt∈CI;Lμ2R+;Vθ;ηtt+ηst∈L∞I;Lμ2R+;H∩L2I;Lμ2R+;Vθis said to be a weak solution of problem (17)-(18) in the time interval I with initial data z(0)=z0, provided that(27)utt,ω+αut,ω+u,ωVθ+ηt,ωμ,Vθ+fu,ω=g,ω,ηtt+ηst,φμ,Vθ=ut,φμ,Vθ,for all ω∈Vθ,φ∈Lμ2R+;Vθ,a.e. t∈I.Then, we start with the following general existence and unique result. The proof is based on the normal Faedo-Galerkin methods introduced in [9, 15].Theorem 5 (existence and uniqueness of solutions).
Letf satisfy (2)-(3), g∈H-1(Ω), and (7)–(9) hold. Then for any given z0=(u0,ut0,η0)∈H1, and any T>0, there is a unique solution z(t)=(u,ut,ηt)∈H1 for problem (17)-(18), satisfying(28)z∈L∞0,T;H1∩C0,T;H1.According to Theorem5 above, we can define the solution operator; that is, (29)St:H1⟶H1,z0=u0,ut0,η0⟶ut,utt,ηt=Stz0.Obviously, {S(t)}t⩾0 can form a solution semigroup.In the remainder of this section, we denote by{S(t)}t⩾0 the semigroup associated with the solution of (17)-(18).


### 3.2. The Existence of Bounded Absorbing Set

Theorem 6 (bounded absorbing inH1).
LetB0 be any bounded subset of H1 and z(t) be a solution of (17)-(18) with initial data z0∈B0. If the nonlinearity f satisfies (2), g∈H-1(Ω), (7)–(9) hold. Then there is a positive constant μ0 such that, for any bounded subset B0⊂H1, there exists t0=t0(B0H1) such that(30)Stz0H1⩽μ0,∀t⩾t0,z0∈B0.Proof.
Taking0<ρ<1, multiplying (17) by ut+ρu, and integrating over Ω, we have(31)12ddtuVθ2+ut+ρu2+ηtμ,Vθ2+α-ρut+ρu2+ρuVθ2+ηt,ηstμ,Vθ-ρα-ρu,ut+ρu+ρηt,uμ,Vθ+fu,ut+ρu=g,ut+ρu.Then, using Poincaré inequality and Cauchy inequality, and taking ρ small enough, we can get that(32)ρuVθ2+α-ρut+ρu2-ρα-ρu,ut+ρu⩾ρuVθ2+α-ρut+ρu2-ραu,ut+ρu+ρ2u,ut⩾ρuVθ2+α-ρut+ρu2-αρ2λ1θuVθ2-α-ρρu,ut⩾ρuVθ2+α-ρut+ρu2-αρ2λ1θuVθ2-α2ut+ρu2=ρ1-αρλ1θuVθ2+12α-ρut+ρu2.
Applying Lemma1, we have(33)ηt,ηstμ,Vθ⩾δ2ηtμ,Vθ2,and, from (8), we see(34)ρηt,uμ,Vθ⩾-ρ∫Ω∫0∞μsAθ/2ηts2ds1/2∫0∞μsAθ/2u2ds1/2dx⩾-δ4ηtμ,Vθ2-k0ρ2δuVθ2.
Therefore,(35)12ddtuVθ2+ut+ρu2+ηtμ,Vθ2+ρ1-αρλ1θ-k0ρδuVθ2+12α-ρut+ρu2+δ4ηtμ,Vθ2+fu,ut+ρu⩽g,ut+ρu.
From (2), we know that there exist λ1θ>λ∗>0 and C0>0 such that(36)fu,u⩾λ∗4∫ΩFudx-C0,Fu⩾-λ∗8u2-C0;here, F(u)=∫0u‍f(r)dr. Then(37)fu,ut+ρu⩾ddt∫ΩFudx+ρλ∗4∫ΩFudx-ρC0.
Evidently, we can get(38)g,ut+ρu=ddtg,u+ρg,u,g,u⩽gH-1Ω∇u⩽2λ1θ-1gH-1Ω2+λ1θ-18∇u2⩽2λ1θ-1gH-1Ω2+18uVθ2.Choose ρ to be sufficiently small, such that(39)1-αρλ1θ-k0ρδ⩾12,12α-ρ⩾α4;thus,(40)ρ1-αρλ1θ-k0ρδuVθ2+12α-ρut+ρu2⩾ρ2uVθ2+α4ut+ρu2.
Substituting (32)–(40) into (31), we can obtain that(41)12ddtuVθ2+ut+ρu2+ηtμ,Vθ2+2∫ΩFudx-2g,u+C~+β02uVθ2+ut+ρu2+ηtμ,Vθ2+2∫ΩFudx-2g,u+C~⩽ρC0+β02C~⩽C,where β0=min⁡{ρ,α/2,δ/2,ρλ∗/4} and C~=2C0+4/λ1θ-1gH-1(Ω)2.
Hence, from (36)–(38), there exist positive constants C1, C2, such that(42)C1ztH12⩽12uVθ2+ut+ρu2+ηtμ,Vθ2+2∫ΩFudx-2g,u+C~⩽C2ztH12.According to Gronwall lemma,(43)ztH12⩽C2C1z0H12e-β0t+Cβ0C1.Assuming that z(0)H12⩽R, as t⩾t0=t0(B0H1), we have(44)ztH1⩽μ0.
We complete the proof.


### 3.3. The Existence of Global Attractor inH1

Now we will establish the necessary asymptotic smoothness, similar to that in [17–19]; we also need to make some decompositions about nonlinearity, forcing term, solution, and solution semigroup.About forcing termg(x), we think about two cases: the weak solutions case g(x)∈H-1(Ω): for every g∈H-1(Ω) and any ϵ>0, there is gϵ∈L2(Ω) which depends on g and ϵ, such that(45)g-gϵH-1Ω<ϵ;the strong solutions case g(x)∈L2(Ω): for every g∈L2(Ω) and any ϵ>0, there is gϵ∈H01(Ω) which depends on g and ϵ, such that(46)g-gϵ<ϵ.For the nonlinearityf, we take into account two cases: for the weak solutions case, critical exponent p=4/n-2; for the strong solutions case, critical exponent p=4/n-4. Similar to that in [5, 18, 19], we know that f allows the following decomposition:(47)f=f0+f1,where f0,f1∈C1(R) with f1(0)=0, f2(0)=0 and satisfy(48)f0uu⩾0,∀u∈R,(49)f0′u⩽C1+up,∀u∈R,(50)f0′u⩽l1,∀u∈R,(51)f1u⩽C1+uγ,∀u∈R,0<γ<p,(52)lim⁡supu→∞⁡f1uu>-λ1θ.Set(53)σ=min⁡1,n+2θ-2nn-2,2θ3,θ-n-2θγ2,where γ is defined by (51).Analogous to that in [17], we decompose the solution z(t)=(u,ut,ηt) of (17) corresponding to initial data z0∈H1 as follows:(54)zt=z1t+z2t;here z1(t)=(vϵ,vtϵ,ζtϵ) and z2(t)=(wϵ,wtϵ,ξtϵ) satisfy the following equations:(55)vttϵ+αvtϵ+Aθvϵ+∫0∞μsAθζtϵsds+f0vϵ=g-gϵ,ζttϵ=-ζstϵ+vtϵ,vϵx,t∂Ω=0,vϵx,0=u0x,vtϵx,0=ut0x,ζtϵx,t∂Ω=0,ζ0ϵx,s=η0x,s,(56)wttϵ+αwtϵ+Aθwϵ+∫0∞μsAθξtϵsds+fu-f0vϵ=gϵ,ξttϵ=-ξstϵ+wtϵ,wϵx,t∂Ω=0,wϵx,0=0,wtϵx,0=0,ξtϵx,s∂Ω=0,ξ0ϵx,s=0.According to the general theorem about the existence of global attractors of infinite-dimensional dynamical systems (see [6, 15]), we also need to prove the asymptotic compactness of {S(t)}t⩾0 in H1.Similar to the proof of Theorem5, we can gain the corresponding existence and uniqueness of solutions for (55) and (56); furthermore, we know that the solutions of (55) also form a semigroup. For the sake of convenience, we denote the solution operators of (55) and (56) by {S1(t)}t⩾0 and {S2(t,·)}t⩾0, respectively. Then, for every z0∈H1, we have(57)zt=Stz0=S1tz0+S2t,z0,∀t⩾0.Hereafter, we will test the necessary condition of asymptotic smoothness.At first, akin to the proof of Theorem6, we can obtain the following result about the solution z1(t) of (55) in H1.Lemma 7.
Assume thatf0 satisfy (48) and (49), g∈H-1(Ω), and (6)–(8) hold. Then, for any ε0>0, there is a constant ϵ=ϵ(ε0,g), such that the solutions of (55) satisfy the following estimates:(58)S1tz0H12=12vϵVθ2+vtϵ2+ζtϵμ,Vθ2⩽Qz0H1e-β1t+ε0,∀t⩾t0,where Q(·) is an increasing function on [0,∞) and β1 only depends on α, ρ, and δ.Proof.
Taking theL2 inner product of (70) with vtϵ+ρvϵ, we have(59)12ddtvϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+ρvϵVθ2+α-ρvtϵ+ρvϵ2-ρα-ρvϵ,vtϵ+ρvϵ+ζtϵ,ζstϵμ,Vθ+ρζtϵ,vϵμ,Vθ=-f0vϵ,vtϵ+ρvϵ+g-gϵ,vtϵ+ρvϵ.For the forcing term, we know(60)-g-gϵ,vtϵ+ρvϵ=-ddtg-gϵ,vϵ-ρg-gϵ,vϵ,g-gϵ,vϵ⩽2λ1θ-1g-gϵH-1Ω2+18vϵVθ2.
About the nonlinearity, from (48), we have(61)-f0vϵ,vtϵ+ρvϵ=-ddt∫ΩF0vϵdx-ρf0vϵ,vϵ⩽-ddt∫ΩF0vϵdx,where F0(vϵ)=∫0vϵf(r)dr.
In view of (49), (13), (14), and (44), we know(62)ρ2∫ΩF0vϵdx⩽ρ2C∫Ωvϵ2+vϵ2n/n-2dx⩽ρ2Cvϵ2+∇vϵ2⩽ρ2C+Cϵvϵ2+ρ2CϵAθ/2vϵ2⩽ρ2Cρ,ϵ,μ0Aθ/2vϵ2;here the constant Cρ,ϵ,μ0 being very small depends not only on ϵ and ρ, but also on the bound μ0 in (44). Choosing ρ to be sufficiently small, we have(63)ρvϵVθ2+α-ρvtϵ+ρvϵ2-ρα-ρvϵ,vtϵ+ρvϵ⩾ρ1-αρλ1θvϵVθ2+12α-ρvtϵ+ρvϵ2⩾ρ2vϵVθ2+α4vtϵ+ρvϵ2.
Thanks to Lemma1 and using (7), we get that(64)ζtϵ,ζstϵμ,Vθ⩾δ2ζtϵμ,Vθ2,ρζtϵ,vϵμ,Vθ⩾-ρ∫Ω∫0∞μsAθ/2ζtϵs2ds1/2∫0∞μsAθ/2vϵ2ds1/2dx⩾-δ4ζtϵμ,Vθ2-k0ρ2δvϵVθ2.
Then,(65)12ddtvϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+2∫ΩF0vϵdx-2g-gϵ,vϵ+Cˇ+ρ2-k0ρ2δ-ρ2Cρ,ϵ,μ0vϵVθ2+α4vtϵ+ρvϵVθ2+δ4ζtϵμ,V2θ2+ρ2∫ΩF0vϵdx-ρg-gϵ,Aθvϵ+ρCˇ⩽ρCˇ;here, Cˇ=4/λ1θ-1g-gϵH-1(Ω)2.
Obviously, we see that(66)ρ2-k0ρ2δ-ρ2Cρ,ϵ,μ0⩾ρ4,so(67)ρ2-k0ρ2δ-ρ2Cρ,ϵ,μ0vϵV2θ2⩾ρ4vϵV2θ2.
Then, combining the above estimates with (59), we see that(68)12ddtvϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+2∫ΩF0vϵdx-2g-gϵ,vϵ+Cˇ+β12vϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+2∫ΩF0vϵdx-2g-gϵ,vϵ+Cˇ⩽ρCˇ<4ρλ1θ-1ϵ2,where β1=min⁡ρ/2,α/2,δ/2,ρ2.
Obviously, there exist constantsCˇ1, Cˇ2, such that(69)Cˇ1z1tH12⩽12vϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+2∫ΩF0vϵdx-2g-gϵ,vϵ+Cˇ⩽Cˇ2z1tH12;thus, using Gronwall lemma and taking 4ρϵ2/λ1θ-1β1Cˇ1<ε0, as t⩾t0, we obtain(70)z1tH12⩽Cˇ2Cˇ1z0H12e-β1t+4ρϵ2λ1θ-1β1Cˇ1<Qz0H1e-β1t+ε0.
The proof is complete.Lemma 8.
Let the nonlinearityf satisfy (2)-(3) and (51)-(52), g∈H-1(Ω), and (7)–(9) hold. Then, for every given T>0 and ϵ>0, there is a position constant(71)N0=N0T,gH-1Ω,z0H1,δ,such that the solutions of (56) satisfy(72)S2T,z0H1+σ2=12wϵVθ+σ2+wtϵVσ2+ξTϵμ,Vθ+σ2⩽N0,where σ=min⁡{1,(n+2)θ-2n/n-2,2θ/3,θ-(n-2θ)γ/2}.Proof.
Multiplying (56) by Aσ(wtϵ+ρwϵ), we get that(73)12ddtwϵVθ+σ2+wtϵ+ρwϵVσ2+ξtϵμ,Vθ+σ2+ρwϵVθ+σ2+α-ρwtϵ+ρwVσ2-ρα-ρwϵ,wtϵ+ρwϵVσ+ξtϵ,ξstϵμ,Vθ+σ+ρξtϵ,wϵμ,Vθ+σ=-fu-f0vϵ,Aσwtϵ+ρwϵ+gϵ,Aσwtϵ+ρwϵ.
Using Lemma1, we can obtain(74)ξtϵ,ξstϵμ,Vθ+σ⩾δ2ξtϵμ,Vθ+σ2,ρξtϵ,wϵμ,Vθ+σ⩾-ρ∫Ω∫0∞μsAθ+σ/2ξtϵAθ+σ/2wϵdsdx⩾-ρ∫Ω∫0∞μsAθ+σ/2ξtϵ2ds1/2∫0∞μsAθ+σ/2wϵ2ds1/2dx⩾-δ4ξtϵμ,Vθ+σ2-k0ρ2δwϵVθ+σ2.
Similarly, we have(75)12ddtwϵVθ+σ2+wtϵ+ρwϵVσ2+ξtϵμ,Vθ+σ2+ρ-k0ρ2δwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-ρα-ρwϵ,wtϵ+ρwϵVσ+δ4ξtϵμ,Vθ+σ2=-fu-f0vϵ,Aσwtϵ+ρwϵ+gϵ,Aσwtϵ+ρwϵ.Choosing ρ to be sufficiently small, such that α/2-ρ⩾α/4 and 1-ρα/λ1θ-k0ρ/δ⩾1/2, we can obtain that(76)ρ1-k0ρδwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-ρα-ρwϵ,wtϵ+ρwϵVσ⩾ρ1-k0ρδwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-ραwϵ,wtϵ+ρwϵVσ+ρ2wϵ,wtϵVσ⩾ρ1-k0ρδwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-αρ2λ1θwϵVθ+σ2-α-ρρwϵ,wtϵVσ⩾ρ1-k0ρδwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-αρ2λ1θwϵVθ+σ2-α2wtϵ+ρwϵVσ2⩾ρ2wϵVθ+σ2+α4wtϵ+ρwϵVσ2.
Sincef=f0+f1, we have(77)fu-f0vϵ,Aσwtϵ+ρwϵ=fu-fvϵ,Aσwtϵ+ρwϵ+f1vϵ,Aσwtϵ+ρwϵ.We deal with the terms in (77) one by one.
First, by virtue of (3), we deduce that(78)-fu-fvϵ,Aσwtϵ+ρwϵ⩽∫Ωf′uA3σ/4u-f′vϵA3σ/4vϵ·Aσ/4wtϵ+ρwϵdx⩽C∫Ω1+u4/n-2+vϵ4/n-2A3σ/4wϵAσ/4wtϵ+ρwϵdx+2C∫Ω1+u4/n-2+vϵ4/n-2A3σ/4vϵAσ/4wtϵ+ρwϵdx.
Second, sinceθ∈[2n/n+2,n/2), we have(79)C∫Ω1+u4/n-2+vϵ4/n-2A3σ/4wϵAσ/4wtϵ+ρwϵdx⩽C∫Ω1+u4/n-2+vϵ4/n-2n/θdxθ/n∫ΩA3σ/4wϵ2n/n-2θ-σ/2dxn-2θ-σ/2/2n·∫ΩAσ/4wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽C1+uL4n/n-2θ4/n-2+vϵL4n/n-2θ4/n-2A3σ/4wϵL2n/n-2θ-σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽C1+uL2n/n-2θ4/n-2+vϵL2n/n-2θ4/n-2A3σ/4wϵL2n/n-2θ-σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽C1+Aθ/2u4/n-2+Aθ/2vϵ4/n-2Aθ+σ/2wϵ·Aσ/2wtϵ+ρwϵ⩽Cμ0,ε0Aθ+σ/2wϵ2+α4Aσ/2wtϵ+ρwϵ2,where 4n/(n-2)θ⩽2n/n-2θ, and we have used the embedding Vθ=D(Aθ/2)↪L2n/n-2θ↪L4n/(n-2)θ, Vσ/2=D(Aσ/4)↪L2n/n-σ, Vθ-σ/2=D(A2θ-σ/4)↪L2n/n-2(θ-σ/2).
Third, from (53), we obtain that(80)2C∫Ω1+u4/n-2+vϵ4/n-2A3σ/4vϵAσ/4wtϵ+ρwϵdx⩽2C∫Ω1+u4/n-2+vϵ4/n-2n/θ-σdxθ-σ/n∫ΩA3σ/4vϵ2n/n-2θ-3σ/2dxn-2θ-3σ/2/2n·∫ΩAσ/4wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽2C1+uL4n/n-2θ-σ4/n-2+vϵL4n/n-2θ-σ4/n-2A3σ/4vϵL2n/n-2θ-3σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽2C1+uL2n/n-2θ4/n-2+vϵL2n/n-2θ4/n-2A3σ/4vϵL2n/n-2θ-σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽2C1+Aθ/2u4/n-2+Aθ/2vϵ4/n-2Aθ/2vϵ·Aσ/2wtϵ+ρwϵ⩽2C1+Aθ/2u4/n-2+Aθ/2vϵ4/n-22Aθ/2vϵ2+α4Aσ/2wtϵ+ρwϵ2⩽Cμ0,ε0+α4Aσ/2wtϵ+ρwϵ2,where 4n/(n-2)(θ-σ)⩽2n/n-2θ, and we have used the embedding Vθ=DAθ/2↪L2n/n-2θ↪L4n/n-2θ-σ, Vσ/2=D(Aσ/4)↪L2n/n-σ, Vθ-3σ/2=D(A2θ-3σ/4)↪L2n/n-2(θ-3σ/2).
Finally, by virtue of (51), we get(81)-f1vϵ,Aσwtϵ+ρwϵ=-∫Ωf1′vϵ·A3σ/4vϵ·Aσ/4wtϵ+ρwϵdx⩽C∫Ω1+vϵγA3σ/4vϵAσ/4wtϵ+ρwϵdx⩽C∫Ω1+vϵγn/θ-σdxθ-σ/n·∫ΩA3σ/4vϵ2n/n-2θ-3σ/2dxn-2θ-3σ/2/2n·∫ΩAσ/4wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽C1+vϵLnγ/θ-σγA3σ/4vϵL2n/n-2θ-3σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽C1+vϵL2n/n-2θγA3σ/4vϵL2n/n-2θ-3σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽C1+Aθ/2vϵγAθ/2vϵAσ/2wtϵ+ρwϵ⩽C1+Aθ/2vϵγ2Aθ/2vϵ2+α4Aσ/2wtϵ+ρwϵ2⩽Cε0+α4Aσ/2wtϵ+ρwϵ2,where we have employed the inequality σ⩽θ-(n-2θ)γ/2,nγ/θ-σ⩽2n/n-2θ and used the embedding Vθ↪L2n/n-2θ↪Lnγ/θ-σ, Vσ/2=D(Aσ/4)↪L2n/n-σ and Vθ-3σ/2=D(A2θ-3σ/4)↪L2n/n-2(θ-3σ/2).
Obviously, we see(82)gϵ,Aσwtϵ+ρwϵ=ddtgϵ,Aσwϵ+ρgϵ,Aσwϵ,and, noting that 0<σ<2θ/3, we know that(83)gϵ,Aσwϵ⩽gϵAσwϵ⩽12λ1θ-σgϵ2+λ1θ-σ2Aσwϵ2⩽12λ1θ-σgϵ2+12Aθ+σ/2wϵ2.
Combining above estimates, we can deduce that(84)12ddtwϵVθ+σ2+wtϵ+ρwϵVσ2+ξtϵμ,Vθ+σ2-2gϵ,Aσwϵ+C˘⩽β22wϵVθ+σ2+wtϵ+ρwϵVσ2+ξtϵμ,Vθ+σ2-2gϵ,Aσwϵ+C˘+C,where C˘=1/λ1θ-σgϵ2, β2=max⁡2Cμ0,ε0-ρ,α,δ/2,ρ.
Applying the Gronwall inequality and integrating over[0,t], we gain that(85)12wϵtVθ+σ2+wtϵt+ρwϵtVσ2+ξtϵsμ,Vθ+σ2-2gϵx,Aσwϵt+C˘⩽12wϵ0Vθ+σ2+wtϵ0+ρwϵ0Vσ2+ξ0ϵsμ,Vθ+σ2-2gϵx,Aσwϵ0+C˘eβ2t+Cβ2eβ2t-1⩽C˘2+Cβ2eβ2t.
Clearly, there exist constantsC˘1, C˘2, such that(86)C˘1z2tH1+σ2⩽12wϵtVθ+σ2+wtϵt+ρwϵtVσ2+ξtϵsμ,Vθ+σ2-2gϵx,Aσwϵt+C˘⩽C˘2z2tH1+σ2,so(87)z2tH1+σ2⩽1C˘1C˘2+Cβ2eβ2t.That is,(88)S2T,z0H1+σ2=12wϵTθ+σ2+wtTVσ2+ξTϵμ,Vθ+σ2⩽N0,where N0=N0(T,g,z0H1,α,δ).
The proof is complete.In addition, for anyξ0∈Lμ2(R+;Vθ), Cauchy problem (see [9, 15, 16])(89)ξtt=-ξst+wt,t>0ξ0=ξ0has a unique solution ξt∈C([0,∞);Lμ2(R+;Vθ)) and explicit expression:(90)ξtx,s=wx,t-wx,t-s,0<s⩽twx,t-w0,s>t.LetB0∗ be the bounded absorbing set in H1 obtained from Theorem 6; we can obtain the following.Lemma 9.
Set(91)KTϵ≔ΠS2T,B0∗.Under the assumption of Lemma 8, for every given T>0,∀ϵ>0, there is a positive constant N1=N1(T,g,ϵ,B0∗H1,α,δ), such that(i)
KTϵ is bounded in Lμ2(R+;Vθ+σ)∩Hμ1(R+;H);(ii)
supξ∈KTϵξ(s)Vθ2⩽N1,where σ=min⁡{1,(n+2)θ-2n/n-2,2θ/3,θ-(n-2θ)γ/2}. S2(t,·) is the solution operators of (56), and Π:Vθ×H×Lμ2(R+;Vθ)→Lμ2(R+;Vθ) is a projection operator.Proof.
Due to the explicit expression (90), we can deduce that(92)ξstx,s=wsx,t-s,0<s⩽t0,s>t,which, combined with Lemma 8, implies that (i) holds.
After that, we can easily know that(93)ξTx,sVθ⩽wT-wT-sVθ,0<s⩽T.wTVθ,s>T.By virtue of (88), we get that (ii) holds.
We complete the proof.Moreover, applying Lemma2, we can obtain that KTϵ is relatively compact in Lμ2(R+;Vθ). Then, using the compact embedding Vθ+σ↪Vθ once again, we can get the following.Lemma 10.
LetS2(t,·) be the corresponding solution operator of (56). Under the assumption of Lemma 8, for any T>0, S2(T,B0∗) is relatively compact in H1.By applying the classical theory (e.g., see [6, 20]), from Lemma 3, Theorem 6, and Lemmas 7 and 10, we can gain the main result.Theorem 11 (global attractor inH1).
Let the nonlinearityf satisfy (2)-(3), g∈H-1(Ω), (7)–(9) hold, and {S(t)}t⩾0 be the solution semigroup of (17)-(18) in H1. Then {S(t)}t⩾0 has a global attractor A0 in H1; namely, A0 is compact and invariant in H1 and attracts every bounded subset of H1 with respect to the H1-norm.


## 3.1. The Existence and Uniqueness of Weak Solution

At first, we define the concept of weak solution for dynamical systems (17)-(18) (cf. [15] about the corresponding definition of solutions for wave equations).Definition 4.
DefineI=[0,T],∀T>0. Let g∈H-1(Ω) and z0∈H2. A ternary form z=(u,ut,ηt) which fulfils(26)u∈CI;Vθ,ut∈L2I;Vθ,ηt∈CI;Lμ2R+;Vθ;ηtt+ηst∈L∞I;Lμ2R+;H∩L2I;Lμ2R+;Vθis said to be a weak solution of problem (17)-(18) in the time interval I with initial data z(0)=z0, provided that(27)utt,ω+αut,ω+u,ωVθ+ηt,ωμ,Vθ+fu,ω=g,ω,ηtt+ηst,φμ,Vθ=ut,φμ,Vθ,for all ω∈Vθ,φ∈Lμ2R+;Vθ,a.e. t∈I.Then, we start with the following general existence and unique result. The proof is based on the normal Faedo-Galerkin methods introduced in [9, 15].Theorem 5 (existence and uniqueness of solutions).
Letf satisfy (2)-(3), g∈H-1(Ω), and (7)–(9) hold. Then for any given z0=(u0,ut0,η0)∈H1, and any T>0, there is a unique solution z(t)=(u,ut,ηt)∈H1 for problem (17)-(18), satisfying(28)z∈L∞0,T;H1∩C0,T;H1.According to Theorem5 above, we can define the solution operator; that is, (29)St:H1⟶H1,z0=u0,ut0,η0⟶ut,utt,ηt=Stz0.Obviously, {S(t)}t⩾0 can form a solution semigroup.In the remainder of this section, we denote by{S(t)}t⩾0 the semigroup associated with the solution of (17)-(18).


## 3.2. The Existence of Bounded Absorbing Set

Theorem 6 (bounded absorbing inH1).
LetB0 be any bounded subset of H1 and z(t) be a solution of (17)-(18) with initial data z0∈B0. If the nonlinearity f satisfies (2), g∈H-1(Ω), (7)–(9) hold. Then there is a positive constant μ0 such that, for any bounded subset B0⊂H1, there exists t0=t0(B0H1) such that(30)Stz0H1⩽μ0,∀t⩾t0,z0∈B0.Proof.
Taking0<ρ<1, multiplying (17) by ut+ρu, and integrating over Ω, we have(31)12ddtuVθ2+ut+ρu2+ηtμ,Vθ2+α-ρut+ρu2+ρuVθ2+ηt,ηstμ,Vθ-ρα-ρu,ut+ρu+ρηt,uμ,Vθ+fu,ut+ρu=g,ut+ρu.Then, using Poincaré inequality and Cauchy inequality, and taking ρ small enough, we can get that(32)ρuVθ2+α-ρut+ρu2-ρα-ρu,ut+ρu⩾ρuVθ2+α-ρut+ρu2-ραu,ut+ρu+ρ2u,ut⩾ρuVθ2+α-ρut+ρu2-αρ2λ1θuVθ2-α-ρρu,ut⩾ρuVθ2+α-ρut+ρu2-αρ2λ1θuVθ2-α2ut+ρu2=ρ1-αρλ1θuVθ2+12α-ρut+ρu2.
Applying Lemma1, we have(33)ηt,ηstμ,Vθ⩾δ2ηtμ,Vθ2,and, from (8), we see(34)ρηt,uμ,Vθ⩾-ρ∫Ω∫0∞μsAθ/2ηts2ds1/2∫0∞μsAθ/2u2ds1/2dx⩾-δ4ηtμ,Vθ2-k0ρ2δuVθ2.
Therefore,(35)12ddtuVθ2+ut+ρu2+ηtμ,Vθ2+ρ1-αρλ1θ-k0ρδuVθ2+12α-ρut+ρu2+δ4ηtμ,Vθ2+fu,ut+ρu⩽g,ut+ρu.
From (2), we know that there exist λ1θ>λ∗>0 and C0>0 such that(36)fu,u⩾λ∗4∫ΩFudx-C0,Fu⩾-λ∗8u2-C0;here, F(u)=∫0u‍f(r)dr. Then(37)fu,ut+ρu⩾ddt∫ΩFudx+ρλ∗4∫ΩFudx-ρC0.
Evidently, we can get(38)g,ut+ρu=ddtg,u+ρg,u,g,u⩽gH-1Ω∇u⩽2λ1θ-1gH-1Ω2+λ1θ-18∇u2⩽2λ1θ-1gH-1Ω2+18uVθ2.Choose ρ to be sufficiently small, such that(39)1-αρλ1θ-k0ρδ⩾12,12α-ρ⩾α4;thus,(40)ρ1-αρλ1θ-k0ρδuVθ2+12α-ρut+ρu2⩾ρ2uVθ2+α4ut+ρu2.
Substituting (32)–(40) into (31), we can obtain that(41)12ddtuVθ2+ut+ρu2+ηtμ,Vθ2+2∫ΩFudx-2g,u+C~+β02uVθ2+ut+ρu2+ηtμ,Vθ2+2∫ΩFudx-2g,u+C~⩽ρC0+β02C~⩽C,where β0=min⁡{ρ,α/2,δ/2,ρλ∗/4} and C~=2C0+4/λ1θ-1gH-1(Ω)2.
Hence, from (36)–(38), there exist positive constants C1, C2, such that(42)C1ztH12⩽12uVθ2+ut+ρu2+ηtμ,Vθ2+2∫ΩFudx-2g,u+C~⩽C2ztH12.According to Gronwall lemma,(43)ztH12⩽C2C1z0H12e-β0t+Cβ0C1.Assuming that z(0)H12⩽R, as t⩾t0=t0(B0H1), we have(44)ztH1⩽μ0.
We complete the proof.


## 3.3. The Existence of Global Attractor inH1

Now we will establish the necessary asymptotic smoothness, similar to that in [17–19]; we also need to make some decompositions about nonlinearity, forcing term, solution, and solution semigroup.About forcing termg(x), we think about two cases: the weak solutions case g(x)∈H-1(Ω): for every g∈H-1(Ω) and any ϵ>0, there is gϵ∈L2(Ω) which depends on g and ϵ, such that(45)g-gϵH-1Ω<ϵ;the strong solutions case g(x)∈L2(Ω): for every g∈L2(Ω) and any ϵ>0, there is gϵ∈H01(Ω) which depends on g and ϵ, such that(46)g-gϵ<ϵ.For the nonlinearityf, we take into account two cases: for the weak solutions case, critical exponent p=4/n-2; for the strong solutions case, critical exponent p=4/n-4. Similar to that in [5, 18, 19], we know that f allows the following decomposition:(47)f=f0+f1,where f0,f1∈C1(R) with f1(0)=0, f2(0)=0 and satisfy(48)f0uu⩾0,∀u∈R,(49)f0′u⩽C1+up,∀u∈R,(50)f0′u⩽l1,∀u∈R,(51)f1u⩽C1+uγ,∀u∈R,0<γ<p,(52)lim⁡supu→∞⁡f1uu>-λ1θ.Set(53)σ=min⁡1,n+2θ-2nn-2,2θ3,θ-n-2θγ2,where γ is defined by (51).Analogous to that in [17], we decompose the solution z(t)=(u,ut,ηt) of (17) corresponding to initial data z0∈H1 as follows:(54)zt=z1t+z2t;here z1(t)=(vϵ,vtϵ,ζtϵ) and z2(t)=(wϵ,wtϵ,ξtϵ) satisfy the following equations:(55)vttϵ+αvtϵ+Aθvϵ+∫0∞μsAθζtϵsds+f0vϵ=g-gϵ,ζttϵ=-ζstϵ+vtϵ,vϵx,t∂Ω=0,vϵx,0=u0x,vtϵx,0=ut0x,ζtϵx,t∂Ω=0,ζ0ϵx,s=η0x,s,(56)wttϵ+αwtϵ+Aθwϵ+∫0∞μsAθξtϵsds+fu-f0vϵ=gϵ,ξttϵ=-ξstϵ+wtϵ,wϵx,t∂Ω=0,wϵx,0=0,wtϵx,0=0,ξtϵx,s∂Ω=0,ξ0ϵx,s=0.According to the general theorem about the existence of global attractors of infinite-dimensional dynamical systems (see [6, 15]), we also need to prove the asymptotic compactness of {S(t)}t⩾0 in H1.Similar to the proof of Theorem5, we can gain the corresponding existence and uniqueness of solutions for (55) and (56); furthermore, we know that the solutions of (55) also form a semigroup. For the sake of convenience, we denote the solution operators of (55) and (56) by {S1(t)}t⩾0 and {S2(t,·)}t⩾0, respectively. Then, for every z0∈H1, we have(57)zt=Stz0=S1tz0+S2t,z0,∀t⩾0.Hereafter, we will test the necessary condition of asymptotic smoothness.At first, akin to the proof of Theorem6, we can obtain the following result about the solution z1(t) of (55) in H1.Lemma 7.
Assume thatf0 satisfy (48) and (49), g∈H-1(Ω), and (6)–(8) hold. Then, for any ε0>0, there is a constant ϵ=ϵ(ε0,g), such that the solutions of (55) satisfy the following estimates:(58)S1tz0H12=12vϵVθ2+vtϵ2+ζtϵμ,Vθ2⩽Qz0H1e-β1t+ε0,∀t⩾t0,where Q(·) is an increasing function on [0,∞) and β1 only depends on α, ρ, and δ.Proof.
Taking theL2 inner product of (70) with vtϵ+ρvϵ, we have(59)12ddtvϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+ρvϵVθ2+α-ρvtϵ+ρvϵ2-ρα-ρvϵ,vtϵ+ρvϵ+ζtϵ,ζstϵμ,Vθ+ρζtϵ,vϵμ,Vθ=-f0vϵ,vtϵ+ρvϵ+g-gϵ,vtϵ+ρvϵ.For the forcing term, we know(60)-g-gϵ,vtϵ+ρvϵ=-ddtg-gϵ,vϵ-ρg-gϵ,vϵ,g-gϵ,vϵ⩽2λ1θ-1g-gϵH-1Ω2+18vϵVθ2.
About the nonlinearity, from (48), we have(61)-f0vϵ,vtϵ+ρvϵ=-ddt∫ΩF0vϵdx-ρf0vϵ,vϵ⩽-ddt∫ΩF0vϵdx,where F0(vϵ)=∫0vϵf(r)dr.
In view of (49), (13), (14), and (44), we know(62)ρ2∫ΩF0vϵdx⩽ρ2C∫Ωvϵ2+vϵ2n/n-2dx⩽ρ2Cvϵ2+∇vϵ2⩽ρ2C+Cϵvϵ2+ρ2CϵAθ/2vϵ2⩽ρ2Cρ,ϵ,μ0Aθ/2vϵ2;here the constant Cρ,ϵ,μ0 being very small depends not only on ϵ and ρ, but also on the bound μ0 in (44). Choosing ρ to be sufficiently small, we have(63)ρvϵVθ2+α-ρvtϵ+ρvϵ2-ρα-ρvϵ,vtϵ+ρvϵ⩾ρ1-αρλ1θvϵVθ2+12α-ρvtϵ+ρvϵ2⩾ρ2vϵVθ2+α4vtϵ+ρvϵ2.
Thanks to Lemma1 and using (7), we get that(64)ζtϵ,ζstϵμ,Vθ⩾δ2ζtϵμ,Vθ2,ρζtϵ,vϵμ,Vθ⩾-ρ∫Ω∫0∞μsAθ/2ζtϵs2ds1/2∫0∞μsAθ/2vϵ2ds1/2dx⩾-δ4ζtϵμ,Vθ2-k0ρ2δvϵVθ2.
Then,(65)12ddtvϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+2∫ΩF0vϵdx-2g-gϵ,vϵ+Cˇ+ρ2-k0ρ2δ-ρ2Cρ,ϵ,μ0vϵVθ2+α4vtϵ+ρvϵVθ2+δ4ζtϵμ,V2θ2+ρ2∫ΩF0vϵdx-ρg-gϵ,Aθvϵ+ρCˇ⩽ρCˇ;here, Cˇ=4/λ1θ-1g-gϵH-1(Ω)2.
Obviously, we see that(66)ρ2-k0ρ2δ-ρ2Cρ,ϵ,μ0⩾ρ4,so(67)ρ2-k0ρ2δ-ρ2Cρ,ϵ,μ0vϵV2θ2⩾ρ4vϵV2θ2.
Then, combining the above estimates with (59), we see that(68)12ddtvϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+2∫ΩF0vϵdx-2g-gϵ,vϵ+Cˇ+β12vϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+2∫ΩF0vϵdx-2g-gϵ,vϵ+Cˇ⩽ρCˇ<4ρλ1θ-1ϵ2,where β1=min⁡ρ/2,α/2,δ/2,ρ2.
Obviously, there exist constantsCˇ1, Cˇ2, such that(69)Cˇ1z1tH12⩽12vϵVθ2+vtϵ+ρvϵ2+ζtϵμ,Vθ2+2∫ΩF0vϵdx-2g-gϵ,vϵ+Cˇ⩽Cˇ2z1tH12;thus, using Gronwall lemma and taking 4ρϵ2/λ1θ-1β1Cˇ1<ε0, as t⩾t0, we obtain(70)z1tH12⩽Cˇ2Cˇ1z0H12e-β1t+4ρϵ2λ1θ-1β1Cˇ1<Qz0H1e-β1t+ε0.
The proof is complete.Lemma 8.
Let the nonlinearityf satisfy (2)-(3) and (51)-(52), g∈H-1(Ω), and (7)–(9) hold. Then, for every given T>0 and ϵ>0, there is a position constant(71)N0=N0T,gH-1Ω,z0H1,δ,such that the solutions of (56) satisfy(72)S2T,z0H1+σ2=12wϵVθ+σ2+wtϵVσ2+ξTϵμ,Vθ+σ2⩽N0,where σ=min⁡{1,(n+2)θ-2n/n-2,2θ/3,θ-(n-2θ)γ/2}.Proof.
Multiplying (56) by Aσ(wtϵ+ρwϵ), we get that(73)12ddtwϵVθ+σ2+wtϵ+ρwϵVσ2+ξtϵμ,Vθ+σ2+ρwϵVθ+σ2+α-ρwtϵ+ρwVσ2-ρα-ρwϵ,wtϵ+ρwϵVσ+ξtϵ,ξstϵμ,Vθ+σ+ρξtϵ,wϵμ,Vθ+σ=-fu-f0vϵ,Aσwtϵ+ρwϵ+gϵ,Aσwtϵ+ρwϵ.
Using Lemma1, we can obtain(74)ξtϵ,ξstϵμ,Vθ+σ⩾δ2ξtϵμ,Vθ+σ2,ρξtϵ,wϵμ,Vθ+σ⩾-ρ∫Ω∫0∞μsAθ+σ/2ξtϵAθ+σ/2wϵdsdx⩾-ρ∫Ω∫0∞μsAθ+σ/2ξtϵ2ds1/2∫0∞μsAθ+σ/2wϵ2ds1/2dx⩾-δ4ξtϵμ,Vθ+σ2-k0ρ2δwϵVθ+σ2.
Similarly, we have(75)12ddtwϵVθ+σ2+wtϵ+ρwϵVσ2+ξtϵμ,Vθ+σ2+ρ-k0ρ2δwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-ρα-ρwϵ,wtϵ+ρwϵVσ+δ4ξtϵμ,Vθ+σ2=-fu-f0vϵ,Aσwtϵ+ρwϵ+gϵ,Aσwtϵ+ρwϵ.Choosing ρ to be sufficiently small, such that α/2-ρ⩾α/4 and 1-ρα/λ1θ-k0ρ/δ⩾1/2, we can obtain that(76)ρ1-k0ρδwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-ρα-ρwϵ,wtϵ+ρwϵVσ⩾ρ1-k0ρδwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-ραwϵ,wtϵ+ρwϵVσ+ρ2wϵ,wtϵVσ⩾ρ1-k0ρδwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-αρ2λ1θwϵVθ+σ2-α-ρρwϵ,wtϵVσ⩾ρ1-k0ρδwϵVθ+σ2+α-ρwtϵ+ρwϵVσ2-αρ2λ1θwϵVθ+σ2-α2wtϵ+ρwϵVσ2⩾ρ2wϵVθ+σ2+α4wtϵ+ρwϵVσ2.
Sincef=f0+f1, we have(77)fu-f0vϵ,Aσwtϵ+ρwϵ=fu-fvϵ,Aσwtϵ+ρwϵ+f1vϵ,Aσwtϵ+ρwϵ.We deal with the terms in (77) one by one.
First, by virtue of (3), we deduce that(78)-fu-fvϵ,Aσwtϵ+ρwϵ⩽∫Ωf′uA3σ/4u-f′vϵA3σ/4vϵ·Aσ/4wtϵ+ρwϵdx⩽C∫Ω1+u4/n-2+vϵ4/n-2A3σ/4wϵAσ/4wtϵ+ρwϵdx+2C∫Ω1+u4/n-2+vϵ4/n-2A3σ/4vϵAσ/4wtϵ+ρwϵdx.
Second, sinceθ∈[2n/n+2,n/2), we have(79)C∫Ω1+u4/n-2+vϵ4/n-2A3σ/4wϵAσ/4wtϵ+ρwϵdx⩽C∫Ω1+u4/n-2+vϵ4/n-2n/θdxθ/n∫ΩA3σ/4wϵ2n/n-2θ-σ/2dxn-2θ-σ/2/2n·∫ΩAσ/4wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽C1+uL4n/n-2θ4/n-2+vϵL4n/n-2θ4/n-2A3σ/4wϵL2n/n-2θ-σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽C1+uL2n/n-2θ4/n-2+vϵL2n/n-2θ4/n-2A3σ/4wϵL2n/n-2θ-σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽C1+Aθ/2u4/n-2+Aθ/2vϵ4/n-2Aθ+σ/2wϵ·Aσ/2wtϵ+ρwϵ⩽Cμ0,ε0Aθ+σ/2wϵ2+α4Aσ/2wtϵ+ρwϵ2,where 4n/(n-2)θ⩽2n/n-2θ, and we have used the embedding Vθ=D(Aθ/2)↪L2n/n-2θ↪L4n/(n-2)θ, Vσ/2=D(Aσ/4)↪L2n/n-σ, Vθ-σ/2=D(A2θ-σ/4)↪L2n/n-2(θ-σ/2).
Third, from (53), we obtain that(80)2C∫Ω1+u4/n-2+vϵ4/n-2A3σ/4vϵAσ/4wtϵ+ρwϵdx⩽2C∫Ω1+u4/n-2+vϵ4/n-2n/θ-σdxθ-σ/n∫ΩA3σ/4vϵ2n/n-2θ-3σ/2dxn-2θ-3σ/2/2n·∫ΩAσ/4wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽2C1+uL4n/n-2θ-σ4/n-2+vϵL4n/n-2θ-σ4/n-2A3σ/4vϵL2n/n-2θ-3σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽2C1+uL2n/n-2θ4/n-2+vϵL2n/n-2θ4/n-2A3σ/4vϵL2n/n-2θ-σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽2C1+Aθ/2u4/n-2+Aθ/2vϵ4/n-2Aθ/2vϵ·Aσ/2wtϵ+ρwϵ⩽2C1+Aθ/2u4/n-2+Aθ/2vϵ4/n-22Aθ/2vϵ2+α4Aσ/2wtϵ+ρwϵ2⩽Cμ0,ε0+α4Aσ/2wtϵ+ρwϵ2,where 4n/(n-2)(θ-σ)⩽2n/n-2θ, and we have used the embedding Vθ=DAθ/2↪L2n/n-2θ↪L4n/n-2θ-σ, Vσ/2=D(Aσ/4)↪L2n/n-σ, Vθ-3σ/2=D(A2θ-3σ/4)↪L2n/n-2(θ-3σ/2).
Finally, by virtue of (51), we get(81)-f1vϵ,Aσwtϵ+ρwϵ=-∫Ωf1′vϵ·A3σ/4vϵ·Aσ/4wtϵ+ρwϵdx⩽C∫Ω1+vϵγA3σ/4vϵAσ/4wtϵ+ρwϵdx⩽C∫Ω1+vϵγn/θ-σdxθ-σ/n·∫ΩA3σ/4vϵ2n/n-2θ-3σ/2dxn-2θ-3σ/2/2n·∫ΩAσ/4wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽C1+vϵLnγ/θ-σγA3σ/4vϵL2n/n-2θ-3σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽C1+vϵL2n/n-2θγA3σ/4vϵL2n/n-2θ-3σ/2Aσ/4wtϵ+ρwϵL2n/n-σ⩽C1+Aθ/2vϵγAθ/2vϵAσ/2wtϵ+ρwϵ⩽C1+Aθ/2vϵγ2Aθ/2vϵ2+α4Aσ/2wtϵ+ρwϵ2⩽Cε0+α4Aσ/2wtϵ+ρwϵ2,where we have employed the inequality σ⩽θ-(n-2θ)γ/2,nγ/θ-σ⩽2n/n-2θ and used the embedding Vθ↪L2n/n-2θ↪Lnγ/θ-σ, Vσ/2=D(Aσ/4)↪L2n/n-σ and Vθ-3σ/2=D(A2θ-3σ/4)↪L2n/n-2(θ-3σ/2).
Obviously, we see(82)gϵ,Aσwtϵ+ρwϵ=ddtgϵ,Aσwϵ+ρgϵ,Aσwϵ,and, noting that 0<σ<2θ/3, we know that(83)gϵ,Aσwϵ⩽gϵAσwϵ⩽12λ1θ-σgϵ2+λ1θ-σ2Aσwϵ2⩽12λ1θ-σgϵ2+12Aθ+σ/2wϵ2.
Combining above estimates, we can deduce that(84)12ddtwϵVθ+σ2+wtϵ+ρwϵVσ2+ξtϵμ,Vθ+σ2-2gϵ,Aσwϵ+C˘⩽β22wϵVθ+σ2+wtϵ+ρwϵVσ2+ξtϵμ,Vθ+σ2-2gϵ,Aσwϵ+C˘+C,where C˘=1/λ1θ-σgϵ2, β2=max⁡2Cμ0,ε0-ρ,α,δ/2,ρ.
Applying the Gronwall inequality and integrating over[0,t], we gain that(85)12wϵtVθ+σ2+wtϵt+ρwϵtVσ2+ξtϵsμ,Vθ+σ2-2gϵx,Aσwϵt+C˘⩽12wϵ0Vθ+σ2+wtϵ0+ρwϵ0Vσ2+ξ0ϵsμ,Vθ+σ2-2gϵx,Aσwϵ0+C˘eβ2t+Cβ2eβ2t-1⩽C˘2+Cβ2eβ2t.
Clearly, there exist constantsC˘1, C˘2, such that(86)C˘1z2tH1+σ2⩽12wϵtVθ+σ2+wtϵt+ρwϵtVσ2+ξtϵsμ,Vθ+σ2-2gϵx,Aσwϵt+C˘⩽C˘2z2tH1+σ2,so(87)z2tH1+σ2⩽1C˘1C˘2+Cβ2eβ2t.That is,(88)S2T,z0H1+σ2=12wϵTθ+σ2+wtTVσ2+ξTϵμ,Vθ+σ2⩽N0,where N0=N0(T,g,z0H1,α,δ).
The proof is complete.In addition, for anyξ0∈Lμ2(R+;Vθ), Cauchy problem (see [9, 15, 16])(89)ξtt=-ξst+wt,t>0ξ0=ξ0has a unique solution ξt∈C([0,∞);Lμ2(R+;Vθ)) and explicit expression:(90)ξtx,s=wx,t-wx,t-s,0<s⩽twx,t-w0,s>t.LetB0∗ be the bounded absorbing set in H1 obtained from Theorem 6; we can obtain the following.Lemma 9.
Set(91)KTϵ≔ΠS2T,B0∗.Under the assumption of Lemma 8, for every given T>0,∀ϵ>0, there is a positive constant N1=N1(T,g,ϵ,B0∗H1,α,δ), such that(i)
KTϵ is bounded in Lμ2(R+;Vθ+σ)∩Hμ1(R+;H);(ii)
supξ∈KTϵξ(s)Vθ2⩽N1,where σ=min⁡{1,(n+2)θ-2n/n-2,2θ/3,θ-(n-2θ)γ/2}. S2(t,·) is the solution operators of (56), and Π:Vθ×H×Lμ2(R+;Vθ)→Lμ2(R+;Vθ) is a projection operator.Proof.
Due to the explicit expression (90), we can deduce that(92)ξstx,s=wsx,t-s,0<s⩽t0,s>t,which, combined with Lemma 8, implies that (i) holds.
After that, we can easily know that(93)ξTx,sVθ⩽wT-wT-sVθ,0<s⩽T.wTVθ,s>T.By virtue of (88), we get that (ii) holds.
We complete the proof.Moreover, applying Lemma2, we can obtain that KTϵ is relatively compact in Lμ2(R+;Vθ). Then, using the compact embedding Vθ+σ↪Vθ once again, we can get the following.Lemma 10.
LetS2(t,·) be the corresponding solution operator of (56). Under the assumption of Lemma 8, for any T>0, S2(T,B0∗) is relatively compact in H1.By applying the classical theory (e.g., see [6, 20]), from Lemma 3, Theorem 6, and Lemmas 7 and 10, we can gain the main result.Theorem 11 (global attractor inH1).
Let the nonlinearityf satisfy (2)-(3), g∈H-1(Ω), (7)–(9) hold, and {S(t)}t⩾0 be the solution semigroup of (17)-(18) in H1. Then {S(t)}t⩾0 has a global attractor A0 in H1; namely, A0 is compact and invariant in H1 and attracts every bounded subset of H1 with respect to the H1-norm.


## 4. Global Attractors inH2

### 4.1. The Existence and Uniqueness of Strong Solution

First of all, we define the concept of strong solution for dynamical systems (17)-(18).Definition 12.
DefineI=[0,T],∀T>0. Let g∈L2(Ω) and z0∈H2. A ternary form z=(u,ut,ηt) is said to be the strong solution of problem (17)-(18) in the time interval I with initial data z(0)=z0, if z satisfies (27), and(94)u∈CI;V2θ,ut∈L2I;V2θ,ηt∈CI;Lμ2R+;V2θ;ηtt+ηst∈L∞I;Lμ2R+;Vθ∩L2I;Lμ2R+;V2θ.Hereafter, similar to Theorem5, we can obtain the following.Theorem 13 (existence and uniqueness of strong solutions).
Assume thatf satisfy (2) and (4), g∈L2(Ω), and (7)–(9) hold. Then, for any given z0=(u0,ut0,η0)∈H2, and any T>0, there exists a unique solution z(t)=(u,ut,ηt)∈H2 for problem (17)-(18), in the sense that(95)z∈L∞0,T;H2∩C0,T;H2.By Theorem13, we can define the strong solution operator as follows: (96)St:H2⟶H2,z0=u0,ut0,η0⟶ut,utt,ηt=Stz0.Evidently, {S(t)}t⩾0 is a solution semigroup.


### 4.2. The Existence of Bounded Absorbing Set

At first, we will prove the existence of bounded absorbing set inH2.Theorem 14 (bounded absorbing inH2).
Let the nonlinearityf satisfy (2), (5), and (6), g∈L2(Ω), (7)–(9) hold, and B1 be bounded subset of H2. If z(t) is a solution of (14) and (17) with initial data z0∈B1, then there is a positive constant μ1 such that, for any bounded subset B⊂H2, there exists t1=t1(BH2) such that(97)Stz0H2⩽μ1,∀t⩾t1,z0∈B.Proof.
Taking theL2 inner product of (17) with Aθut+ρAθu, we get that(98)12ddtuV2θ2+ut+ρuVθ2+ηtμ,V2θ2+ρuV2θ2+α-ρut+ρuVθ2-ρα-ρu,ut+ρuVθ+ηt,ηstμ,V2θ+ρηt,uμ,V2θ+fu,Aθut+ρu=g,Aθut+ρu.Similar to (35), we have(99)12ddtuV2θ2+ut+ρuVθ2+ηtμ,V2θ2+ρ1-αρλ1θ-k0ρδuV2θ2+12α-ρut+ρuVθ2+δ4ηtμ,V2θ2+fu,Aθut+ρu⩽g,Aθut+ρu.
About the nonlinearity, from (5), (6), and (44), these yield(100)-fu,Aθut+ρAθu=-f′τu,Aθut+ρu⩽lu,Aθut+ρu=lAθ/2u,Aθ/2ut+ρu⩽ρ2ut+ρuVθ2+l22ρuVθ2⩽ρ2ut+ρuVθ2+C,where 0⩽τ⩽u (or u⩽τ⩽0) and 0⩽f′(τ)⩽l.
For the forcing term, we see easily(101)-g,Aθut+ρu=-ddtg,Aθu-ρg,Aθu,and since g∈L2(Ω), then(102)g,Aθu⩽g2+14uV2θ2.
Takingρ small enough, we then obtain that(103)1-αρλ1θ-k0ρδ⩾12,12α-3ρ2⩾α4.In combination with the above estimates, we can deduce that, for all t⩾t0,(104)12ddtuV2θ2+ut+ρuVθ2+ηtμ,V2θ2-2g,Aθu+C¯+β32uV2θ2+ut+ρuVθ2+ηtμ,V2θ2-2g,Aθu+C¯⩽C,where C¯=2g2 and β3=min⁡{ρ,α/2,δ/2}.
From (102), there exist constants C¯1, C¯2, such that(105)C¯1ztH22⩽12uV2θ2+ut+ρuVθ2+ηtμ,V2θ2-2g,Aθu+C¯⩽C¯2ztH22;therefore, applying Gronwall lemma, we have(106)ztH22⩽C¯2C¯1z0H22e-β3t+Cβ3C¯1.
Presuming thatz(0)H22⩽R, as t⩾t1=t1(B1H2)⩾t0, we get that(107)ztH2⩽μ1.
The proof is complete.


### 4.3. The Existence of Global Attractor inH2

According to the general theorem about the existence of global attractors of infinite-dimensional dynamical systems (see [6, 15]), we also need to test the asymptotic compactness of {S(t)}t⩾0 in H2.Similarly, we also gain the corresponding existence and uniqueness of solutions for (55) and (56); moreover, we can deduce that the solutions of (55) also form a semigroup. For the sake of convenience, we denote the solution operators of (55) and (56) by {S1(t)}t⩾0 and {S2(t,·)}t⩾0, respectively. Then, for every z0∈H2, we have(108)zt=Stz0=S1tz0+S2t,z0,∀t⩾0.After that, the necessary condition of asymptotic smoothness will be verified.At first, akin to the proof of Lemma7, the following results about the solution z1(t) of (55) in H2 can be obtained.Lemma 15.
Letf0 satisfy (48)–(50), g∈L2(Ω), and (7)–(9) hold. Then, for any ε1>0, there is a constant ϵ=ϵ(ε1,g) and t1⩾t0∗>t0, such that the solutions of (55) satisfy the following estimates:(109)S1tz0H22=12vϵV2θ2+vtϵVθ2+ζtϵμ,V2θ2⩽Qz0H2e-β4t+ε1,∀t⩾t1;here Q(·) is an increasing function on [0,∞) and β4 only depends on α, ρ, and δ.Proof.
Multiplying (55) by Aθvtϵ+ρAθvϵ, we get that(110)12ddtvϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2+ρvϵV2θ2+α-ρvtϵ+ρvϵVθ2-ρα-ρvϵ,vtϵ+ρvϵVθ+ζtϵ,ζstϵμ,V2θ+ρζtϵ,vϵμ,V2θ=-f0vϵ,Aθvtϵ+ρvϵ+g-gϵ,Aθvtϵ+ρvϵ.For the forcing term, we have(111)-g-gϵ,Aθvtϵ+ρvϵ=-ddtg-gϵ,Aθvϵ-ρg-gϵ,Aθvϵ,g-gϵ,Aθvϵ⩽g-gϵ2+14vϵV2θ2.
About the nonlinearity, similar to (100) and from (48), (50), and (70), as t⩾t0∗>t0, we see(112)-f0vϵ,Aθvtϵ+ρAθvϵ⩽α8vtϵ+ρvϵVθ2+2l12αvϵVθ2⩽α8vtϵ+ρvϵVθ2+Cϵ.Similarly,(113)12ddtvϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2-2g-gϵ,Aθvϵ+Cˇ+ρ2-k0ρ2δvϵV2θ2+α8vtϵ+ρvϵVθ2+δ4ζtϵμ,V2θ2-ρg-gϵ,Aθvϵ+ρCˇ⩽ρCˇ+Cϵ<Cϵ;here, Cˇ=2g-gϵ2<2ϵ2.
Then, substituting the above estimates into (110), this yields(114)12ddtvϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2-2g-gϵ,Aθvϵ+Cˇ+β42vϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2-2g-gϵ,Aθvϵ+Cˇ⩽Cϵ,where ρ/2-k0ρ2/δ⩾ρ/4, β4=min⁡{ρ/2,α/4,δ/2}.
Obviously, there exist constantsCˇ1, Cˇ2, such that(115)Cˇ1z1tH22⩽12vϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2-2g-gϵ,Aθvϵ+Cˇ⩽Cˇ2z1tH22.From Gronwall lemma, for all t⩾t1⩾t0∗, we obtain(116)z1tH22⩽Cˇ2Cˇ1z0H22e-β4t+Cϵβ4Cˇ1⩽Qz0H2e-β4t+ε1.
The proof is complete.Lemma 16.
Let the nonlinearityf satisfy (2)–(6) and (51)-(52), g∈L2(Ω), and (7)–(9) hold. Then, for any given T>0 and ϵ>0, there exists a position constant(117)N1=N1T,g,z0H2,δ,such that the solutions of (56) satisfy(118)S2T,z0H2+σ2=12wϵV2θ+σ2+wtϵVθ+σ2+ξTϵμ,V2θ+σ2⩽N1,where σ=min⁡{1,(n+2)θ-2n/n-2,2θ/3,θ-(n-2θ)γ/2}.Proof.
Multiplying (56) by Aθ+σ(wtϵ+ρwϵ), we obtain that(119)12ddtwϵV2θ+σ2+wtϵ+ρwϵVθ+σ2+ξtϵμ,V2θ+σ2+ρwϵV2θ+σ2+α-ρwtϵ+ρwVθ+σ2-ρα-ρwϵ,wtϵ+ρwϵVθ+σ+ξtϵ,ξstϵμ,V2θ+σ+ρξtϵ,wϵμ,V2θ+σ=-fu-f0vϵ,Aθ+σwtϵ+ρwϵ+gϵ,Aθ+σwtϵ+ρwϵ.
Similarly, choosingρ to be sufficiently small, we can obtain that(120)12ddtwϵV2θ+σ2+wtϵ+ρwϵVθ+σ2+ξtϵμ,V2θ+σ2+ρ2wϵV2θ+σ2+α4wtϵ+ρwVθ+σ2+δ4ξtϵμ,V2θ+σ2=-fu-f0vϵ,Aθ+σwtϵ+ρwϵ+gϵ,Aθ+σwtϵ+ρwϵ.
Clearly(121)fu-f0vϵ,Aθ+σwtϵ+ρwϵ=fu-fvϵ,Aθ+σwtϵ+ρwϵ+f1vϵ,Aθ+σwtϵ+ρwϵ.We deal with the terms in (121) one by one.
First, from (4), we obtain(122)-fu-fvϵ,Aθ+σwtϵ+ρwϵ⩽∫Ωf′uA3σ/4+θ/2u-f′vϵA3σ/4+θ/2vϵ·Aσ/4+θ/2wtϵ+ρwϵdx⩽C∫Ω1+u4/n-4+vϵ4/n-4A3σ/4+θ/2wϵAσ/4+θ/2wtϵ+ρwϵdx+2C∫Ω1+u4/n-4+vϵ4/n-4A3σ/4+θ/2vϵAσ/4+θ/2wtϵ+ρwϵdx.
Second, forθ∈[2n/n+2,n/2), we have(123)C∫Ω1+u4/n-4+vϵ4/n-4A3σ/4+θ/2wϵAσ/4+θ/2wtϵ+ρwϵdx⩽C∫Ω1+u4/n-4+vϵ4/n-4n/θdxθ/n∫ΩA3σ/4+θ/2wϵ2n/n-2θ-σ/2dxn-2θ-σ/2/2n·∫ΩAσ/4+θ/2wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽C1+uL4n/n-4θ4/n-4+vϵL4n/n-4θ4/n-4A3σ/4+θ/2wϵL2n/n-2θ-σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽C1+uL2n/n-4θ4/n-4+vϵL2n/n-4θ4/n-4A3σ/4+θ/2wϵL2n/n-2θ-σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽C1+Aθu4/n-4+Aθvϵ4/n-4A2θ+σ/2wϵ·Aθ+σ/2wtϵ+ρwϵ⩽Cμ1,ε1A2θ+σ/2wϵ2+α4Aθ+σ/2wtϵ+ρwϵ2,where 4n/(n-4)θ⩽2n/n-4θ, and we have used the embedding V2θ=D(Aθ)↪L2n/n-4θ↪L4n/(n-4)θ,Vθ-σ/2=D(A2θ-σ/4)↪L2n/n-2(θ-σ/2),Vσ/2=D(Aσ/4)↪L2n/n-σ.
Third, from (53), we obtain that(124)2C∫Ω1+u4/n-4+vϵ4/n-4A3σ/4+θ/2vϵAσ/4+θ/2wtϵ+ρwϵdx⩽2C∫Ω1+u4/n-4+vϵ4/n-4n/θ-σdxθ-σ/n∫ΩA3σ/4+θ/2vϵ2n/n-2θ-3σ/2dxn-2θ-3σ/2/2n·∫ΩAσ/4+θ/2wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽2C1+uL4n/n-4θ-σ4/n-4+vϵL4n/n-4θ-σ4/n-4A3σ/4+θ/2vϵL2n/n-2θ-3σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽2C1+uL2n/n-4θ4/n-4+vϵL2n/n-4θ4/n-4A3σ/4+θ/2vϵL2n/n-2θ-3σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽2C1+Aθu4/n-4+Aθvϵ4/n-4Aθvϵ·Aθ+σ/2wtϵ+ρwϵ⩽Cμ1,ε1+α4Aθ+σ/2wtϵ+ρwϵ2,where 4n/(n-4)(θ-σ)⩽2n/n-4θ, and we have used the embedding V2θ=D(Aθ)↪L2n/n-4θ↪L4n/(n-4)(θ-σ), Vθ-3σ/2=D(A2θ-3σ/4)↪L2n/n-2(θ-3σ/2), Vσ/2=D(Aσ/4)↪L2n/n-σ.
Finally, by virtue of (51), we get(125)-f1vϵ,Aθ+σwtϵ+ρwϵ=-∫Ωf1′vϵ·A3σ/4+θ/2vϵ·Aσ/4+θ/2wtϵ+ρwϵdx⩽C∫Ω1+vϵγA3σ/4+θ/2vϵAσ/4+θ/2wtϵ+ρwϵdx⩽C∫Ω1+vϵγn/θ-σdxθ-σ/n·∫ΩA3σ/4+θ/2vϵ2n/n-2θ-3σ/2dxn-2θ-3σ/2/2n·∫ΩAσ/4+θ/2wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽C1+vϵLnγ/θ-σγA3σ/4+θ/2vϵL2n/n-2θ-3σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽C1+vϵL2n/n-4θγA3σ/4+θ/2vϵL2n/n-2θ-3σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽C1+AθvϵγAθvϵAθ+σ/2wtϵ+ρwϵ⩽Cε1+α4Aθ+σ/2wtϵ+ρwϵ2,where we have employed the inequality σ⩽θ-n-4θγ/2,nγ/θ-σ⩽2n/n-4θ and used the embedding V2θ↪L2n/n-4θ↪Lnγ/θ-σ, Vσ/2=D(Aσ/4)↪L2n/n-σ, and Vθ-3σ/2=D(A2θ-3σ/4)↪L2n/n-2(θ-3σ/2).
At the same time, due to0<σ<1, we have(126)gϵ,Aθ+σwtϵ+ρwϵ=ddtgϵ,Aθ+σwϵ+ρgϵ,Aθ+σwϵ,gϵ,Aθ+σwϵ⩽gϵV1Aθ+σ-1/2wϵ⩽12λ11-σgϵV12+λ11-σ2Aθ+σ-1/2wϵ2⩽12λ11-σgϵV12+12A2θ+σ/2wϵ2.
Combining the above estimates, we can deduce that(127)12ddtwϵV2θ+σ2+wtϵ+ρwϵVθ+σ2+ξtϵμ,V2θ+σ2-2gϵ,Aθ+σwϵ+C˘⩽β52wϵV2θ+σ2+wtϵ+ρwϵVθ+σ2+ξtϵμ,V2θ+σ2-2gϵ,Aθ+σwϵ+C˘+C,where C˘=1/λ11-σgϵV12, β5=max⁡{2Cμ1,ε1-ρ,α,δ/2}, and the constant C only depends on gϵ, z0H2.
Applying the Gronwall inequality and integrating over[0,t], we deduce that(128)12wϵtV2θ+σ2+wtϵt+ρwϵtVθ+σ2+ξtϵsμ,V2θ+σ2-2gϵx,Aθ+σwϵt+C˘⩽12wϵ0V2θ+σ2+wtϵ0+ρwϵ0Vθ+σ2+ξ0ϵsμ,V2θ+σ2-2gϵx,Aθ+σwϵ0+C˘eβ5t+Cβ5eβ5t-1⩽C˘2+Cβ5eβ5t.
So, there exist constantsC˘1, C˘2, such that(129)C˘1z2tH2+σ2⩽12wϵtV2θ+σ2+wtϵt+ρwϵtVθ+σ2+ξtϵsμ,V2θ+σ2-2gϵx,Aθ+σwϵt+C˘⩽C˘2z2tH2+σ2;then(130)z2tH2+σ2⩽1C˘1C˘2+Cβ5eβ5t.That is,(131)S2T,z0H2+σ2=12wϵT2θ+σ2+wtTVθ+σ2+ξTϵμ,V2θ+σ2⩽N1,where N1=N1(T,g,z0H2,α,δ).
The proof is complete.LetB1∗ be the bounded absorbing set obtained from Theorem 14, and(132)KTϵ∗≔ΠS2T,B1∗.Thanks to Lemma 2, we can obtain that KTϵ∗ is relatively compact in Lμ2(R+;V2θ). Thus, using the compact embedding V2θ+σ↪V2θ once again, we can get the following.Lemma 17.
LetS2(t,·) be the corresponding solution operator of (56). Under the assumption of Lemma 16, for any T>0, S2(T,B1∗) is relatively compact in H2.By applying the classical theory (e.g., see [6, 20]), from Lemma 3, Theorem 14, and Lemmas 15 and 17, we can gain the main result.Theorem 18 (global attractor inH2).
Let the nonlinearityf satisfy (2)–(6), g∈L2(Ω), (7)–(9) hold, and {S(t)}t⩾0 be the solution semigroup of (17)-(18) in H2. Then {S(t)}t⩾0 has a global attractor A1 in H2; namely, A1 is compact and invariant in H2 and attracts every bounded subset of H2 with respect to the H2-norm.


## 4.1. The Existence and Uniqueness of Strong Solution

First of all, we define the concept of strong solution for dynamical systems (17)-(18).Definition 12.
DefineI=[0,T],∀T>0. Let g∈L2(Ω) and z0∈H2. A ternary form z=(u,ut,ηt) is said to be the strong solution of problem (17)-(18) in the time interval I with initial data z(0)=z0, if z satisfies (27), and(94)u∈CI;V2θ,ut∈L2I;V2θ,ηt∈CI;Lμ2R+;V2θ;ηtt+ηst∈L∞I;Lμ2R+;Vθ∩L2I;Lμ2R+;V2θ.Hereafter, similar to Theorem5, we can obtain the following.Theorem 13 (existence and uniqueness of strong solutions).
Assume thatf satisfy (2) and (4), g∈L2(Ω), and (7)–(9) hold. Then, for any given z0=(u0,ut0,η0)∈H2, and any T>0, there exists a unique solution z(t)=(u,ut,ηt)∈H2 for problem (17)-(18), in the sense that(95)z∈L∞0,T;H2∩C0,T;H2.By Theorem13, we can define the strong solution operator as follows: (96)St:H2⟶H2,z0=u0,ut0,η0⟶ut,utt,ηt=Stz0.Evidently, {S(t)}t⩾0 is a solution semigroup.


## 4.2. The Existence of Bounded Absorbing Set

At first, we will prove the existence of bounded absorbing set inH2.Theorem 14 (bounded absorbing inH2).
Let the nonlinearityf satisfy (2), (5), and (6), g∈L2(Ω), (7)–(9) hold, and B1 be bounded subset of H2. If z(t) is a solution of (14) and (17) with initial data z0∈B1, then there is a positive constant μ1 such that, for any bounded subset B⊂H2, there exists t1=t1(BH2) such that(97)Stz0H2⩽μ1,∀t⩾t1,z0∈B.Proof.
Taking theL2 inner product of (17) with Aθut+ρAθu, we get that(98)12ddtuV2θ2+ut+ρuVθ2+ηtμ,V2θ2+ρuV2θ2+α-ρut+ρuVθ2-ρα-ρu,ut+ρuVθ+ηt,ηstμ,V2θ+ρηt,uμ,V2θ+fu,Aθut+ρu=g,Aθut+ρu.Similar to (35), we have(99)12ddtuV2θ2+ut+ρuVθ2+ηtμ,V2θ2+ρ1-αρλ1θ-k0ρδuV2θ2+12α-ρut+ρuVθ2+δ4ηtμ,V2θ2+fu,Aθut+ρu⩽g,Aθut+ρu.
About the nonlinearity, from (5), (6), and (44), these yield(100)-fu,Aθut+ρAθu=-f′τu,Aθut+ρu⩽lu,Aθut+ρu=lAθ/2u,Aθ/2ut+ρu⩽ρ2ut+ρuVθ2+l22ρuVθ2⩽ρ2ut+ρuVθ2+C,where 0⩽τ⩽u (or u⩽τ⩽0) and 0⩽f′(τ)⩽l.
For the forcing term, we see easily(101)-g,Aθut+ρu=-ddtg,Aθu-ρg,Aθu,and since g∈L2(Ω), then(102)g,Aθu⩽g2+14uV2θ2.
Takingρ small enough, we then obtain that(103)1-αρλ1θ-k0ρδ⩾12,12α-3ρ2⩾α4.In combination with the above estimates, we can deduce that, for all t⩾t0,(104)12ddtuV2θ2+ut+ρuVθ2+ηtμ,V2θ2-2g,Aθu+C¯+β32uV2θ2+ut+ρuVθ2+ηtμ,V2θ2-2g,Aθu+C¯⩽C,where C¯=2g2 and β3=min⁡{ρ,α/2,δ/2}.
From (102), there exist constants C¯1, C¯2, such that(105)C¯1ztH22⩽12uV2θ2+ut+ρuVθ2+ηtμ,V2θ2-2g,Aθu+C¯⩽C¯2ztH22;therefore, applying Gronwall lemma, we have(106)ztH22⩽C¯2C¯1z0H22e-β3t+Cβ3C¯1.
Presuming thatz(0)H22⩽R, as t⩾t1=t1(B1H2)⩾t0, we get that(107)ztH2⩽μ1.
The proof is complete.


## 4.3. The Existence of Global Attractor inH2

According to the general theorem about the existence of global attractors of infinite-dimensional dynamical systems (see [6, 15]), we also need to test the asymptotic compactness of {S(t)}t⩾0 in H2.Similarly, we also gain the corresponding existence and uniqueness of solutions for (55) and (56); moreover, we can deduce that the solutions of (55) also form a semigroup. For the sake of convenience, we denote the solution operators of (55) and (56) by {S1(t)}t⩾0 and {S2(t,·)}t⩾0, respectively. Then, for every z0∈H2, we have(108)zt=Stz0=S1tz0+S2t,z0,∀t⩾0.After that, the necessary condition of asymptotic smoothness will be verified.At first, akin to the proof of Lemma7, the following results about the solution z1(t) of (55) in H2 can be obtained.Lemma 15.
Letf0 satisfy (48)–(50), g∈L2(Ω), and (7)–(9) hold. Then, for any ε1>0, there is a constant ϵ=ϵ(ε1,g) and t1⩾t0∗>t0, such that the solutions of (55) satisfy the following estimates:(109)S1tz0H22=12vϵV2θ2+vtϵVθ2+ζtϵμ,V2θ2⩽Qz0H2e-β4t+ε1,∀t⩾t1;here Q(·) is an increasing function on [0,∞) and β4 only depends on α, ρ, and δ.Proof.
Multiplying (55) by Aθvtϵ+ρAθvϵ, we get that(110)12ddtvϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2+ρvϵV2θ2+α-ρvtϵ+ρvϵVθ2-ρα-ρvϵ,vtϵ+ρvϵVθ+ζtϵ,ζstϵμ,V2θ+ρζtϵ,vϵμ,V2θ=-f0vϵ,Aθvtϵ+ρvϵ+g-gϵ,Aθvtϵ+ρvϵ.For the forcing term, we have(111)-g-gϵ,Aθvtϵ+ρvϵ=-ddtg-gϵ,Aθvϵ-ρg-gϵ,Aθvϵ,g-gϵ,Aθvϵ⩽g-gϵ2+14vϵV2θ2.
About the nonlinearity, similar to (100) and from (48), (50), and (70), as t⩾t0∗>t0, we see(112)-f0vϵ,Aθvtϵ+ρAθvϵ⩽α8vtϵ+ρvϵVθ2+2l12αvϵVθ2⩽α8vtϵ+ρvϵVθ2+Cϵ.Similarly,(113)12ddtvϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2-2g-gϵ,Aθvϵ+Cˇ+ρ2-k0ρ2δvϵV2θ2+α8vtϵ+ρvϵVθ2+δ4ζtϵμ,V2θ2-ρg-gϵ,Aθvϵ+ρCˇ⩽ρCˇ+Cϵ<Cϵ;here, Cˇ=2g-gϵ2<2ϵ2.
Then, substituting the above estimates into (110), this yields(114)12ddtvϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2-2g-gϵ,Aθvϵ+Cˇ+β42vϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2-2g-gϵ,Aθvϵ+Cˇ⩽Cϵ,where ρ/2-k0ρ2/δ⩾ρ/4, β4=min⁡{ρ/2,α/4,δ/2}.
Obviously, there exist constantsCˇ1, Cˇ2, such that(115)Cˇ1z1tH22⩽12vϵV2θ2+vtϵ+ρvϵVθ2+ζtϵμ,V2θ2-2g-gϵ,Aθvϵ+Cˇ⩽Cˇ2z1tH22.From Gronwall lemma, for all t⩾t1⩾t0∗, we obtain(116)z1tH22⩽Cˇ2Cˇ1z0H22e-β4t+Cϵβ4Cˇ1⩽Qz0H2e-β4t+ε1.
The proof is complete.Lemma 16.
Let the nonlinearityf satisfy (2)–(6) and (51)-(52), g∈L2(Ω), and (7)–(9) hold. Then, for any given T>0 and ϵ>0, there exists a position constant(117)N1=N1T,g,z0H2,δ,such that the solutions of (56) satisfy(118)S2T,z0H2+σ2=12wϵV2θ+σ2+wtϵVθ+σ2+ξTϵμ,V2θ+σ2⩽N1,where σ=min⁡{1,(n+2)θ-2n/n-2,2θ/3,θ-(n-2θ)γ/2}.Proof.
Multiplying (56) by Aθ+σ(wtϵ+ρwϵ), we obtain that(119)12ddtwϵV2θ+σ2+wtϵ+ρwϵVθ+σ2+ξtϵμ,V2θ+σ2+ρwϵV2θ+σ2+α-ρwtϵ+ρwVθ+σ2-ρα-ρwϵ,wtϵ+ρwϵVθ+σ+ξtϵ,ξstϵμ,V2θ+σ+ρξtϵ,wϵμ,V2θ+σ=-fu-f0vϵ,Aθ+σwtϵ+ρwϵ+gϵ,Aθ+σwtϵ+ρwϵ.
Similarly, choosingρ to be sufficiently small, we can obtain that(120)12ddtwϵV2θ+σ2+wtϵ+ρwϵVθ+σ2+ξtϵμ,V2θ+σ2+ρ2wϵV2θ+σ2+α4wtϵ+ρwVθ+σ2+δ4ξtϵμ,V2θ+σ2=-fu-f0vϵ,Aθ+σwtϵ+ρwϵ+gϵ,Aθ+σwtϵ+ρwϵ.
Clearly(121)fu-f0vϵ,Aθ+σwtϵ+ρwϵ=fu-fvϵ,Aθ+σwtϵ+ρwϵ+f1vϵ,Aθ+σwtϵ+ρwϵ.We deal with the terms in (121) one by one.
First, from (4), we obtain(122)-fu-fvϵ,Aθ+σwtϵ+ρwϵ⩽∫Ωf′uA3σ/4+θ/2u-f′vϵA3σ/4+θ/2vϵ·Aσ/4+θ/2wtϵ+ρwϵdx⩽C∫Ω1+u4/n-4+vϵ4/n-4A3σ/4+θ/2wϵAσ/4+θ/2wtϵ+ρwϵdx+2C∫Ω1+u4/n-4+vϵ4/n-4A3σ/4+θ/2vϵAσ/4+θ/2wtϵ+ρwϵdx.
Second, forθ∈[2n/n+2,n/2), we have(123)C∫Ω1+u4/n-4+vϵ4/n-4A3σ/4+θ/2wϵAσ/4+θ/2wtϵ+ρwϵdx⩽C∫Ω1+u4/n-4+vϵ4/n-4n/θdxθ/n∫ΩA3σ/4+θ/2wϵ2n/n-2θ-σ/2dxn-2θ-σ/2/2n·∫ΩAσ/4+θ/2wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽C1+uL4n/n-4θ4/n-4+vϵL4n/n-4θ4/n-4A3σ/4+θ/2wϵL2n/n-2θ-σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽C1+uL2n/n-4θ4/n-4+vϵL2n/n-4θ4/n-4A3σ/4+θ/2wϵL2n/n-2θ-σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽C1+Aθu4/n-4+Aθvϵ4/n-4A2θ+σ/2wϵ·Aθ+σ/2wtϵ+ρwϵ⩽Cμ1,ε1A2θ+σ/2wϵ2+α4Aθ+σ/2wtϵ+ρwϵ2,where 4n/(n-4)θ⩽2n/n-4θ, and we have used the embedding V2θ=D(Aθ)↪L2n/n-4θ↪L4n/(n-4)θ,Vθ-σ/2=D(A2θ-σ/4)↪L2n/n-2(θ-σ/2),Vσ/2=D(Aσ/4)↪L2n/n-σ.
Third, from (53), we obtain that(124)2C∫Ω1+u4/n-4+vϵ4/n-4A3σ/4+θ/2vϵAσ/4+θ/2wtϵ+ρwϵdx⩽2C∫Ω1+u4/n-4+vϵ4/n-4n/θ-σdxθ-σ/n∫ΩA3σ/4+θ/2vϵ2n/n-2θ-3σ/2dxn-2θ-3σ/2/2n·∫ΩAσ/4+θ/2wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽2C1+uL4n/n-4θ-σ4/n-4+vϵL4n/n-4θ-σ4/n-4A3σ/4+θ/2vϵL2n/n-2θ-3σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽2C1+uL2n/n-4θ4/n-4+vϵL2n/n-4θ4/n-4A3σ/4+θ/2vϵL2n/n-2θ-3σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽2C1+Aθu4/n-4+Aθvϵ4/n-4Aθvϵ·Aθ+σ/2wtϵ+ρwϵ⩽Cμ1,ε1+α4Aθ+σ/2wtϵ+ρwϵ2,where 4n/(n-4)(θ-σ)⩽2n/n-4θ, and we have used the embedding V2θ=D(Aθ)↪L2n/n-4θ↪L4n/(n-4)(θ-σ), Vθ-3σ/2=D(A2θ-3σ/4)↪L2n/n-2(θ-3σ/2), Vσ/2=D(Aσ/4)↪L2n/n-σ.
Finally, by virtue of (51), we get(125)-f1vϵ,Aθ+σwtϵ+ρwϵ=-∫Ωf1′vϵ·A3σ/4+θ/2vϵ·Aσ/4+θ/2wtϵ+ρwϵdx⩽C∫Ω1+vϵγA3σ/4+θ/2vϵAσ/4+θ/2wtϵ+ρwϵdx⩽C∫Ω1+vϵγn/θ-σdxθ-σ/n·∫ΩA3σ/4+θ/2vϵ2n/n-2θ-3σ/2dxn-2θ-3σ/2/2n·∫ΩAσ/4+θ/2wtϵ+ρwϵ2n/n-σdxn-σ/2n⩽C1+vϵLnγ/θ-σγA3σ/4+θ/2vϵL2n/n-2θ-3σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽C1+vϵL2n/n-4θγA3σ/4+θ/2vϵL2n/n-2θ-3σ/2Aσ/4+θ/2wtϵ+ρwϵL2n/n-σ⩽C1+AθvϵγAθvϵAθ+σ/2wtϵ+ρwϵ⩽Cε1+α4Aθ+σ/2wtϵ+ρwϵ2,where we have employed the inequality σ⩽θ-n-4θγ/2,nγ/θ-σ⩽2n/n-4θ and used the embedding V2θ↪L2n/n-4θ↪Lnγ/θ-σ, Vσ/2=D(Aσ/4)↪L2n/n-σ, and Vθ-3σ/2=D(A2θ-3σ/4)↪L2n/n-2(θ-3σ/2).
At the same time, due to0<σ<1, we have(126)gϵ,Aθ+σwtϵ+ρwϵ=ddtgϵ,Aθ+σwϵ+ρgϵ,Aθ+σwϵ,gϵ,Aθ+σwϵ⩽gϵV1Aθ+σ-1/2wϵ⩽12λ11-σgϵV12+λ11-σ2Aθ+σ-1/2wϵ2⩽12λ11-σgϵV12+12A2θ+σ/2wϵ2.
Combining the above estimates, we can deduce that(127)12ddtwϵV2θ+σ2+wtϵ+ρwϵVθ+σ2+ξtϵμ,V2θ+σ2-2gϵ,Aθ+σwϵ+C˘⩽β52wϵV2θ+σ2+wtϵ+ρwϵVθ+σ2+ξtϵμ,V2θ+σ2-2gϵ,Aθ+σwϵ+C˘+C,where C˘=1/λ11-σgϵV12, β5=max⁡{2Cμ1,ε1-ρ,α,δ/2}, and the constant C only depends on gϵ, z0H2.
Applying the Gronwall inequality and integrating over[0,t], we deduce that(128)12wϵtV2θ+σ2+wtϵt+ρwϵtVθ+σ2+ξtϵsμ,V2θ+σ2-2gϵx,Aθ+σwϵt+C˘⩽12wϵ0V2θ+σ2+wtϵ0+ρwϵ0Vθ+σ2+ξ0ϵsμ,V2θ+σ2-2gϵx,Aθ+σwϵ0+C˘eβ5t+Cβ5eβ5t-1⩽C˘2+Cβ5eβ5t.
So, there exist constantsC˘1, C˘2, such that(129)C˘1z2tH2+σ2⩽12wϵtV2θ+σ2+wtϵt+ρwϵtVθ+σ2+ξtϵsμ,V2θ+σ2-2gϵx,Aθ+σwϵt+C˘⩽C˘2z2tH2+σ2;then(130)z2tH2+σ2⩽1C˘1C˘2+Cβ5eβ5t.That is,(131)S2T,z0H2+σ2=12wϵT2θ+σ2+wtTVθ+σ2+ξTϵμ,V2θ+σ2⩽N1,where N1=N1(T,g,z0H2,α,δ).
The proof is complete.LetB1∗ be the bounded absorbing set obtained from Theorem 14, and(132)KTϵ∗≔ΠS2T,B1∗.Thanks to Lemma 2, we can obtain that KTϵ∗ is relatively compact in Lμ2(R+;V2θ). Thus, using the compact embedding V2θ+σ↪V2θ once again, we can get the following.Lemma 17.
LetS2(t,·) be the corresponding solution operator of (56). Under the assumption of Lemma 16, for any T>0, S2(T,B1∗) is relatively compact in H2.By applying the classical theory (e.g., see [6, 20]), from Lemma 3, Theorem 14, and Lemmas 15 and 17, we can gain the main result.Theorem 18 (global attractor inH2).
Let the nonlinearityf satisfy (2)–(6), g∈L2(Ω), (7)–(9) hold, and {S(t)}t⩾0 be the solution semigroup of (17)-(18) in H2. Then {S(t)}t⩾0 has a global attractor A1 in H2; namely, A1 is compact and invariant in H2 and attracts every bounded subset of H2 with respect to the H2-norm.



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*Source: 1410408-2017-06-14.xml*