# Global Attractor for Second-Order Nonlinear Evolution Differential Inclusions

Authors: Guangwang Su; Funing Lin
Journal: Complexity
 (2021)

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## Abstract

In this paper, we address the model of global attractor formulated in the form of evolution differential inclusions with second order in Banach spaces. Firstly, based on the fixed point theorem, the existence result of mild solutions is deduced. Then, by implementing the measure of noncompactness, the existence of global attractor associated withm-semiflow is validated. Finally, a concrete application of the main result is demonstrated to enhance the practical signification.


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## Body

## 1. Introduction

Consider a Banach spaceE,⋅. We attempt to handle the following problem:(1)u″t−Aut∈Ft,ut,t>0,u0=u0,u′0=v0,where A is a closed, linear, and densely defined operator from E to E that generates a strongly continuous cosine family Ctt≥0; F:0,+∞×E⟶PE is measurable and upper semicontinuous in terms of both first-order and second-order variables, and it also satisfies the natural growth condition.The nonlinear evolution differential inclusions in Banach spaces have drawn more and more attention of a large number of experts and scholars during the past decades, see [1]. At present, there are substantial outcomes focusing on the existence of solutions for differential equations with second or higher order [2, 3]. For instance, semilinear evolution differential inclusions were explored by Cardinali and Rubbioni in [4]. Liu, Li, and Migórski et al. discussed the problems of one and second evolution differential variational inequalities, and they attained several existence results about the mild solution, see [5, 6] and references therein.It is known that global attractor has played an important role in studying the asymptotic feature of solutions for diverse types of differential systems. Melnik, Valero, and Tran et al. were devoted to researching on global attractor of multivalued semiflows and differential inclusions, see [7–13] and references therein. In [10], the authors studied the global attractor of the following functional differential inclusions:(2)u′t−Aut∈Fut,ut,t>0,us=φs,s∈−h,0,where u represents the state function and ut describes the history of the state function, that is, uts=ut+s for s∈−h,0. They proved the existences of solutions of system (2) and global attractor by applying noncompactness measure. Global attractor is a gradual state of solution set, and it is also an important research content of a dynamic system. Compared with the first-order differential inclusions, the cosine family generated by operator A is more complex and more meaningful. There are relatively few studies on this type of second-order differential inclusions, so it is more important to study the attractor of the second-order differential inclusions.The first aim of the work is to explore the existence of solutions in terms of system (1) in infinite dimensional spaces. Secondly, we are devoted to show the existence of a global attractor for system (1). To our knowledge, there is currently a research gap in the aforementioned problem. However, it is not easy to establish a connection between mild solution and m-semiflows for second-order differential inclusions.The work is structured as follows. Section2 presents several relevant definitions involving noncompactness measures and m-semiflows with its global attractor. In Section 3, several sufficient conditions with respect to the mild solutions are invoked. In Section 4, the presence of a global attractor of system (1) is examined in Banach spaces. Finally, a concrete application is presented to emphasize the magnitude of the main results.


## 2. Preliminaries

This section recalls some fundamental definitions and properties of nonlinear analysis, for more details, see [14]. The symbol ℝ (resp., ℝ+) represents the set of real (resp., positive real) numbers. The strong and weak convergences of uk to u are denoted by uk⟶u and uk⇀u, respectively.Consider a Banach spaceE whose dual space is denoted by E∗. Let I=0,T with 0<T<∞ and let L1I,E be the Banach space of all (Bochner) integrable functions on I endowed with the norm f=∫IfsEds. The notations mentioned as follows will be employed:(3)PfcE≔Q⊂E:Qis nonempty,closedconvex,PωkcE≔Q⊂E:Qis nonempty,weakly compactconvex.Definition 1.
Suppose thatE1 and E2 are Banach spaces. A multivalued mapping F:E1⟶PE2 is referred to as follows:(1)
Upper semicontinuous (u.s.c), provided that, for an arbitrary closed setQ in E2,(4)F−1Q=e∈E1:Fe∩Q≠∅,is closed inE1.(2)
Weakly upper semicontinuous (weakly u.s.c), if for arbitrary weakly closed setQ in E2,(5)F−1Q=e∈E1:Fe∩Q≠∅,is closed inE1.(3)
Closed, if its graphΓF=e1,e2:e2∈Fe1 is closed in E1×E2.(4)
Compact, ifFD is a relatively compact subset of E2 for every bounded set D.
A weakly upper semicontinuous mapping with a weakly compact convex value has good properties. The following conclusion can be found in [15].Lemma 1.
Suppose thatE1 and E2 are Banach spaces and F:E1⟶PwkcE2 is a mapping. Then, F is weakly u.s.c if only and if for any sequence un of E1 satisfying un⟶u0∈E1 and vn∈Fun, it implies vn⇀v0∈Fu0 up to a subsequence.
In the subsequent arguments, we consider the presence of mild solutions about system (1) from the viewpoint of evolution differential inclusions with second order. We only need to deal with the following differential equation:(6)u″t=Aut+ft,t>0,u0=u0,u′0=v0,where ft∈Ft,ut.Definition 2.
Assume thatLE is the space of all bounded linear operators defined from E to E∗. A mapping C:ℝ⟶ℒE is called a strongly continuous cosine family, provided that C0=I0 (I0 is an identity operator) and Cr1+r2+Cr1−r2=2Cr1Cr2 for every r1,r2∈ℝ.
Given a strongly continuous cosine familyCt with t∈ℝ. Then, its generator is an operator A:E⟶E formulated as(7)Au=d2dt2Ctu|t=0,∀u∈DA,where the domain DA=e∈E:Cte∈C2ℝ,E. By the definition of generator A, it is known that A is closed, linear, and densely defined over E, see [2]. The sine operator S:ℝ⟶ℒE, which is closely relevant to the cosine operator C, is expressed by(8)Stu=∫0tCsuds,∀t∈ℝ,u∈E.Definition 3.
A mappingu∈CI,E is referred to as a mild solution of system (6), provided that there is an f∈L1I,E with ft∈Ft,ut and(9)ut=Ctu0+Stv0+∫0tSt−sfsds,∀t∈I.
Given an arbitrary bounded setB in E, the Hausdorff measure of noncompactness (MNC, for short) is given by(10)χB=infε>0:Bhas a finiteε−net.
LetΩ⊂L1I,E. We say that Ω is integrably bounded, if any element f of Ω satisfies that ft≤gt for a.e. t∈I with some g∈L1I,ℝ+. An integrably bounded Ω is said to be semicompact, provided that Ωt is relatively compact for a.e. t∈I.
The norm of product spacesE×E is represented by(11)u,v=maxuE,vE,∀u,v∈E×E.
It is straightforward to see that, for any bounded setB1×B2⊂E×E,(12)χ˜B1×B2=maxχB1,χB2,where χ˜ is measure of noncompactness on E×E.Definition 4.
Presume thatχ is a Hausdorff measure of noncompactness and 0≤κ<1. A multivalued mapping F:X⟶PkE is referred to as κ-condensing, if(13)χFB≤κ⋅χB,∀B⊂E.
The open (resp., closed) ball with centrum 0 and radiusr>0 is denoted by Br0 (resp., Br0¯). The fixed point theorem used in our main results is introduced in [16] as follows.Lemma 2.
Define two operatorsA:Br0¯⟶X and B:Br0¯⟶PkcE by coupling with the following properties:(1)
A is a contraction mapping with a single value and a coefficient λ<1/2(2)
B is compact and u.s.c
Then, each of the following conclusions holds:(i)
There exists an elementω∈Br0¯∖Br0 such that ρω∈Aω+Bω for some ρ>1(ii)
The operator inclusionu∈Au+Bu admits a solution in Br0¯
Next, we introduce related theories involving global attractors andm-semiflow, see [8, 10, 11]. Suppose that Γ is a nontrivial subgroup of the additive group of ℝ and Γ+=Γ∩ℝ+.Definition 5.
A mappingΦ:Γ×E⟶PE is said to be a multivalued flow (m-flow, in short) if the following conditions are guaranteed:(1)
Φ0,u=u for every u∈E(2)
Φt+s,u⊂Φt,Φs,u for any pair t,s∈Γ and u∈E with Φτ,B=∪u∈BΦτ,uRemark 1.
A mappingΦ:Γ+×E⟶PE is referred to as an m-semiflow, if conditions (1) and (2) are satisfied for any t1,t1∈Γ+.
A mappingΦ is said to be a strict m-semiflow if for every t,s∈Γ+, u∈E, we have Φt+s,u=Φt,Φs,u. A map u⋅:Γ+⟶E is called a trajectory of m-semiflow with respect to the initial condition u0=u0, if ut+s∈Φt,us for all t,s∈Γ+.Definition 6.
LetΦ be an m-semiflow. A bounded set B0⊂E is referred to as an absorbing set for Φ, provided that, for every bounded B1⊂E, γTB1+⊂B0 holds for some T=TB1≥0, where γTB1+=∪t≥TB1Φt,B1.
For a givene∈E and B1,B2⊂E, we define de,Bi=infx∈Bide,x. The metric excess of B1 over B2 is given as ηB1,B2=supe∈B1de,B2. It is well known that the Hausdorff distance between B1 and B2 can be represented in the following way:(14)dHB1,B2=maxηB1,B2,ηB2,B1.Definition 7.
LetΦ be an m-semiflow. A set ℜ is said to be a global attractor of Φ, provided that the following conditions are satisfied:(1)
ℜ attracts any bounded set B in E, namely, ηΦt,B,ℜ⟶0 as t⟶+∞(2)
ℜ is negatively semi-invariant, namely, ℜ⊂Φt,ℜ, for all t∈Γ+
We now give an existence result about a global attractor for a givenm-semiflow Φ, see [11].Lemma 3.
Assume that a mappingΦ:Γ+×E⟶PE satisfies the following conditions:(i)
For everyt∈Γ+, u⟶Φt,u is u.s.c. with closed values(ii)
Φ has an absorbing set(iii)
Φ is asymptotically upper semicompact, i.e., for every bounded B⊂E with bounded γTB+ for some positive value TB, any sequence ξn∈E:ξn∈Φtn,B is relatively compact as tn⟶+∞
Then,Φ admits a compact global attractor ℜ in E. In particular, ℜ is invariant if it is a strict m-semiflow, i.e., ℜ=Φt,ℜ,∀t∈Γ+.


## 3. The Existence Result of Mild Solutions

This section is based on the idea of [3] for nonlinear evolution hemivariational inequalities with second order, which is adapted for evolution differential inclusions in our context. We will handle problem (6) under the following basic assumptions:(A) The operatorA:DA⊂E⟶E is a generator of a cosine operator Ctt∈ℝ, and Ct,St≤MAewt for some MA≥1 and w≥0. Moreover, St is compact for arbitrary t∈I.(F) The multivalued mappingF:I×E⟶PkcE is u.s.c., and there are a∈L2I,ℝ+ and b>0 with(15)Ft,u=supξ:ξ∈Ft,u≤at+bufor everyt,u∈I×E.By combining the uniform boundedness principle with the assumptionA, it is inferred that Ct (resp., St) is uniformly bounded on I subject to some upper-bound MC (resp., MS). We set M=maxMC,MS.In the sequel, let a multivalued operatorPF:CI,E⟶PL2I,E be(16)PFu=f∈L2I,E:ft∈Ft,utfora.e.t∈I.It is well known thatFt,u has a measurable selection for every u∈CI,E and a.e. t∈I under the assumption F, which means that PF is nonempty and well-defined.We also construct a multivalued operatorB:CI,E⟶CI,E by(17)Bu=ϕ∈CI,E:ϕt=∫0tSt−sfsds,f∈PFu.SetW:L2I,E⟶CI,E by(18)Wft=∫0tSt−sfsds.Lemma 4.
Assume thatA and F hold. Then, for an arbitrary sequence fnn=1∞⊂L2I,E with semicompactness, the sequence Wfnn=1∞ is relatively compact. In particular, if fn⇀f0, then Wfn⟶Wf0.Proof.
It is shown by analogizing the proof of Theorem2 of [4].
Aiming to validate the presence of solutions for system (1), we now clarify a property of the multivalued operator B, which is inspired by Lemma 5 of [3].Lemma 5.
Assume thatA and F hold. Then, the multivalued mapping B:CI,E⟶CI,E is compact and u.s.c. with convex and compact values.Proof.
Note that the operatorB has convex values for any u∈CI,E due to the convexity of PF. The verification process will be divided into four steps as follows:Step 1:B is a bounded operator on CI,E. Given u∈Br0 and ϕ∈Bu, from F and Hölder inequality, it follows that(19)ϕtE≤∫0tSt−sfsEds≤M∫0tas+busEds≤MaL2I,ℝ+T+brT.Hence,BBr0 is a bounded subset of CI,E, i.e., the operator B is a bounded operator.Step 2:Bu:u∈Br0 is equicontinuous. Presume that 0<t1<t2≤T, δ>0 is sufficiently small, then(20)ϕt2−ϕt1E=∫0t2St2−sfsds−∫0t1St1−sfsdsE≤∫0t1St2−s−St1−sfsds+∫t1t2‖St2−sfs‖ds≤∫0t1St2−s−St1−sas+brds+M∫t1t2as+brds≤sups∈0,t1−δSt2−s−St1−saL2I,ℝ+T+brT+MaL2I,ℝ+2δ+t2−t1+br2δ+t2−t1.From the continuity of the sine operatorSt, we get easily that the above inequality is independent of u and it converges to 0 as t2⟶t1. This implies the equicontinuity of Bu:u∈Br0 in CI,E.Step 3:Πt=ϕt:ϕ∈BBr0 is relatively compact in E for a.e. t∈I. Firstly, the compactness of Π0=0 is trivially checked. Then, fix t∈0,T. For any u∈Br0 and ϕ∈BBr0, there exists f∈PFu with(21)ϕt=∫0tSt−sfsds.For everyε∈0,t, we define(22)ϕεt=∫0t−εSt−sfsds=Sε∫0t−εSt−s−εfsds.It can be seen that the compactness ofS⋅ with the boundedness of ∫0t−εSt−s−εfsds together assures that Πεt=ϕεt:ϕ∈BBr0 is relatively compact in E. Furthermore, we have(23)ϕt−ϕεtE≤M∫t−εtas+busEds≤MaL2I,ℝ+ε+brε.Thereby,Πtt>0 is totally bounded, and thus it is relatively compact in E.To this end, by applying Ascoli–Arzelà theorem and taking account of Steps 1–2, it can be verified that the mappingB is compact.Step 4:B has a closed graph. In fact, let un,ϕn⊂CI,E,un⟶u¯,ϕn∈W°PFun, and ϕn⟶ϕ¯. Choose any sequence fn⊂L2I,E such that fn∈PFun,ϕn=Wfn. From F, we conclude that the sequence fn is integrable bounded. Hence, fn is weakly compact in L2I,E. It is rational to presume that fn⇀f¯. From Lemma 4, we have(24)ϕn=Wfn⟶Wf¯=ϕ¯.
Besides,f¯∈PFu¯ due to Lemma 1. Therefore, B has a closed graph.Theorem 1.
Presume thatA and F hold. Then, there is at least one mild solution related to system (6) for each pair of initial values u0,v0∈E.Proof.
Construct a multivalued mappingℱ:CI,E⟶CI,E by(25)ℱu=φ∈CI,E:φt=Ctu0+Stv0+∫0tSt−sfsds,f∈PFu.
For convenience, we writeℱ=A+B, where B is given by (17) and A is defined as(26)Au=Ctu0+Stv0,t∈I.
Obviously,u is a fixed point of ℱ if only and if it is a mild solution of system (6). According to Lemma 2, we need to prove that (i) of Lemma 2 is not satisfied.
In fact, suppose thatρu∈Au+Bu with ρ>1 and there exists η∈PFu with(27)ρut=Ctu0+Stv0+∫0tSt−sηsds.
This yields that(28)utE≤Ctu0E+Stv0E+∫0tSt−sηsdsE≤Mu0E+Mv0E+M∫0tas+busEds≤d+Mb∫0tusEds,where d=Mu0E+Mv0E+MaL2I,ℝ+T. Employing Gronwall inequality, one gets(29)utE≤deMbt.
This implies(30)uCI,E≤deMbT≕r.
Furthermore, we write(31)Qr=u∈CI,E:uCI,E<r+1.
Obviously,Qr is open in CI,E. Utilizing Lemma 5, we know that B:Qr¯⟶PkcE is compact and u.s.c. We also see easily that A:Qr¯⟶E is a contraction mapping with single value and coefficient λ<1/2. Therefore, according to the choice of Qr¯, none of u∈E with u=r ensures that λu∈Au+Bu with some ρ>1.
Applying Lemma3, one knows that ℱ has a fixed point due to all the conditions being fulfilled. Hence, there is a mild solution for system (6).
Next, let us discuss some features of the solution set. For everyT>0, we say that πT is a truncation operator form C0,+∞,E to C0,T,E if for any z∈C0,+∞,E, we have πTz∈C0,T,E. Put(32)Σξ=u∈C0,+∞,E:uis a mild solution of1.1on0,T,ξ=u0,v0,where u0=u0,u′0=v0. Obviously, πT°Σξ=Fixℱ and πT°Σξ⊂S⋅ξ+W°PFπT°Σξ. Here, Fixℱ stands for the solution set of the fixed points of system (6).Lemma 6.
LetU=E×E and all the assumptions of Theorem 1 be fulfilled. Then, πT°Σξn is relatively compact in C0,T,E for arbitrary sequence ξn⟶ξ in U. Especially, for every ξ∈U, πT°Σξ is compact in C0,T,E.Proof.
It is verified by analogizing the proof of Lemma 3.7 of [8].
Them-semiflow associated with system (1) is given as follows:(33)Φ:ℝ+×U⟶PUΦt,ξ=ut,ut:xis a mild solution of1.1,ξ=u0,v0.
By applying a similar discussion in [11], we can obtain that Φ is a strict m-semiflow, i.e., for arbitrary t,s∈ℝ+ and ξ∈U, it holds that(34)Φt+s,ξ=Φt,Φs,ξ.Lemma 7.
Assume that all the hypotheses of Theorem1 are fulfilled. For all t>0, Φt,⋅ is u.s.c. with compact values.Proof.
It is verified by analogizing the proof of Lemma 3.8 of [8].


## 4. The Existence Result of Global Attractors

In the sequel, we will present a severe validation for the existence of a global attractor ofm-semiflow Φ induced by system (1). The conditions A and (F) are replaced by (A′) and (F′) as follows, respectively:(A′) The cosine and sine functions are exponentially stable with exponent α>0, that is, there exist M0≥1 and α≥0 with(35)CtℒE≤M0e−αt,StℒE≤M0e−αt,∀t>0.(F′) The multivalued mapping F:ℝ+×E⟶PkcE is u.s.c. and satisfies the following conditions:(1)
There existsa1∈L2ℝ+,ℝ+ and b1>0, ensuring that(36)Ft,x=supξ:ξ∈Ft,u≤a1t+b1u,∀t,u∈ℝ×E.(2)
There existsq∈L2ℝ+,ℝ+ ensuring that, for every bounded Ω in E,(37)χFt,Ω≤qtχΩfor almost allt∈R+.GivenT>0, for convenience, the translation operator ΦT,⋅ is defined by ΦT. The following condensing property of ΦT is an essential result to reveal that Φ is asymptotically upper semicompact.Lemma 8.
Suppose thatA′ and F′ are satisfied. Then, there are ζ∈0,1 and T0>0, guaranteeing that, for all T≥T0 and all bounded B⊂U, it is inferred that(38)χ˜ΦTB≤ζ⋅χ˜B,whenever(39)α−4M0qL2ℝ+,ℝ+>0.Proof.
LetU=E×E, B=B1×B2 be bounded in U. Denote D=ΣB, we obtain(40)Dt⊂CtB1+StB2+∫0tSt−sPFDsds,t>0.
Therefore, we get(41)χ˜ΦtB=χDt≤M0e−αtχB1+M0e−αtχB2+χ∫0tSt−sPFDsds≤2M0e−αtχ˜B+4M0∫0te−αt−sχPFDsds≤2M0e−αtχ˜B+4M0e−αt∫0teαsqsχDsds≤2M0e−αtχ˜B+4M0qL2ℝ+,ℝ+e−αt∫0teαsχDsds.
Hence,(42)eαtχDt≤2M0χ˜B+4M0qL2ℝ+,ℝ+∫0teαsχDsds.
Applying the Gronwall inequality, we have(43)eαtχDt≤2M0e4M0qL2ℝ+,ℝ+tχ˜B.
Equivalently,(44)χDt≤2M0e−α−4M0qL2ℝ+,ℝ+tχ˜B.
Note thatχ˜ΦtB=χDt, we get(45)χ˜ΦTB≤ζt⋅χ˜B,where ζt=2M0e−α−4M0qL2ℝ+,ℝ+t.
TakingT0>ln2M0/α−4M0qL2ℝ+,ℝ+ and ζ=ζT0, we conclude the proof.Lemma 9.
Presume thatA′ and F′ are fulfilled. Then, Φ is asymptotically upper semicompact.Proof.
LetU=E×E and B=B1×B2 be a bounded subset of U and ΔB be a collection consisted of sequences ξnwithξn∈Φtn,B,tn⟶∞. Put(46)η=supχ˜Q:Q∈ΔB.
We claim thatη=0 by using reduction. Assume that there exists Qθ=ξn∈ΔB such that(47)χQθ>η−θ,∀θ∈0,1−ζη,where ζ is the same as in Lemma 8. Choose T>0 as described in Lemma 8; then, there exists mn∈ℕ such that tn=mnT+rn,rn∈0,T for any tn∈T,∞. Let τn=mn−1T+rn. Then,(48)ξn∈Φtn,B=ΦT+τn,B=ΦTΦτn,B.
By takingσn∈Φτn,B such that ξn∈ΦT,σn, we obtain(49)χ˜Qθ=χ˜ξn≤χ˜Φtσn≤ζχ˜σn≤ζη<η−θ.
This is a contradiction.Lemma 10.
Presume thatA′ and F′ are fulfilled. If α>bM0, then an absorbing set exists for m-semiflow Φ.Proof.
Lett>0 and B=B1×B2 be bounded in U=E×E. Then, for each ξ=u0,v0∈U and u∈Σξ, there is MB>0 with(50)ut=Ctu0+Stv0+∫0tSt−sfsds,u0≤MB,v0≤MB,for some f∈PFu. The combination of Ho¨lder inequality, A′ and the part (1) of F′ gives(51)utX≤2MBM0e−αt+M0∫0te−αt−sa1s+b1usEds≤2MBM0e−αt+M0∫0te−αt−sa1sds+b1M0∫0te−αt−susEds.≤2MBM0e−αt+M0e−αta1L2ℝ+,ℝ+12αeαt+b1M0∫0te−αt−susEds≤2MBM0e−αt+M0a1L2ℝ+,ℝ+12α+b1M0∫0te−αt−susEds.
Puttingc1=M0a1L2ℝ+,ℝ+1/2α and c2=b1M0, we obtain(52)eαtutE≤2MBM0+c1eαt+c2∫0teαsusXds.
The employment of Gronwall inequality yields(53)eαtutE≤2MBM0+c1eαt+c2∫0t2MBM0+c1eαsec2t−sds≤2MBM0+c1eαt+c2∫0t2MBM0ec2t−sds+∫0tc1eαs+c2t−c2sds≤2MBM0+c1eαt+2MBM0ec2−1+c1c2α−c2eαt−ec2t.
This implies(54)utE≤2MBM0e−αt+c1+2MBM0ec2−αt+c1c2α−c21−ec2−αt.
The above inequality holds subject to the ball with the radiusR as(55)R=c1+c1c2α−c2+1.
Moreover, it is actually an absorbing set in terms ofΦ.Theorem 2.
Assume thatA′ and F′ are fulfilled. Then, the m-semiflow Φ induced by system (1) has a compact global attractor, if the following holds:(56)minα−4M0qL2ℝ+,ℝ+,α−b1M0>0.Proof.
By Lemmas7, 9, and 10, we obtain that all the conditions of Lemma 3 are satisfied. Therefore, the m-semiflow Φ admits a compact global attractor.


## 5. An Illustrative Example

In this section, a concrete example is illustrated for the application of evolution partial different equations.Take an open setΩ which is bounded in ℝn and denote its boundary by ∂Ω. The following system is worthy of survey:(57)Zttt,u−ΔZt,u=ft,u,t>0,u∈Ω,ft,u=λf1t,Zt,u+1−λf2t,Zt,u,λ∈0,1,t>0,u∈Ω,Zt,u=0,t≥0,u∈∂Ω,Z0,u=Z0u,Zt0,u=Z1u,where f1,f2:ℝ+×ℝ⟶ℝ are continuous functions. This system describes a classical wave equation with a free bounded issue [1].PutE=L2Ω. The multivalued mapping F:ℝ+×E⟶PE is described as(58)FZ˜t=λf1t,Z˜t+1−λf2t,Z˜t:λ∈0,1.Then, (57) can be reformulated as(59)Z″t−AZt∈Ft,Zt,t≥0,Z0=Z0,Zt0=Z1,in which A=Δ, DA=H2Ω∩H01Ω, and Zt∈E.Suppose that there existsa1,a2∈L2Ω and positive constants b1andb2 such that(60)f1t,p≤a1t+b1p,f2t,p≤a2t+b2p,∀x∈Ω,p∈ℝ.Note that the mappingF has convex, closed, and compact values. In addition,(61)Ft,Z˜≤maxa1t,a2t+maxb1,b2Z˜X.By the continuity off1 and f2, we can obtain that there is a closed graph for F. Furthermore, by exploiting Lebesgue dominated convergence theorem, one can verify the quasicompactness of F. Therefore, F is u.s.c., and F′1 is testified. If S⋅ is exponentially stable and compact, then A′ and F′2 is also satisfied. This indicates the presence of a global attractor, which belongs to the m-semiflow induced by (57).



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