# Strong Pullback Attractors of the Nonautonomous Suspension Bridge Equation with Variable Delay

Authors: Suping Wang; Qiaozhen Ma
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

In this paper, we consider the suspension bridge equation with variable delay. The long-time dynamics of the solutions for the suspension equations without delay effects have been investigated by many authors. But there are few works on suspension equations with delay. Moreover there are not many studies on attractors for other systems with delay. Thus, we study the existence of pullback attractor for the suspension equation with variable delay by using the theory of attractors for multivalued dynamical systems.


---

## Body

## 1. Introduction

The suspension bridge equations were presented by Lazer and McKenna in [1] as a new problem in the field of nonlinear analysis. Ma and Zhong et al. have investigated systematically the long-time behavior of solution for both the single and the coupled suspension bridge equations in [2–6]. Bochicchio, Giorgi, and Vuk studied the existence of the global attractor for the Kirchhoff suspension bridge equation and obtained a regularity result of attractor; see [7] for details. The existence of global attractors for the suspension bridge with linear memory was achieved by Kang in [8]. However, none of the above conclusions contain time delay. In regard to wave equation with delay, Wang in [9] studied the wave equation with variable delay; they derived the existence of pullback attractors under some assumptions on delay term. For the related works of equations with variable delay, we also refer to [10–13] and the references therein. Motivated by the above mentioned works, our goal is to prove the existence of pullback attractor for suspension bridge with variable delay. To the best our knowledge, there is no results on suspension bridge equation with variable delay. Here we study the existence of pullback attractors for the suspension bridge equation with variable delay. That is, we consider the following nonautonomous suspension bridge equations in [0,L]:(1)∂ttu+∂xxxxu+μ∂tu+ku+=Ft,ut-ρt+gx,t,x∈0,L,t>τ,u0,t=uL,t=∂xxu0,t=∂xxuL,t=0,x∈0,L,t>τ,ux,t=φx,t-τ,x∈0,L,t∈τ-h,τ,∂tux,t=∂tφx,t-τ,x∈0,L,t∈τ-h,τ.Model (1) describes the vibration of the road bed in the vertical direction, where u(x,t) denotes the deflection in the downward direction, μ∂tu represents the viscous damping, and μ is a positive constant. ku+ represents the restoring force, k>0 denotes the spring constant, the function u+(x,t)=max{u(x,t),0}. τ is the initial time, φ is the initial datum on the interval [τ-h,τ], where h>0, and source term F contains some hereditary characteristic and satisfies the following conditions:(I) The functionsF∈C1(R×R,R) and F(t,0)=0, and ρ∈C1(R;[0,h]) satisfies(2)ρ′t≤ρ∗<1,∀t∈R;(II)g∈Lloc2(R;L2(0,L)) satisfies (3)∫-∞teβsgs2ds<∞,∀t∈R,where β>0 will be determined later.We will denote byCX the Banach space C([-h,0];X), equipped with the sup-norm. For an element φ∈CX, its norm will be written as φCX=supθ∈[-h,0]φ(θ)X.Let(X,.),(Y,.) be two Banach spaces with continuous compact embedding X↪Y and CX,Y denote Banach space CX∩C1([-h,0];Y). For an element φ∈CX,Y, its norm will be written as (4)φCX,Y2=φCX2+φ′CY2,∀φ∈CX,Y.Also, we will use the spaces CH and CW in our analysis. Given T≥τ and u:[τ-h,T)→H, for each t∈[τ,T], we denote by ut the function defined on [-h,0] by the relation ut(θ)=u(t+θ),θ∈[-h,0].The rest of this article consists of three sections. In the next section, we define some functions setting and iterate some useful lemmas and abstracts. In Section3, we show the weak and stronger weak solutions. In Sections 4 and 5, we prove the existence of pullback attractor for (1) in CH, CW, respectively.


## 2. Preliminaries

In this section, we iterate some notations and abstract results.We define Hilbert spaceVs=D(As/4); the scalar product and the norm are denoted by(5)u,vs=As/4u,As/4v,uVs2=As/4u2,where A=△2. When s=0, denote V0=H=L2(0,L); when s=2, V2=H01(0,L)∩H2(0,L), the scalar product and the norm on V0 and V2 are denoted as follows:(6)u,v0=u,v,uV02=u2,u,v2=A1/2u,A1/2v,uV22=A1/2u2=△u2.DefineD(A)={ν∈V,Aν∈H:u(0,t)=u(L,t)=∂xxu(0,t)=∂xxu(L,t)=0}; here Au=△2u. In particular, we denote as Au the norm on D(A). It is obvious that V4⊂V2⊂V0=V0∗⊂V2∗; here V0∗,V2∗ are the dual space of V0,V2 respectively, and each space is dense in the following one and the injections are continuous. By thePoincaré inequality, we have (7)λ1us2≤us+12,∀u∈Vs+1,where λ12 is the first eigenvalue of A.We first need the following definitions and abstract results about pullback attractors.LetX be a complete metric space with metric dX(·,·); we denote by HX∗(·,·) and HX(·,·), respectively, the Hausdorff semidistance and Hausdorff distance between two nonempty subsets of a complete metric space X, which are defined by(8)HX∗A,B=supa∈A⁡distXa,Bwhere distX(a,B)=infb∈BdX(a,b) and(9)HXA,B=max⁡HX∗A,B,HX∗B,A.Finally, denote by N(A,r) the open neighborhood y∈X∣distXy,A<r of radius r>0 of subset A of X.LetD be a nonempty class of parameterized sets D^={D(t);t∈R}⊂P(X), and P(X) be a class of nonempty closed subsets of X.Definition 1 (see [10]).
A family of mappingsS(t,τ):X→P(X),t≥τ,τ∈R, is called a multivalued process if it satisfies(1)
S(τ,τ)x={x},∀τ∈R,x∈X;(2)
S(t,s)S(s,τ)x=U(t,τ)x,∀t≥s≥τ,τ∈R,x∈X.A collectionD of some families of nonempty closed subsets of X is said to be inclusion-closed if for each D^={D(t);t∈R}∈D, {D¯(t):D¯(t) is a nonempty subset of D(t),∀t∈R} and also belongs to D; see [14].Let{S(t,τ)} be a multivalued process on X. Q={Q(t)}t∈R∈D is called a D- pullback absorbing set for {S(t,τ)} if, for any B={B(t)}t∈R∈D and each t∈R, there exists a t0=t0(B,t)∈R+ such that(10)St,t-sBt-s⊂Qt,∀s≥t0.Definition 2 (see [12]).
A family of nonempty compact subsetsA={A(t)}t∈R∈D is called to be a D- pullback attractor for the multivalued process {S(t,τ)} if it satisfies the following:(1)
A={A(t)}t∈R is invariant; i.e., (11)St,τAτ=At,∀t≥τ,τ∈R;(2)
A attracts every member of D; that is, for every B={B(t)}t∈R∈D and any fixed t∈R, (12)lims→+∞HX∗St,t-sBt-s,At=0.Definition 3 (See [11]).
Let{S(t,τ)} be a multivalued process on X. {S(t,τ)} is said to be D- pullback asymptotically upper-semicompact in X with respect to B if, for any fixed t∈R, any sequence yn∈S(t,t-sn)xn has a convergent subsequence in X whenever sn→+∞(n→∞),xn∈B(t-sn) with B={B(t)}t∈R∈D.Lemma 4 (see [15, 16]).
LetX and Y be two Banach spaces such that X⊂Y with a continuous injection. If a function φ belongs to L∞(0,T;X) and is weakly continuous with values in Y, then φ is weakly continuous with values in X.Theorem 5 (see [9]).
Let{S(t,τ)} be a multivalued process on Banach space X, and let Q={Q(t)}t∈R be a D- pullback absorbing set for {S(t,τ)} in D. Suppose that S can be written as (13)St,τ=S1t,τ+S2t,τ,∀t≥τ,and for any fixed t∈R,(1)
lims→∞S2(t,t-s)Q(t-s)X=0;(2)
for any fixeds>0, every sequence yn∈S1(t,t-s)Q(t-s) is a Cauchy sequence in X.Then{S(t,τ)} is D-pullback asymptotically upper-semicompact in X.Theorem 6 (see [12]).
LetD be an inclusion-closed collection of some families of nonempty closed subsets of X and {S(t,τ)} be a multivalued process on X. Also S has closed values, and let S(t,τ)x be upper-semicontinuous in x for fixed t≥τ,τ∈R. Suppose that {S(t,τ)} is D- pullback asymptotically upper-semicompact in X, {S(t,τ)} has a D- pullback absorbing set Q={Q(t)}t∈R in D, and Q(t) is closed for every t∈R. Then, the D- pullback attractor A={A(t)}t∈R is unique and is given by(14)At=⋂T≥0⋃s≥TSt,t-sQt-s¯,∀t∈R.


## 3. Well-Posedness

In this section, we will construct the multivalued processes associated with our problem (1), and in the sequel C,Ci denotes arbitrary positive constants, which may be different from line to line and even in the same line. For convenience, let us define the following Hilbert space:(15)H=V2×V0,zH=u,utH=12△u2+ut21/2,W=V3×V1,zW=u,utW=12∇△u2+∇ut21/2.Theorem 7.
Suppose that(I) holds true and g∈Lloc2(R;V0). Then for each τ∈R, it follows that
(i) for anyφ∈CV2,V0, there exists a solution u(t) corresponding to (1) and u(t) satisfies(16)u∈Cτ-h,T;V2∩C1τ-h,T;V0,∀T>τ.
(ii) If in additiong∈Lloc2(R;V1) and the function F∈C1(R×R;R), then for any φ∈CV3,V1, problem (1) admits a stronger weak solution(17)u∈Cτ-h,T;V3∩C1τ-h,T;V1,∀T>τ.Proof.
We exploit the Galerkin Approximation method. First, we assume that there exists an orthogonal basis ofH consisting of eigenvectors χi of A in H; the corresponding eigenvalues are λi,i=1,2,…, satisfying(18)Aχi=λiχi,∀i∈N.For each m, according to the basic theory of ordinary differential equations, there exists an approximate solution um of the form(19)umt=∑i=1mumiχiand it satisfies(20)d2umdt2+μdumdt+PmAum+kPmum+=PmFt,umt-ρt+Pmgt,umt=Pmφt-τ,t∈τ-h,τ,dumtdt=dPmφt-τdt,t∈τ-h,τ,where Pm:V0→Vm is the orthogonal projector in Vm, and Vm=span{χ1,χ2,…,χm}. We next derive a priori estimate for the Galerkin approximate solutions.
Multiplying (20) by vm=um′+δum yields(21)12ddtvm2+△um2+δ△um2+μ-δvm2-δμ-δum,vm+kum+,vm=Ft,umt-ρt,vm+gt,vm.Note that △u2≥λ1u2. We choose δ>0 small enough such that μ-δ-δμ2/2λ1>3μ/4, and find that(22)δ△um2+μ-δvm2-δμ-δum,vm≥δ2△um2+3μ4vm2,and (23)kum+,vm=12ddtkum+2+δkum+2,(24)gt,vm≤μ4vm21μgt2.Exploiting Young inequality, we can see that(25)Ft,umt-ρt,vmt=Ft,umt-ρt-Ft,0,vmt≤∫0LF′t,θumt-ρtumt-ρtvmtdx≤K12μumt-ρt2+μ4vmt2,and in equality (25), we use F′(t,um(t-ρ(t))L∞≤K1, where K1>0. Taking 0<δ′=min⁡{δ,μ/2} and then combining with (22)–(25), from (21) we get the following:(26)ddt△um2+vm2+kum+2+δ′△um2+vm2+kum+2≤2K12μumt-ρt2+2μgt2.Set(27)Emt=△um2+vm2+kum+2,then (28)ddtEmt+δ′Emt≤2K12μumt-ρt2+2μgt2.Integrating (28) from τ to t, we get(29)Emt-Emτ+δ′∫τtEmsds≤2K12μ∫τtums-ρs2ds+2μ∫τtgs2ds.Exploiting assumption (I), let r=s-ρ(s). Note that ρ(s)∈[0,h] and, for ∀s∈R,1/1-ρ′s≤1/1-ρ∗, we arrive at(30)∫τtums-ρs2ds≤11-ρ∗∫τ-htumr2dr=11-ρ∗∫τ-hτumr2dr+∫τtumr2dr.Thus, together with (30), we get(31)Emt≤Emτ+2μ∫τtgs2ds+2K12μ1-ρ∗∫τ-hτums2ds+∫τtums2ds≤Emτ+2K12hμ1-ρ∗λ1φCV2,V02+2K12μ1-ρ∗λ1∫τtEmsds+2μ∫τtgs2ds.By Gronwall’s lemma, we can deduce that(32)△umt2+vmt2+kum+t2≤φCV2,V02+∫τtgs2dseCt-τ.
Thus we can easily infer from (32) that (33)umt∈L∞τ,T;V2,um′t∈L∞τ,T;V0,∀T>τ.Thus there exists a subsequence, still denoted by um(t) and u(t), such that(34)ut∈L∞τ,T;V2,u′t∈L∞τ,T;V0,and(35)umt→utweakly-starinL∞τ,T;V2,um′t→utweakly-starinL∞τ,T;V0,asm→∞.
By passing to the limit in (20) we immediately obtain that u is a solution of (1) which satisfies u(t)∈L∞(τ,T;V2),u′(t)∈L∞(τ,T;V0).
Moreover, similar to the proof of Theorem 4.1 in [14] and Lemma 4, we obtain(36)ut∈Cτ,T;V2u′t∈Cτ,T;V0,furthermore,(37)ut∈Cτ-h,T;V2u′t∈Cτ-h,T;V0.
Next, we prove the existence of stronger weak solution.
Assume that there exists an orthonormal basis ofV3 consisting of eigenvectors wi of A in V3. The corresponding eigenvalues are νi,i=1,2,···, satisfying the following.(38)Awi=νiwi,∀i∈NFor each m, according to the basic theory of ordinary differential equations, there exists an approximate solution un of the form (39)unt=∑j=1nuniwiand it satisfies(40)d2undt2+μdundt+QNAun+kQNun+=QNFt,unt-ρt+QNgt,unt=QNφt-τ,t∈τ-h,τ,duntdt=dQNφt-τdt,t∈τ-h,τ,where QN:V3→Vn is the orthogonal projector in Vn, and Vn=span{w1,w2,···,wn}.
Take the inner product inV0 of (20) with -△vn=-△∂tun-ε△un to get(41)12ddt∇vn2+∇△un2+2k∇un+2+ε∇△un2+kε∇un+2+μ-ε∇vn2-εμ-εun,-△vn=Ft,unt-ρt,-△vn+gt,-△vn.
By using theHölder and Young’s inequalities, we have the estimates(42)ε∇△un2+μ-ε∇v2-εμ-εun,-△vn≥ε2∇△un2+μ-ε-εμ22λ1∇v2,
and, by using Young’s inequality, we have(43)Ft,unt-ρt,-△vnt≤∇Ft,unt-ρt,∇vnt≤∫0LF′t,unt-ρt∇unt-ρt∇vnt≤K12μ∇unt-ρt2+μ4∇vnt2,and (44)gt,-△vnt≤μ4∇vnt2+1μ∇gt2.Choose ε>0 small enough, such that μ-ε-εμ2/2λ1>3μ/4. Combining with (42)–(44), from (41),(45)ddt∇△un2+∇vn2+2k∇un+2+ε∇△un2+μ2∇vn2+2εk∇un+2≤2K12μ∇unt-ρt2+2μ∇gt2.Set the following.(46)Vmt=∇△unt2+∇vnt2+2k∇un+t2,Taking 0<ε′=min{ε,μ/2}, then (45) can be rewritten as(47)ddtVmt+ε′Vt≤2K12μ∇unt-ρt2+2μ∇gt2.Integrating (47) from τ to t, we obtain(48)Vmt≤Vmτ+2K12μ∫τt∇uns-ρs2ds+2μ∫τt∇gs2ds.Exploiting (32), we get(49)Vmt≤Vmτ+CφCV2,V02+∫τtgs2dseCt-τ+2μ∫τt∇gs2ds,so we can deduce that(50)∇△unt2+∇vnt2+2k∇un+t2≤Vmτ+CφCV2,V02+∫τtgs2dseCt-τ+2μ∫τt∇gs2ds.
Thus we can infer from (50) that {un(t)},{un′(t)} are uniformly bounded in L∞(τ-h,T;V3) and L∞(τ-h,T;V1), respectively, as n→∞. Combining with the results of (1) in Theorem 7 and exploiting Theorem 4.1 in [16] Sec. IV.4.4, by a standard argument we obtain the existence of strong solutions in u∈C([τ-h,T];V3)∩C1([τ-h,T];V1). We can also refer to [4].


## 4. Pullback Attractors inCH

### 4.1. Pullback Absorbing Set inCH

Owing to Theorem7, we can define a family of multivalued mappings as follows: S(t,τ):CH→CH;S ( t , τ ) φ = { u t ( · ; τ , φ ) ∣ u ( · ) is a solution of (1) as in (i) with φ∈CH} and S(t,τ):CW→CW;S ( t , τ ) φ = { u t ( · ; τ , φ ) ∣ u ( · ) is a solution of (1) as in (ii) with φ∈CW}, where ut is defined for θ∈[-h,0] as ut(θ)=u(t+θ). Then {S(t,τ)} are multivalued processes on CH and CW, respectively.we will denote byDCH all the families of nonempty subsets D={D(t);t∈R}⊂P(CH) such that (51)limt→-∞⁡eβtminu∈Dt⁡uCH2=0,where β>0 will be determined later.Lemma 8.
Suppose that(I)-(II) hold true. Then the multivalued process {S(t,τ)} corresponding to (1) possesses a DCH-pullback absorbing set QCH in DCH.Proof.
Taking the scalar product inV0 of (1) with v=∂tu+αu, we have(52)12ddtv2+△u2+α△u2+μ-αv2-αμ-αu,v+ku+,v=Ft,ut-ρt,v+gt,v.Similar to the estimate of (26), we have(53)ddt△u2+v2+ku+2+α′△u2+v2+ku+2≤2K12μut-ρt2+2μgt2,where 0<α′=min{α,μ/2}.Set(54)Et=△ut2+vt2+ku+t2,and then (53) can be rewritten as (55)ddtEt+α′Et≤2K12μut-ρt2+2μgt2.Multiplying (55) by eα′t and integrating it from τ to t, we obtain(56)eα′tEt≤eα′τEτ+2K12μ∫τteα′sus-ρs2ds+2μ∫τteα′sgs2ds.Let r=s-ρ(s). In view of ρ(s)∈[0,h] and 1/1-ρ′s≤1/(1-ρ∗) for all s∈R, consequently, we deduce the following.(57)∫τteα′sus-ρs2ds≤∫τ-hteα′heα′r1-ρ∗ur2dr=eα′h1-ρ∗∫τ-hτeα′rur2dr+∫τteα′rur2dr≤eα′h+τhφCV2,V021-ρ∗λ1+eα′h1-ρ∗λ1∫τteα′r△ur2drMaking use of (57), we get(58)eα′tEt≤eα′τEτ+2K12eα′h+τhφCV2,V02μ1-ρ∗λ1+2μ∫τteα′rgr2dr+2K12eα′hμ1-ρ∗λ1∫τteα′r△ur2dr≤eα′τEτ+2K12eα′h+τhφCV2,V02μ1-ρ∗λ1+2μ∫τteα′rgr2dr+2K12eα′hμ1-ρ∗λ1∫τteα′rErdr.LetC1=2K12heα′h/μ(1-ρ∗)λ1,C2=2K12eα′h/μ(1-ρ∗)λ1,C3=2/μ, and then using Gronwall lemma to (58), we arrive at(59)Et≤Eτe-α′-C2t-τ+C1e-α′-C2t-τφCV2,V02+C3e-α′-C2t∫τteα′-C2rgr2dr.Furthermore, let μ2e-μh/2≥4K12/1-ρ∗λ1, such that α′-C2=β>0, which yields(60)△ut2+vt2+ku+t2≤△uτ2+vτ2+ku+τ2e-βt-τ+C1e-βt-τφCV2,V02+C3e-βt∫τteβrgr2dr.Now settingt+θ instead of t (where θ∈[-h,0]), there holds(61)△ut+θ2+vt+θ2+ku+t+θ2≤△uτ2+vτ2+ku+τ2e-βt+θ-τ+C1e-βt+θ-τφCV2,V02+C3e-βt+θ∫τt+θeβrgr2dr≤△uτ2+vτ2+ku+τ2e-βt-h-τ+C1e-βt-h-τφCV2,V02+C3e-βt-h∫τteβrgr2dr.Let C4=max{2,2α2/λ1+1}, exploiting (II) and (61) implies that(62)utCH2=△utCV22+ut′CH2=supθ∈-h,0⁡u′t+θ2+supθ∈-h,0⁡△ut+θ2≤2supθ∈-h,0vt+θ2+2α2λ1+1supθ∈-h,0⁡△ut+θ2≤C4△uτ2+vτ2+ku+τ2e-βt-h-τ+C1C4e-βt-h-τφCV2,V02+C3C4e-βt-h∫-∞teβsgs2ds.Denote byr(t) the nonnegative number given for each t∈R by (63)r2t=2C1C4e-βt-h-τφCV2,V02+C3C4e-βt-h∫-∞teβrgr2dr,and consider the family of closed bounded balls QCH={Q(t)}t∈R in CH defined by(64)Qt=ϕ∈CH:ϕCH≤rt.It is straightforward to check that QCH∈DCH, and moreover, by (51) and (62), the family of QCH is DCH-pullback absorbing for the multivalued process {S(t,τ)} on CH.


### 4.2. Pullback Attractors inCH

Theorem 9.
Suppose that(I),(II) hold true. Then the multivalued process {S(t,τ)} on CH possesses a unique pullback attractor {ACH(t)}t∈R in DCH.Proof.
Firstly, We prove that the multivalued process{S(t,τ)} on CH is DCH- pullback asymptotically upper-semicompact. We observe that, for any T≥t-s with s≥0,
S ( T , t - s ) φ = { u T ( · ; t - s , φ ) ∣ u ( · ) is a solution of (1) with φ∈Q(t-s)},
whereQCH={Q(t)}t∈R is a DCH-pullback absorbing set in CH. Let u=v+w; we decompose (1) as follows:(65)∂TTv+∂xxxxv+μ∂Tv+ku+=FT,uT-ρT+gT,t,T≥t-s,v0,T=vL,T=∂xxv0,T=∂xxvL,T=0,T≥t-s,vT=0,T∈t-s-h,t-s,∂TvT=0,T∈t-s-h,t-s,and(66)∂TTw+∂xxxxw+μ∂Tw=0,T≥t-s,w0,T=wL,T=∂xxw0,T=∂xxwL,T=0,T≥t-s,wT=φT-t+s,T∈t-s-h,t-s,∂TwT=∂TφT-t+s,T∈t-s-h,t-s.Thanks to (62), but with F=g=0, we have(67)wt′CV02+△wtCV22≤Ce-βsφCV2,V02.According to Theorem5, we consider a couple of solutions u1 and u2 of system (1) corresponding to initial data φ1 and φ2, respectively. Let y(T)=v1(T)-v2(T), where v1(T) and v2(T) are solutions of (65), respectively. Then y(T) satisfies(68)∂TTy+∂xxxxy+μ∂Ty=ku2+-ku1++FT,u1T-ρT-FT,u2T-ρT,T≥t-s,y0,T=yL,T=∂xxy0,T=∂xxyL,T=0,T≥t-s,yT=0,T∈t-s-h,t-s,∂TyT=0,T∈t-s-h,t-s.Taking the inner product in [0,L] of (68) with y′(T), we get(69)12ddty′T2+△yT2+μy′T2+ku2+T-u1+T,y′T=FT,u1T-ρT-FT,u2T-ρT,y′T.Integrating from t-s to t+θ, we have(70)y′t+θ2+△yt+θ2≤2∫t-st+θku2+T-u1+T,y′TdT+2∫t-st+θFT,u1T-ρT-FT,u2T-ρT,y′TdT≤2FT,u1T-ρT-FT,u2T-ρTL20,L×t-s,ty′TL20,L×t-s,t+2klu1T-u2TL20,L×t-s,ty′TL20,L×t-s,t,where k(u1+(T)-u2+(T))≤kl(u1(T)-u2(T))(l>0) is a constant.LetunT∈S(T,t-s)φn with φn∈Q(t-s). Thanks to (62), we can assume that(71)un→uweaklystarinL∞t-s-h,t;V2un′→u′weaklystarinL∞t-s-h,t;V0.Hence(72)un→uinL2t-s-h,t;V0SinceF∈C(R×R;R), thenF ( T , u n ( T , x ) ) → F ( T , u ( T , x ) )   n → ∞ for almost every (T,x)∈[t-s-h,t]×[0,L].By Lebesgue dominated convergence theorem, we have(73)limn→∞⁡limm→∞⁡FT,unT-ρT-FT,umT-ρTL20,L×t-s,t=0.It follows from (70) that(74)vnt′-vmt′CV02+△vnt-△vmtCV22=supθ∈-h,0⁡vn′t+θ-vm′t+θ2+△vnt+θ-△vmt+θ2≤2FT,unT-ρT-FT,umT-ρTL20,L×t-s,tvn′T-vm′TL20,L×t-s,t+2klum1T-um2TL20,L×t-s,tvn′T-vm′TL20,L×t-s,t.Then combining (73) and (74), from (70)(75)limn→∞⁡limm→∞⁡vnt′-vmt′CV02+△vnt-△vmtCV22=0.Secondly, by Lemma8 and the above arguments, we can find that the multivalued process {S(t,τ)} is DCH- pullback asymptotically upper-semicompact in CHand {S(t,τ)} has a closed bounded DCH- pullback absorbing set QCH in CH. Let(76)ACHt=ωQCH,t=⋂T∈R+⋃s≥TSt,t-sQt-s¯∀t∈R.In order to show that{ACH(t)}t∈R is a DCH-pullback attractor, according to the Theorem 6, we only prove the negative invariance of {ACH(t)}t∈R. The proof is similar to [9].Then{S(t,τ)} is D-pullback asymptotically upper-semicompact in CH.


## 4.1. Pullback Absorbing Set inCH

Owing to Theorem7, we can define a family of multivalued mappings as follows: S(t,τ):CH→CH;S ( t , τ ) φ = { u t ( · ; τ , φ ) ∣ u ( · ) is a solution of (1) as in (i) with φ∈CH} and S(t,τ):CW→CW;S ( t , τ ) φ = { u t ( · ; τ , φ ) ∣ u ( · ) is a solution of (1) as in (ii) with φ∈CW}, where ut is defined for θ∈[-h,0] as ut(θ)=u(t+θ). Then {S(t,τ)} are multivalued processes on CH and CW, respectively.we will denote byDCH all the families of nonempty subsets D={D(t);t∈R}⊂P(CH) such that (51)limt→-∞⁡eβtminu∈Dt⁡uCH2=0,where β>0 will be determined later.Lemma 8.
Suppose that(I)-(II) hold true. Then the multivalued process {S(t,τ)} corresponding to (1) possesses a DCH-pullback absorbing set QCH in DCH.Proof.
Taking the scalar product inV0 of (1) with v=∂tu+αu, we have(52)12ddtv2+△u2+α△u2+μ-αv2-αμ-αu,v+ku+,v=Ft,ut-ρt,v+gt,v.Similar to the estimate of (26), we have(53)ddt△u2+v2+ku+2+α′△u2+v2+ku+2≤2K12μut-ρt2+2μgt2,where 0<α′=min{α,μ/2}.Set(54)Et=△ut2+vt2+ku+t2,and then (53) can be rewritten as (55)ddtEt+α′Et≤2K12μut-ρt2+2μgt2.Multiplying (55) by eα′t and integrating it from τ to t, we obtain(56)eα′tEt≤eα′τEτ+2K12μ∫τteα′sus-ρs2ds+2μ∫τteα′sgs2ds.Let r=s-ρ(s). In view of ρ(s)∈[0,h] and 1/1-ρ′s≤1/(1-ρ∗) for all s∈R, consequently, we deduce the following.(57)∫τteα′sus-ρs2ds≤∫τ-hteα′heα′r1-ρ∗ur2dr=eα′h1-ρ∗∫τ-hτeα′rur2dr+∫τteα′rur2dr≤eα′h+τhφCV2,V021-ρ∗λ1+eα′h1-ρ∗λ1∫τteα′r△ur2drMaking use of (57), we get(58)eα′tEt≤eα′τEτ+2K12eα′h+τhφCV2,V02μ1-ρ∗λ1+2μ∫τteα′rgr2dr+2K12eα′hμ1-ρ∗λ1∫τteα′r△ur2dr≤eα′τEτ+2K12eα′h+τhφCV2,V02μ1-ρ∗λ1+2μ∫τteα′rgr2dr+2K12eα′hμ1-ρ∗λ1∫τteα′rErdr.LetC1=2K12heα′h/μ(1-ρ∗)λ1,C2=2K12eα′h/μ(1-ρ∗)λ1,C3=2/μ, and then using Gronwall lemma to (58), we arrive at(59)Et≤Eτe-α′-C2t-τ+C1e-α′-C2t-τφCV2,V02+C3e-α′-C2t∫τteα′-C2rgr2dr.Furthermore, let μ2e-μh/2≥4K12/1-ρ∗λ1, such that α′-C2=β>0, which yields(60)△ut2+vt2+ku+t2≤△uτ2+vτ2+ku+τ2e-βt-τ+C1e-βt-τφCV2,V02+C3e-βt∫τteβrgr2dr.Now settingt+θ instead of t (where θ∈[-h,0]), there holds(61)△ut+θ2+vt+θ2+ku+t+θ2≤△uτ2+vτ2+ku+τ2e-βt+θ-τ+C1e-βt+θ-τφCV2,V02+C3e-βt+θ∫τt+θeβrgr2dr≤△uτ2+vτ2+ku+τ2e-βt-h-τ+C1e-βt-h-τφCV2,V02+C3e-βt-h∫τteβrgr2dr.Let C4=max{2,2α2/λ1+1}, exploiting (II) and (61) implies that(62)utCH2=△utCV22+ut′CH2=supθ∈-h,0⁡u′t+θ2+supθ∈-h,0⁡△ut+θ2≤2supθ∈-h,0vt+θ2+2α2λ1+1supθ∈-h,0⁡△ut+θ2≤C4△uτ2+vτ2+ku+τ2e-βt-h-τ+C1C4e-βt-h-τφCV2,V02+C3C4e-βt-h∫-∞teβsgs2ds.Denote byr(t) the nonnegative number given for each t∈R by (63)r2t=2C1C4e-βt-h-τφCV2,V02+C3C4e-βt-h∫-∞teβrgr2dr,and consider the family of closed bounded balls QCH={Q(t)}t∈R in CH defined by(64)Qt=ϕ∈CH:ϕCH≤rt.It is straightforward to check that QCH∈DCH, and moreover, by (51) and (62), the family of QCH is DCH-pullback absorbing for the multivalued process {S(t,τ)} on CH.


## 4.2. Pullback Attractors inCH

Theorem 9.
Suppose that(I),(II) hold true. Then the multivalued process {S(t,τ)} on CH possesses a unique pullback attractor {ACH(t)}t∈R in DCH.Proof.
Firstly, We prove that the multivalued process{S(t,τ)} on CH is DCH- pullback asymptotically upper-semicompact. We observe that, for any T≥t-s with s≥0,
S ( T , t - s ) φ = { u T ( · ; t - s , φ ) ∣ u ( · ) is a solution of (1) with φ∈Q(t-s)},
whereQCH={Q(t)}t∈R is a DCH-pullback absorbing set in CH. Let u=v+w; we decompose (1) as follows:(65)∂TTv+∂xxxxv+μ∂Tv+ku+=FT,uT-ρT+gT,t,T≥t-s,v0,T=vL,T=∂xxv0,T=∂xxvL,T=0,T≥t-s,vT=0,T∈t-s-h,t-s,∂TvT=0,T∈t-s-h,t-s,and(66)∂TTw+∂xxxxw+μ∂Tw=0,T≥t-s,w0,T=wL,T=∂xxw0,T=∂xxwL,T=0,T≥t-s,wT=φT-t+s,T∈t-s-h,t-s,∂TwT=∂TφT-t+s,T∈t-s-h,t-s.Thanks to (62), but with F=g=0, we have(67)wt′CV02+△wtCV22≤Ce-βsφCV2,V02.According to Theorem5, we consider a couple of solutions u1 and u2 of system (1) corresponding to initial data φ1 and φ2, respectively. Let y(T)=v1(T)-v2(T), where v1(T) and v2(T) are solutions of (65), respectively. Then y(T) satisfies(68)∂TTy+∂xxxxy+μ∂Ty=ku2+-ku1++FT,u1T-ρT-FT,u2T-ρT,T≥t-s,y0,T=yL,T=∂xxy0,T=∂xxyL,T=0,T≥t-s,yT=0,T∈t-s-h,t-s,∂TyT=0,T∈t-s-h,t-s.Taking the inner product in [0,L] of (68) with y′(T), we get(69)12ddty′T2+△yT2+μy′T2+ku2+T-u1+T,y′T=FT,u1T-ρT-FT,u2T-ρT,y′T.Integrating from t-s to t+θ, we have(70)y′t+θ2+△yt+θ2≤2∫t-st+θku2+T-u1+T,y′TdT+2∫t-st+θFT,u1T-ρT-FT,u2T-ρT,y′TdT≤2FT,u1T-ρT-FT,u2T-ρTL20,L×t-s,ty′TL20,L×t-s,t+2klu1T-u2TL20,L×t-s,ty′TL20,L×t-s,t,where k(u1+(T)-u2+(T))≤kl(u1(T)-u2(T))(l>0) is a constant.LetunT∈S(T,t-s)φn with φn∈Q(t-s). Thanks to (62), we can assume that(71)un→uweaklystarinL∞t-s-h,t;V2un′→u′weaklystarinL∞t-s-h,t;V0.Hence(72)un→uinL2t-s-h,t;V0SinceF∈C(R×R;R), thenF ( T , u n ( T , x ) ) → F ( T , u ( T , x ) )   n → ∞ for almost every (T,x)∈[t-s-h,t]×[0,L].By Lebesgue dominated convergence theorem, we have(73)limn→∞⁡limm→∞⁡FT,unT-ρT-FT,umT-ρTL20,L×t-s,t=0.It follows from (70) that(74)vnt′-vmt′CV02+△vnt-△vmtCV22=supθ∈-h,0⁡vn′t+θ-vm′t+θ2+△vnt+θ-△vmt+θ2≤2FT,unT-ρT-FT,umT-ρTL20,L×t-s,tvn′T-vm′TL20,L×t-s,t+2klum1T-um2TL20,L×t-s,tvn′T-vm′TL20,L×t-s,t.Then combining (73) and (74), from (70)(75)limn→∞⁡limm→∞⁡vnt′-vmt′CV02+△vnt-△vmtCV22=0.Secondly, by Lemma8 and the above arguments, we can find that the multivalued process {S(t,τ)} is DCH- pullback asymptotically upper-semicompact in CHand {S(t,τ)} has a closed bounded DCH- pullback absorbing set QCH in CH. Let(76)ACHt=ωQCH,t=⋂T∈R+⋃s≥TSt,t-sQt-s¯∀t∈R.In order to show that{ACH(t)}t∈R is a DCH-pullback attractor, according to the Theorem 6, we only prove the negative invariance of {ACH(t)}t∈R. The proof is similar to [9].Then{S(t,τ)} is D-pullback asymptotically upper-semicompact in CH.


## 5. Pullback Attractors inCW

### 5.1. Pullback Absorbing Set inCW

Lemma 10.
Suppose that the conditions in Lemma8 hold true and g∈Lloc2(R;V1) satisfies (77)∫-∞teα′r∇gr2dr<∞,∀t∈R,α′>0.Then the multivalued process {S(t,τ)} corresponding to (1) possesses a pullback absorbing set DCW-pullback absorbing set QCW in DCW.Proof.
Taking the scalar product inV0 of (1) with -△v=-△∂tu-α△u, we have(78)12ddt∇v2+∇△u2+2k∇u+2+α∇△u2+kα∇u+2+μ-α∇v2-αμ-αu,-△v=Ft,ut-ρt,-△v+gt,-△v.
By using theHölder and Young’s inequalities, we have the estimates(79)α∇△u2+μ-α∇v2-αμ-αu,-△v≥α2∇△u2+μ-α-αμ22λ1∇v2,
and, by using Young’s inequality, we have(80)Ft,ut-ρt,-△vt≤∇Ft,ut-ρt,∇vt≤∫0LF′t,ut-ρt∇ut-ρt∇vt≤K12μ∇ut-ρt2+μ4∇vt2,and (81)gt,-△vt≤μ4∇vt2+1μ∇gt2.Choose α>0 small enough, such that μ-α-αμ2/2λ1>3μ/4. Combining with (79)–(81), from (78)(82)ddt∇△u2+∇v2+2k∇u+2+α∇△u2+μ2∇v2+2αk∇u+2≤2K12μ∇ut-ρt2+2μ∇gt2.Set the following.(83)Vt=∇△ut2+∇vt2+2k∇u+t2Taking0<α′=min{α,μ/2}, then (82) can be rewritten as(84)ddtVt+α′Vt≤2K12μ∇ut-ρt2+2μ∇gt2.Multiplying (84) by eα′t and integrating it from τ to t, we obtain(85)eα′tVt≤eα′τVτ+2K12μ∫τteα′s∇us-ρs2ds+2μ∫τteα′s∇gs2ds.In view of (62) and conditions (77), it follows that(86)eα′tVt≤eα′τVτ+2K12μ∫τteα′s∇us-ρs2ds+2μ∫τteα′s∇gs2ds≤eα′τVτ+Ce-βt-τφCV2,V02+Ce-βt∫-∞teβrgr2dr+C∫-∞teα′s∇gs2ds,and thus(87)∇△ut2+∇vt2+2k∇u+t2≤e-βt-τ∇△uτ2+∇vτ2+2k∇u+τ2+Ce-βteβτφCV2,V02+Ce-βt∫-∞teβrgr2dr+Ce-βt∫-∞teα′s∇gs2ds.Setting nowt+θ instead of t (where θ∈[-h,0]), we can see that(88)∇△ut+θ2+∇vt+θ2+2k∇u+t+θ2≤e-βteβheβτ∇△uτ2+∇vτ2+2k∇u+τ2+Ce-βteβh+τφCV2,V02+Ce-βteβh∫-∞teβrgr2dr+Ce-βteβh∫-∞teα′s∇gs2ds.Therefore,(89)utCW2=∇△utCV32+∇ut′CV12=supθ∈-h,0∇△ut+θ2+supθ∈-h,0∇u′t+θ2≤Ce-βteβτ∇△uτ2+∇vτ2+2k∇u+τ2+Ce-βteβτφCV2,V02+Ce-βt∫-∞teβrgr2dr+Ce-βt∫-∞teα′s∇gs2ds.Similar to (51), we denote by DCW all the families of nonempty subsets D={D(t);t∈R}⊂P(CW) such that (90)limt→-∞⁡eβtsupu∈Dt⁡uCW2=0.Consider the family of closed bounded balls QCW={Q(t)}t∈R in CW defined by(91)Qt=ψ∈CW:ψCW≤rt.It is straightforward to check that QCW∈DCW, and moreover, by (89) and (90), the family of QCW is DCW-pullback absorbing for the multivalued process {S(t,τ)} on CW.


### 5.2. Pullback Attractors inCW

Theorem 11.
Assume that the hypotheses in Lemma10 hold. Let {ACH(t)}t∈R be the DCH-pullback attractor of the multivalued process {S(t,τ)} corresponding to (1) on CH. Then ACH(t) is bounded in CW for all t∈R.Proof.
From the proof of Theorem9, we see that(92)ut=vt+wt,∀τ∈R,t≥τ-h.Thus, S(t,τ) can be written as (93)St,τφ=S1t,τφ+S2t,τφ,∀φ∈CH,τ∈R,t≥τ-h,where S1(t,τ)φ={vt(·)}={vt(·;τ,φ)} and S2(t,τ)φ={wt(·)}={wt(·;τ,φ)} are the solutions of (65) and (66), respectively.From (89), with φ=0, we can see that(94)∇ut′CV12+∇△utCV32≤ρt2=Ce-βt∫-∞teβrgr2dr+Ce-βt∫-∞teα′s∇gs2ds.Note that for any fixed t∈R, the closed ball B0(t)=BCW(0,ρ(t)) is a bounded set in CW; clearly B0(t) is bounded in CH, (95)lims→∞⁡HCH∗S1t,t-sACHt-s,B0t=0,and, on the other hand, (67) implies that(96)lims→∞⁡supφ∈ACHt-s⁡S2t,τφCH=0.Hence, (97)lims→∞⁡HCH∗St,t-sACHt-s,B0t=0.By the invariance of {ACH(t)}t∈R, we have ACH(t)⊂B0(t), and then the boundedness of {ACH(t)} in CW follows immediately.Theorem 12.
Suppose that the hypotheses in Lemma10 hold true. Then there exists a unique DCW-pullback attractor {ACW(t)}t∈R for the multivalued process {S(t,τ)} corresponding to (1) in DCW.Proof.
According to Lemma10 and Theorem 5, we only show that the multivalued process {S(t,τ)} on DCW is DCW-pullback asymptotically upper-semicompact.
Exploiting (65), (66), and (89), with F=g=0 yields(98)∇wt′CV12+∇△wtCV32≤Ce-βsϕCV3,V12.This implies (1) in Theorem 5; now it only remains to check (2) in Theorem 5.Taking the inner product inV0 of (68) with -△y′(T), we get(99)12ddt∇y′T2+∇△yT2+μ∇y′T2=ku2+T-u1+T,-△y′T+FT,u1T-ρT-FT,u2T-ρT,-∇y′T,and, integrating (99) from t-s to t+θ (where θ∈[-h,0]), we have(100)∇y′t+θ2+∇△yt+θ2≤2k∫t-st+θu2+T-u1+T,-△y′T+2∫t-st+θFT,u1T-ρT-FT,u2T-ρT,-△y′TdT≤2k∫t-st∫0L∇u2+T-u1+T∇y′TdxdT+2∫t-st∫0L∇FT,u1T-ρT-FT,u2T-ρT∇y′TdxdT≤2kl∇u1T-∇u2TL20,L×t-s,t∇y′TL20,L×t-s,t+2K1∇u1T-ρT-∇u2T-ρTL20,L×t-s,t∇y′TL20,L×t-s,t+2F′T,u1T-ρT-F′T,u2T-ρTL30,L×t-s,t×∇y′TL20,L×t-s,t∇u2T-ρTL60,L×t-s,t.Let unT∈S(T,t-s)ϕn with ϕn∈Q(t-s). By (89), we can obtain(101)un→uweaklystarinL∞t-s-h,t;V3un′→u′weaklystarinL∞t-s-h,t;V1.Hence (102)un→uinL2t-s-h,t;V1and, since F∈C1(R×R;R), thenF ′ ( T , u n ( T , x ) ) → F ′ ( T , u ( T , x ) )   n → ∞ for almost every (T,x)∈[t-s-h,t]×[0,L].By Lebesgue dominated convergence theorem, we get(103)limn→∞⁡limm→∞⁡F′T,unT-ρT-F′T,umT-ρTL30,L×t-s,t=0.Note that (104)∇y′TL20,L×t-s,t,∇u2T-ρTL60,L×t-s,tarebounded.It follows from (102)–(104) the following.(105)limn→∞⁡limm→∞⁡∇vnt′-∇vmt′CV12+∇△vnt-∇△vmtCV32=0Then{S(t,τ)} is D-pullback asymptotically upper-semicompact in CW.


## 5.1. Pullback Absorbing Set inCW

Lemma 10.
Suppose that the conditions in Lemma8 hold true and g∈Lloc2(R;V1) satisfies (77)∫-∞teα′r∇gr2dr<∞,∀t∈R,α′>0.Then the multivalued process {S(t,τ)} corresponding to (1) possesses a pullback absorbing set DCW-pullback absorbing set QCW in DCW.Proof.
Taking the scalar product inV0 of (1) with -△v=-△∂tu-α△u, we have(78)12ddt∇v2+∇△u2+2k∇u+2+α∇△u2+kα∇u+2+μ-α∇v2-αμ-αu,-△v=Ft,ut-ρt,-△v+gt,-△v.
By using theHölder and Young’s inequalities, we have the estimates(79)α∇△u2+μ-α∇v2-αμ-αu,-△v≥α2∇△u2+μ-α-αμ22λ1∇v2,
and, by using Young’s inequality, we have(80)Ft,ut-ρt,-△vt≤∇Ft,ut-ρt,∇vt≤∫0LF′t,ut-ρt∇ut-ρt∇vt≤K12μ∇ut-ρt2+μ4∇vt2,and (81)gt,-△vt≤μ4∇vt2+1μ∇gt2.Choose α>0 small enough, such that μ-α-αμ2/2λ1>3μ/4. Combining with (79)–(81), from (78)(82)ddt∇△u2+∇v2+2k∇u+2+α∇△u2+μ2∇v2+2αk∇u+2≤2K12μ∇ut-ρt2+2μ∇gt2.Set the following.(83)Vt=∇△ut2+∇vt2+2k∇u+t2Taking0<α′=min{α,μ/2}, then (82) can be rewritten as(84)ddtVt+α′Vt≤2K12μ∇ut-ρt2+2μ∇gt2.Multiplying (84) by eα′t and integrating it from τ to t, we obtain(85)eα′tVt≤eα′τVτ+2K12μ∫τteα′s∇us-ρs2ds+2μ∫τteα′s∇gs2ds.In view of (62) and conditions (77), it follows that(86)eα′tVt≤eα′τVτ+2K12μ∫τteα′s∇us-ρs2ds+2μ∫τteα′s∇gs2ds≤eα′τVτ+Ce-βt-τφCV2,V02+Ce-βt∫-∞teβrgr2dr+C∫-∞teα′s∇gs2ds,and thus(87)∇△ut2+∇vt2+2k∇u+t2≤e-βt-τ∇△uτ2+∇vτ2+2k∇u+τ2+Ce-βteβτφCV2,V02+Ce-βt∫-∞teβrgr2dr+Ce-βt∫-∞teα′s∇gs2ds.Setting nowt+θ instead of t (where θ∈[-h,0]), we can see that(88)∇△ut+θ2+∇vt+θ2+2k∇u+t+θ2≤e-βteβheβτ∇△uτ2+∇vτ2+2k∇u+τ2+Ce-βteβh+τφCV2,V02+Ce-βteβh∫-∞teβrgr2dr+Ce-βteβh∫-∞teα′s∇gs2ds.Therefore,(89)utCW2=∇△utCV32+∇ut′CV12=supθ∈-h,0∇△ut+θ2+supθ∈-h,0∇u′t+θ2≤Ce-βteβτ∇△uτ2+∇vτ2+2k∇u+τ2+Ce-βteβτφCV2,V02+Ce-βt∫-∞teβrgr2dr+Ce-βt∫-∞teα′s∇gs2ds.Similar to (51), we denote by DCW all the families of nonempty subsets D={D(t);t∈R}⊂P(CW) such that (90)limt→-∞⁡eβtsupu∈Dt⁡uCW2=0.Consider the family of closed bounded balls QCW={Q(t)}t∈R in CW defined by(91)Qt=ψ∈CW:ψCW≤rt.It is straightforward to check that QCW∈DCW, and moreover, by (89) and (90), the family of QCW is DCW-pullback absorbing for the multivalued process {S(t,τ)} on CW.


## 5.2. Pullback Attractors inCW

Theorem 11.
Assume that the hypotheses in Lemma10 hold. Let {ACH(t)}t∈R be the DCH-pullback attractor of the multivalued process {S(t,τ)} corresponding to (1) on CH. Then ACH(t) is bounded in CW for all t∈R.Proof.
From the proof of Theorem9, we see that(92)ut=vt+wt,∀τ∈R,t≥τ-h.Thus, S(t,τ) can be written as (93)St,τφ=S1t,τφ+S2t,τφ,∀φ∈CH,τ∈R,t≥τ-h,where S1(t,τ)φ={vt(·)}={vt(·;τ,φ)} and S2(t,τ)φ={wt(·)}={wt(·;τ,φ)} are the solutions of (65) and (66), respectively.From (89), with φ=0, we can see that(94)∇ut′CV12+∇△utCV32≤ρt2=Ce-βt∫-∞teβrgr2dr+Ce-βt∫-∞teα′s∇gs2ds.Note that for any fixed t∈R, the closed ball B0(t)=BCW(0,ρ(t)) is a bounded set in CW; clearly B0(t) is bounded in CH, (95)lims→∞⁡HCH∗S1t,t-sACHt-s,B0t=0,and, on the other hand, (67) implies that(96)lims→∞⁡supφ∈ACHt-s⁡S2t,τφCH=0.Hence, (97)lims→∞⁡HCH∗St,t-sACHt-s,B0t=0.By the invariance of {ACH(t)}t∈R, we have ACH(t)⊂B0(t), and then the boundedness of {ACH(t)} in CW follows immediately.Theorem 12.
Suppose that the hypotheses in Lemma10 hold true. Then there exists a unique DCW-pullback attractor {ACW(t)}t∈R for the multivalued process {S(t,τ)} corresponding to (1) in DCW.Proof.
According to Lemma10 and Theorem 5, we only show that the multivalued process {S(t,τ)} on DCW is DCW-pullback asymptotically upper-semicompact.
Exploiting (65), (66), and (89), with F=g=0 yields(98)∇wt′CV12+∇△wtCV32≤Ce-βsϕCV3,V12.This implies (1) in Theorem 5; now it only remains to check (2) in Theorem 5.Taking the inner product inV0 of (68) with -△y′(T), we get(99)12ddt∇y′T2+∇△yT2+μ∇y′T2=ku2+T-u1+T,-△y′T+FT,u1T-ρT-FT,u2T-ρT,-∇y′T,and, integrating (99) from t-s to t+θ (where θ∈[-h,0]), we have(100)∇y′t+θ2+∇△yt+θ2≤2k∫t-st+θu2+T-u1+T,-△y′T+2∫t-st+θFT,u1T-ρT-FT,u2T-ρT,-△y′TdT≤2k∫t-st∫0L∇u2+T-u1+T∇y′TdxdT+2∫t-st∫0L∇FT,u1T-ρT-FT,u2T-ρT∇y′TdxdT≤2kl∇u1T-∇u2TL20,L×t-s,t∇y′TL20,L×t-s,t+2K1∇u1T-ρT-∇u2T-ρTL20,L×t-s,t∇y′TL20,L×t-s,t+2F′T,u1T-ρT-F′T,u2T-ρTL30,L×t-s,t×∇y′TL20,L×t-s,t∇u2T-ρTL60,L×t-s,t.Let unT∈S(T,t-s)ϕn with ϕn∈Q(t-s). By (89), we can obtain(101)un→uweaklystarinL∞t-s-h,t;V3un′→u′weaklystarinL∞t-s-h,t;V1.Hence (102)un→uinL2t-s-h,t;V1and, since F∈C1(R×R;R), thenF ′ ( T , u n ( T , x ) ) → F ′ ( T , u ( T , x ) )   n → ∞ for almost every (T,x)∈[t-s-h,t]×[0,L].By Lebesgue dominated convergence theorem, we get(103)limn→∞⁡limm→∞⁡F′T,unT-ρT-F′T,umT-ρTL30,L×t-s,t=0.Note that (104)∇y′TL20,L×t-s,t,∇u2T-ρTL60,L×t-s,tarebounded.It follows from (102)–(104) the following.(105)limn→∞⁡limm→∞⁡∇vnt′-∇vmt′CV12+∇△vnt-∇△vmtCV32=0Then{S(t,τ)} is D-pullback asymptotically upper-semicompact in CW.



---

*Source: 1930369-2019-03-06.xml*