# Existence and Upper Semicontinuity of Attractors for Nonautonomous Stochastic Sine-Gordon Lattice Systems with Random Coupled Coefficients

Authors: Zhaojuan Wang; Shengfan Zhou
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

We study nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise. We first consider the existence of random attractors in a weighted space for this system and then establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.


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## Body

## 1. Introduction

Stochastic lattice differential equations arise naturally in a wide variety of applications where the spatial structure has a discrete character and uncertainties are taken into account. In recent years, some works have been done regarding the existence of random attractors for stochastic lattice differential equations (see, e.g., [1–11]). Of those works, Bates et al. [2] considered the existence of random attractors for first-order nonautonomous stochastic lattice system driven by multiplicative white noise. Zhou and Wei [11] considered the existence of random attractors for second-order lattice system with random coupled coefficients and multiplicative white noise.Motivated by [2, 11], we will study the asymptotic behavior of solutions of the following nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise: for every τ∈R and t>τ,(1)d2uidt2+αduidt=-λui-βsin⁡ui+git+∑j=-qqηi,jθtωui+j+cui∘dwtdt,i∈Z,with the initial data (2)uiτ=uτ,i,u˙iτ=u1τ,i,i∈Z,where λ,α,β, and c are positive constants; ui∈R; gi(t)∈R; ∘ denotes the Stratonovich sense of the stochastic term; ηi,j(ω)j∈-q,…,0,…,q,q∈N are random variables; w(t) is a Brownian motion (Wiener process) on a probability space (Ω,F,P), where Ω=ω∈CR,R:ω0=0, F is the Borel σ-algebra induced by the compact open topology of Ω, and P is the Wiener measure on (Ω,F) (see [12]); and {θt}t∈R is a metric dynamical system defined on (Ω,F,P), where here θtω(·)=ω(t+·)-ω(·) for all t∈R,ω∈Ω. Then (Ω,F,P,{θt}t∈R) is an ergodic metric dynamical system.In practice, the coupled mode between two nodes (say, adjacent nodes) is usually random. It is then of great importance to investigate stochastic lattice system with random coupled coefficients. To the best of our knowledge, there are no results on nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise.In this paper, we first consider the existence of a tempered random attractor in a weighted spacelρ2×lρ2 for stochastic sine-Gordon lattice systems (1), which attracts the random tempered bounded sets in pullback sense. Then, we consider the dependence of attractors on the parameters c of the system (1) and establish the upper semicontinuity of the random attractor as the intensity c of noise approaches zero.The rest of this paper is organized as follows. In the next section, we present some mathematical setting for system (1). In Section 3, we mainly consider the existence of a tempered random attractor in a weighted space of infinite sequences for system (1). Then in Section 4, we consider the upper semicontinuity of the tempered random attractor for system (1) as c→0.


## 2. Mathematical Settings

We first introduce a weighted spacelρ2 of infinite sequences. Let ρ be a positive-valued function from Z into (0,M0]⊂R+ which satisfies(3)0<ρi⩽M0,ρi⩽c0ρi±1,∀i∈Z,where M0 and c0 are positive constants. For example, for i∈Z, ρ(i)=1/(1+γ2i2)q(q>1/2) [13] and ρ(i)=e-γi satisfy condition (3), where γ>0. Define ρi=ρ(i),∀i∈Z and (4)lρ2=u=uii∈Z:∑i∈Zρiui2<∞,ui∈Rwith norm uρ=∑i∈Zρiui21/2 and inner product (u,v)ρ=∑i∈Zρiuivi for u=(ui)i∈Z,v=(vi)i∈Z∈lρ2. We write ·ρ as · if ρ(i)≡1. Then lρ2 is a separable Hilbert space with the norm ·ρ.For anyu=(ui)i∈Z,v=(vi)i∈Z∈lρ2, define an inner product on lρ2 by (5)u,vλ,ρ=λ∑i∈Zρiuivi;then the norm ·λ,ρ induced by inner product (·,·)λ,ρ is equivalent to the norm ·ρ induced by the inner product (·,·)ρ. We write lλ,ρ2=(lρ2,(·,·)λ,ρ,·λ,ρ); then lρ2×lρ2 and lλ,ρ2×lρ2 are both separable Hilbert spaces.Note that system (1) can be rewritten as(6)d2udt2+αdudt=-λu-βsin⁡u+gt+Bθtωu+cu∘dwtdt,i∈Z,uτ=uτ,u˙τ=u1τ,where u=(ui)i∈Z, g(t)=(gi(t))i∈Z is a given time dependent sequence, and B(ω) is a linear operator defined by(7)Bωui=∑j=-qqηi,jωui+j.To convert problem (6) into a random differential equation, let z(θtω)≔-λ∫-∞0eλs(θtω)(s)ds, t∈R and ω∈Ω, which is an Ornstein-Uhlenbeck process on (Ω,F,P) and solve the Ornstein-Uhlenbeck equation dz+αzdt=dw(t,ω), where w(t,ω)=ω(t) for ω∈Ω. From [12, 14, 15], it is known that the random variables zω are tempered, and there is a θt-invariant set Ω~⊂Ω of full P measure such that t↦z(θtω) is continuous in t for every ω∈Ω~.Letv(t,ω)=u˙(t,ω)+σu(t,ω)-cuz(θtω),ω∈Ω,t∈R; then (6) can be written as the following equivalent random system with random coefficients: for every τ∈R and t>τ,(8)dudt=-σu+v+cuzθtω,dvdt=σα-σ-λu+σ-αv+Bθtω+2cσzθtω-c2z2θtωu-czθtωv-βsin⁡u+gt,uτ=uτ,vτ=vτ=u1τ+σuτ-cuτzω.We will consider (8) for ω∈Ω~ and write Ω~ as Ω from now on. In order to obtain the existence and uniqueness of solutions to problem (8), we make the following assumptions on gi and the coefficients ηi,jω,j∈-q,…,0,…,q, for i∈Z:(A1)
Lettingη(ω)=sup⁡ηi,jω:j∈{-q,…,0,…,q},q∈Nandi∈Z⩾0,η(θtω) belongs to Lloc1(R) with respect to t∈R for each ω∈Ω,(9)limt→±∞⁡1t∫0tηθsωds=0;andη(ω) is tempered.(A2)
g=(gi)i∈Z∈Lloc2(R,lρ2).(A3)
Letλ=min⁡{α/2-σ,σ/2}, and(10)ϵ=λ-λ+1σα-σ+β2λ>0.There exists a positive constanta∈(0,ϵ) such that(11)∫-∞0easgs+τρ2ds<∞,∀τ∈R.We callϕ:[τ,τ+T)→lρ2×lρ2 a mild solution of the following random lattice differential equations:(12)dϕdt=Gϕ,t,θtω,ϕ=ϕii∈Z,G=Gii∈Z,t∈τ,τ+T,τ∈R,where ω∈Ω, if ϕ∈C([τ,τ+T),lρ2×lρ2) and(13)ϕit=ϕiτ+∫τtGiϕs,s,θsωds,i∈Z,t∈τ,τ+T,τ∈R.Theorem 1.
LetT>0 and (A1)-(A2) hold. Then for every τ∈R and ω∈Ω and any initial data (uτ,vτ)∈lρ2×lρ2, problem (8) admits a unique mild solution (u(·,τ,ω,uτ,vτ),v(·,τ,ω,uτ,vτ))∈C([τ,τ+T),lρ2×lρ2) with (u(τ,τ,ω,uτ,vτ),v(τ,τ,ω,uτ,vτ))=(uτ,vτ), (u(t,τ,ω,uτ,vτ),v(t,τ,ω,uτ,vτ)) being continuous in (uτ,vτ)∈lρ2×lρ2; (u(t,τ,ω,uτ,vτ),v(t,τ,ω,uτ,vτ))∈l2×l2 if (uτ,vτ)∈l2×l2. Moreover, (8) generates a continuous cocycle Ψc over R and (Ω,F,P,{θt}t∈R) with state space lρ2×lρ2: for (uτ,vτ)∈lρ2×lρ2,t∈R+,τ∈R, and ω∈Ω,(14)Ψct,τ,ω,uτ,vτ≔ut+τ,τ,θ-τω,uτ,vτ,vt+τ,τ,θ-τω,uτ,vτ.We can prove Theorem1 by Theorem  6.1.7 in [16] and Definition  2.1 in [2]. We omit it here.


## 3. Existence of Random Attractors

We first provide some sufficient conditions for the existence of random attractors for nonautonomous RDSs in weighted spaces of infinite sequences in [2].In the following, let(X,·X) be a separable Banach space and let D(X) be the collection of all tempered families of nonempty bounded subsets of X.Definition 2.
LetΦ be a continuous cocycle on X over R and (Ω,F,P,{θt}t∈R).(1)
A familyK={K(τ,ω):τ∈R,ω∈Ω}∈D(X) is called a random absorbing set for Φ if, for all τ∈R and ω∈Ω and for every D∈D(X), there exists T=T(D,τ,ω)>0 such that(15)Φt,τ-t,θ-tω,Dτ-t,θ-tω⊆Kτ,ω∀t≥T.(2)
A familyA={A(τ,ω):τ∈R,ω∈Ω}∈D(X) is called a random attractor for Φ if for all t∈R+, τ∈R, and ω∈Ω, (i) A(τ,ω) is compact in X and is measurable in ω with respect to F; (ii) A is invariant, that is, Φ(t,τ,ω,A(τ,ω))=A(τ+t,θtω); and (iii) for every D={D(τ,ω):τ∈R,ω∈Ω}∈D(X),(16)limt→∞⁡dHΦt,τ-t,θ-tω,Dτ-t,θ-tω,Aτ,ω=0,wheredH is the Hausdorff semidistance given by dH(F,G)=supu∈Finfv∈Gu-vX, for any F,G⊂X.Theorem 3.
LetΦ be a continuous cocycle on lρ2×lρ2 over R and (Ω,F,P,{θt}t∈R). Suppose the following.(a)
There exists a bounded closed random absorbing setB0={B0(τ,ω):τ∈R,ω∈Ω}∈D(lρ2×lρ2) such that, for any τ∈R, ω∈Ω, and B={B(τ,ω):τ∈R,ω∈Ω}∈D(lρ2×lρ2), there exists T1=T1(τ,ω,B)>0 yielding Φ(t,τ-t,θ-tω,B(τ-t,θ-tω))⊂B0(τ,ω),∀t⩾T1;.(b)
For eachτ∈R, ω∈Ω and for any ε>0, there exist T2=T2(τ,ε,ω,B0)>0 and I0=I0(τ,ε,ω,B0)∈N such that(17)∑i>I0ρiΦit,τ-t,θ-tω,uτ-t2⩽ε,∀t⩾T2,uτ-t∈B0τ-t,θ-tω.Then Φ possesses a unique random attractor A in D(lρ2×lρ2) given by, for every τ∈R and ω∈Ω,(18)Aτ,ω=⋂τ⩾T1⋃t⩾τΦt,τ-t,θ-tω,B0τ-t,θ-tω¯.Next, we will use Theorem3 to prove the existence of a random attractor for the continuous cocycle Ψc in lρ2×lρ2 under conditions (A1)–(A3).Theorem 4.
If (A1)–(A3) hold, then, for every c>0, τ∈R, and ω∈Ω and for any B={B(τ,ω):τ∈R,ω∈Ω}∈D(lρ2×lρ2), there exists T=T(τ,ω,B,c)>0 such that, for all t⩾T and (uτ-t,vτ-t)∈B(τ-t,θ-tω), the solution (u,v) of (8) satisfies(19)uτ,τ-t,θ-τω,uτ-t,vτ-tρ2+vτ,τ-t,θ-τω,uτ-t,vτ-tρ2+σ2∫-t0e∫s0Hr,ωdrus,τ-t,θ-τω,uτ-t,vτ-tρ2ds⩽Ic,τ,ω,where(20)Ic,τ,ω=1+λ2λe∫-∞0Hr,ωdruτ-tρ2+vτ-tρ2+1+λαλ∫-∞0e∫s0Hr,ωdrgs+τρ2ds,(21)Ht,ω=-2ϵ+2cλ+σλ+1λzθtω+c2λ+1λz2θtω+2λ+11+q+q~ληθtω.Proof.
For eachω∈Ω, there exists a sequence ηi,j(m)(t,ω) of continuous functions in t∈R (see [17]) such that(22)limm→∞⁡∫τtηi,jms,ω-ηi,jθsωds=0,∀t>0,τ∈R,j∈-q,…,0,…,q,q∈N,i∈Z,and ηi,jmt,ω⩽ηi,jθtω⩽η(θtω) for t∈R. Consider the following random differential equations:(23)dumdt=-σum+vm+cumzθtω,dvmdt=σα-σ-λum+σ-αvm+Bmt,ωum+2cσzθtω-c2z2θtωum-czθtωvm-βsin⁡um+gt,umτ=uτ,vmτ=vτ=u1τ+σuτ-cuτzω,where (Bm(t,ω)u(m))i=∑j=-qq‍ηi,j(m)(t,ω)ui+j(m). It is easy to see that (23) has a unique mild solution (u(m)(·,τ,ω,uτ,vτ),v(m)(·,τ,ω,uτ,vτ))∈C([τ,+∞),l2×l2)∩C1((τ,+∞),l2×l2) satisfying (23). Taking the inner product of the first equation in (23) with u(m)(t) in lλ,ρ2 and the second equation with v(m)(t) in lρ2, then we have that(24)12ddtumλ,ρ2=-σum,umλ,ρ+vm,umλ,ρ+czθtωum,umλ,ρ,12ddtvmρ2=σα-σ-λum,vmρ+σ-αvm,vmρ+Bmt,ωum,vmρ+2cσzθtω-c2z2θtωum,vmρ-czθtωvm,vmρ-βsin⁡um,vmρ+gt,vmρ.Summing the two equations of (24), we find that(25)12ddtumλ,ρ2+vmρ2=-σ+czθtωumλ,ρ2+σα-σum,vmρ+σ-α-czθtωvmρ2+Bmt,ωum,vmρ+2cσzθtω-c2z2θtωum,vmρ-βsin⁡um,vmρ+gt,vmρ.Note that(26)σα-σum,vmρ⩽σα-σ21λvmρ2+λumρ2⩽σα-σ21λvmρ2+umλ,ρ2⩽σα-σ21λ+1vmρ2+umλ,ρ2,(27)2cσzθtω-c2z2θtωum,vmρ⩽cσzθtω+c2z2θtω21λvmρ2+λumρ2⩽cσzθtω+c2z2θtω21λvmρ2+umλ,ρ2⩽cσzθtω+c2z2θtω21λ+1vmρ2+umλ,ρ2,(28)-βsin⁡um,vmρ=-β∑i∈Zρivimsin⁡uim⩽β∑i∈Zρivim·uim⩽β21λvmρ2+λumρ2⩽β21λ+1vmρ2+umλ,ρ2,(29)gt,vmρ⩽12αgtρ2+α2vmρ2.By (3), we get(30)Bmt,ωum,vmρ=∑i∈Zρivim·∑j=-qqηi,jmt,ωui+jm⩽ηθtω∑i∈Zρivim·∑j=-qqui+jm⩽12ηθtω1+2qvmρ2+1+2q~umρ2⩽ηθtω1+q+q~1λ+1vmρ2+umλ,ρ2,where q~=∑k=1q‍c0k.
From (25)–(28), we obtain that, for t>0,(31)12ddtumλ,ρ2+vmρ2+σ2umλ,ρ2⩽1λ+1cσzθtω+c2z2θtω2+σα-σ2+β2+ηθtω1+q+q~vmρ2+umλ,ρ2+-σ2+czθtωumλ,ρ2+σ-α-czθtωvmρ2+12αgtρ2+α2vmρ2.Recalling that λ=min{α/2-σ,σ/2} and ϵ in (10), then we have(32)12ddtumλ,ρ2+vmρ2+σ2umλ,ρ2⩽1λ+1cσzθtω+12c2z2θtω+σα-σ2+β2+ηθtω1+q+q~vmρ2+umλ,ρ2+12αgtρ2+-σ2+czθtωumλ,ρ2+σ-α2-czθtωvmρ2⩽-λ+czθtω+λ+1λcσzθtω+c2z2θtω2+σα-σ2+β2+ηθtω1+q+q~·vmρ2+umλ,ρ2+12αgtρ2=-ϵ+cλ+σλ+1λzθtω+c2λ+12λz2θtω+λ+11+q+q~ληθtω·vmρ2+umλ,ρ2+12αgtρ2=Ht,ω2vmρ2+umλ,ρ2+12αgtρ2,where H(t,ω) is as in (21). Then we obtain from (32) that, for t>0,(33)umτ,τ-t,ω,uτ-t,vτ-tλ,ρ2+vmτ,τ-t,ω,uτ-t,vτ-tρ2+σ∫τ-tτe∫sτHr,ωdrums,τ-t,ω,uτ-t,vτ-tλ,ρ2ds⩽e∫τ-tτHr,ωdruτ-tλ,ρ2+vτ-tρ2+1α∫τ-tτe∫sτHr,ωdrgsρ2ds.From (33) and by replacing ω by θ-τω, we have(34)umτ,τ-t,θ-τω,uτ-t,vτ-tλ,ρ2+vmτ,τ-t,θ-τω,uτ-t,vτ-tρ2+σ∫-t0e∫s0Hr,ωdrums,τ-t,θ-τω,uτ-t,vτ-tλ,ρ2ds⩽e∫τ-tτHr-τ,ωdruτ-tλ,ρ2+vτ-tρ2+1α∫τ-tτe∫sτHr-τ,ωdrgsρ2ds⩽e∫-t0Hr,ωdruτ-tλ,ρ2+vτ-tρ2+1α∫-t0e∫s0Hr,ωdrgs+τρ2ds.Note that (34) holds with u(m)(τ,τ-t,θ-τω,uτ-t,vτ-t) and v(m)(τ,τ-t,θ-τω,uτ-t,vτ-t) being replaced by u(τ,τ-t,θ-τω,uτ-t,vτ-t) and v(τ,τ-t,θ-τω,uτ-t,vτ-t); then we have(35)uτ,τ-t,θ-τω,uτ-t,vτ-tρ2+vτ,τ-t,θ-τω,uτ-t,vτ-tρ2+σ∫-t0e∫s0Hr,ωdrus,τ-t,θ-τω,uτ-t,vτ-tρ2ds⩽1+λ2λe∫-t0Hr,ωdruτ-tρ2+vτ-tρ2+1+λαλ∫-t0e∫s0Hr,ωdrgs+τρ2ds.Let(36)H1t,ω=cλ+σλ+1λzθtω+c2λ+12λz2θtω+λ+11+q+q~ληθtω;then H(t,ω)=2(-ϵ+H1(t,ω)). By (9), we find that there exists T1=T1(ω)>0 such that, for t>T1, (37)∫-t0zθsωds⩽ϵλ4cλ+σλ+1t,∫-t0zθsω2ds⩽ϵλ2c2λ+1t,∫-t0ηθsωds⩽ϵλ4λ+11+q+q~t, from which, along with (uτ-t,vτ-t)∈B(τ-t,θ-tω)∈D(lρ2×lρ2), we have(38)limt→+∞⁡1+λ2λe∫-t0Hr,ωdruτ-tρ2+vτ-tρ2⩽limsupt→+∞⁡1+λ2λe-ϵt/4Bτ-t,θ-tωρ2=0.Therefore, there exists T2=T2(τ,ω,B,c)>0 such that, for all t⩾T2,(39)1+λ2λe∫-t0Hr,ωdruτ-tρ2+vτ-tρ2⩽1.Since z(θtω) and η(θtω) are tempered, then H(t,ω) is tempered. Then by (A3), we can verify that the following integrals are convergent:(40)1+λαλ∫-t0e∫s0Hr,ωdrgs+τρ2ds<∞.Thus the theorem follows from (35), (39), and (40).Theorem 5.
Assume that (A1)–(A3) hold. Then the continuous cocycle Ψc associated with (8) has a unique random attractor Ac={Ac(τ,ω):τ∈R,ω∈Ω}∈D(lρ2×lρ2).Proof.
By Theorem3, it suffices to prove that, for every ε>0, c>0, τ∈R, and ω∈Ω and for any B={B(τ,ω):τ∈R,ω∈Ω}∈D(lρ2×lρ2), there exist T=T(τ,ω,B,c,ε)>0 and R=R(τ,ω,c,ε)>1 such that, for all t⩾T and (uτ-t,vτ-t)∈B(τ-t,θ-tω), the solution (u,v) of (8) satisfies(41)∑i>Rρiuiτ,τ-t,θ-τω,uτ-t,vτ-t2+viτ,τ-t,θ-τω,uτ-t,vτ-t2⩽ε.
Choose a smooth increasing functionχ:R+→[0,1] such that(42)χs=0,0⩽s⩽1,1,s⩾2,and there exists a positive constant cχ such that χ′s⩽cχ for s∈R+.
Let(u(τ,τ-t,ω,uτ-t,vτ-t),v(τ,τ-t,ω,uτ-t,vτ-t)) be a mild solution of (8) with (uτ-t,vτ-t)∈lρ2×lρ2. For any given N>0 define QN:lρ2×lρ2→l2×l2, (u,v)=(ui,vi)i∈Z↦QN(u,v)=((QNu)i,(QNv)i)i∈Z by ((QNu)j,(QNv)j)=(uj,vj) if j⩽N and ((QNu)j,(QNv)j)=(0,0) otherwise.
For anyn⩾1, let (u(m),v(m))=(u(m)(τ,τ-t,ω,Qnuτ-t,Qnvτ-t), v(m)(τ,τ-t,ω,Qnuτ-t,Qnvτ-t))=(ui(m),vi(m))i∈Z be the solution of (23).
Then taking the inner product of the first equation in (23) with χ^(r)u(m)=χi/ruimi∈Z in lλ,ρ2 and the second equation with χ^(r)v(m)=χi/rvimi∈Z in lρ2, we have that(43)12ddt∑i∈Zρiχirλuim2+vim2=-σum,χ^rumλ,ρ+vm,χ^rumλ,ρ+czθtωum,χ^rumλ,ρ+σα-σ-λum,χ^rvmρ+σ-αvm,χ^rvmρ+Bmt,ωum,χ^rvmρ+2cσzθtω-c2z2θtωum,χ^rvmρ-czθtωvm,χ^rvmρ-βsin⁡um,χ^rvmρ+gt,χ^rvmρ=λ-σ+czθtω∑i∈Zρiχiruim2+σα-σum,χ^rvmρ+σ-α-czθtω∑i∈Zρiχirvim2+Bmt,ωum,χ^rvmρ+2cσzθtω-c2z2θtωum,χ^rvmρ-βsin⁡um,χ^rvmρ+gt,χ^rvmρ.For each term of (43), it has been checked that(44)um,χ^rvmρ=∑i∈Zρiχiruim·vim⩽1+λ2λ∑i∈Zρiχirλuim2+vim2,Bmt,ωum,χ^rvmρ=∑i∈Zρiχirvim∑j=-qqηi,jmt,ωui+jm⩽ηθtω2∑i∈Zρiχir1+2qvim2+∑j=-qqui+jm2⩽ηθtω2∑i∈Zρi∑j=-qqχir-χi+jrui+jm2+χi+jrui+jm2+1+2qηθtω2∑i∈Zρiχirvim2⩽cχq1+2q~ηθtω2rumρ2+1+2q~ηθtω2∑i∈Zρiχiruim2+1+2qηθtω2∑i∈Zρiχirvim2⩽cχq1+2q~ηθtω2rumρ2+1+λ1+q+q~ηθtωλ∑i∈Zρiχirλuim2+vim2,-βsin⁡um,χ^rvmρ=-β∑i∈Zρiχirsin⁡uim·vim⩽β∑i∈Zρiχiruim·vim⩽β1+λ2λ∑i∈Zρiχirλuim2+vim2,gt,χ^rvmρ=∑i∈Zρiχirgit·vim⩽12α∑i∈Zρiχirgit2+α2∑i∈Zρiχirvim2.Putting above inequalities into (43), we obtain(45)12ddt∑i∈Zρiχirλuim2+vim2⩽1+λλcσzθtω+12c2z2θtω+σα-σ2+β2+ηθtω1+q+q~·∑i∈Zρiχirλuim2+vim2+λ-σ+czθtω∑i∈Zρiχiruim2+σ-α2-czθtω∑i∈Zρiχirvim2+12α∑i∈Zρiχirgit2+cχq1+2q~ηθtω2rumρ2⩽Ht,ω2∑i∈Zρiχirλuim2+vim2+12α∑i∈Zρiχirgit2+cχq1+2q~ηθtω2rumρ2,where H(t,ω) is as in (21). Multiplying (45) by e∫0t‍H(r,ω)dr and then integrating over [τ-t,τ] with t>0, we obtain(46)∑i∈Zρiχirλuimτ,τ-t,ω,Qnuτ-t,Qnvτ-t2+vimτ,τ-t,ω,Qnuτ-t,Qnvτ-t2⩽e∫τ-tτHr,ωdrλQnuτ-tρ2+Qnvτ-tρ2+1α∑i∈Zρiχir∫τ-tτe∫sτHr,ωdrgis2ds+cχq1+2q~r∫τ-tτe∫sτHr,ωdrηθsωus,τ-t,ω,Qnuτ-t,Qnvτ-tρ2ds.Replacing ω and m in (46) by θ-τω and mk, respectively, and letting k→∞, then we have that(47)∑i∈Zρiχirλuiτ,τ-t,θ-τω,Qnuτ-t,Qnvτ-t2+viτ,τ-t,θ-τω,Qnuτ-t,Qnvτ-t2⩽e∫-t0Hr,ωdrλQnuτ-tρ2+Qnvτ-tρ2+1α∑i∈Zρiχir∫-t0e∫s0Hr,ωdrgis+τ2ds+cχq1+2q~r∫-t0e∫s0Hr,ωdrηθsωus+τ,τ-t,θ-τω,Qnuτ-t,Qnvτ-tρ2ds.We now estimate each term on the right-hand side of (47). For the first term on the right-hand side of (47), since (Qnuτ-t,Qnvτ-t)∈B(τ-t,θ-tω) and B is tempered, then there exists T1=T1(τ,ε,ω,B)>0 such that if t>T1, then(48)e∫-t0Hr,ωdrλQnuτ-tρ2+Qnvτ-tρ2⩽ε.For the second term on the right-hand side of (47), by (A3), there exist R1=R1(ε,ω)>0 and T2=T2(ε,ω)>0, such that if r>R1 and t>T2, then(49)1α∑i∈Zρiχir∫-t0e∫s0Hr,ωdrgis+τ2ds⩽ε.For the third term on the right-hand side of (47), by (A1) and Theorem 4, there exist T3=T3(τ,ε,ω,B)>0 and R2=R2(ε,ω)>0 such that, for all t>T3 and r>R2,(50)cχq1+2q~r∫-t0e∫s0Hr,ωdrηθsωus+τ,τ-t,θ-τω,Qnuτ-t,Qnvτ-tρ2ds⩽ε.Let T=max⁡{T1,T2,T3} and R=2max⁡{R1,R2}. Then it follows from (48)–(50) that, for all t>T and r>R, we obtain(51)∑i>Rρiλuiτ,τ-t,θ-τω,Qnuτ-t,Qnvτ-t2+viτ,τ-t,θ-τω,Qnuτ-t,Qnvτ-t2⩽3ε,for any n⩾1. Let n→∞; we have that (41) holds. This completes the proof.


## 4. Upper Semicontinuity of Random Attractors

First, we present a criteria concerning upper semicontinuity of nonautonomous random attractors with respect to a parameter in [18].Theorem 6.
LetΦc be a continuous cocycle on X over R and (Ω,F,P,{θt}t∈R). Suppose the following.
(i)Φc has a closed measurable random absorbing set Kc={Kc(τ,ω):τ∈R,ω∈Ω} in D(X) and a unique random attractor Ac={Ac(τ,ω):τ∈R,ω∈Ω} in D(X).
(ii) For eachτ∈R and ω∈Ω, K0(τ,ω)={u∈X:uX⩽r0(τ,ω)} and(52)limsupc→0⁡Kcτ,ωX=limsupc→0⁡supx∈Kcτ,ω⁡xX⩽r0τ,ω,where r0(τ,ω) is a positive-valued tempered random variable.
(iii) There existsε>0 such that, for every τ∈R and ω∈Ω, ⋃c⩽ε‍Ac(τ,ω) is precompact in X.
(iv) Fort>0, τ∈R, ω∈Ω, cn→0 when n→∞, and xn,x0∈X with xn→x0 when n→∞, it holds:(53)limn→∞⁡Φcnt,τ,ω,xn=Φ0t,τ,ω,x0.
Then forτ∈R and ω∈Ω,(54)dHAcτ,ω,A0τ,ω=supu∈Acτ,ω⁡infv∈A0τ,ω⁡u-vX⟶0asc⟶0.Next, we will use Theorem6 to consider an upper semicontinuity of random attractors Ac(τ,ω) to A0(τ,ω) as c→0.Theorem 7.
Assume that (A1)–(A3) hold. Then for every τ∈R and ω∈Ω, one has(55)dHAcτ,ω,A0τ,ω=supuc,vc∈Acτ,ω⁡infu0,v0∈A0τ,ω⁡uc-u0ρ2+vc-v0ρ21/2⟶0asc⟶0.Proof.
LetI(c,τ,ω) be as in Theorem 4.
(i) By Theorems4 and 5, Φc has a closed measurable random absorbing set Bc={Bc(τ,ω):τ∈R,ω∈Ω}∈D(lρ2×lρ2), where Bc(τ,ω)={(uc,vc)∈lρ2×lρ2:ucρ2+vcρ2⩽I(c,τ,ω)}, and a unique random attractor Ac={Ac(τ,ω):τ∈R,ω∈Ω} in D(lρ2×lρ2), for each τ∈R and ω∈Ω, Ac(τ,ω)⊆Bc(τ,ω).
(ii) Givenc<1. By (20), we have(56)Ic,τ,ω⩽I1,τ,ω<∞,limsupc→0⁡Ic,τ,ω⩽I1,τ,ω.So, for every τ∈R and ω∈Ω,(57)limsupc→0Bcτ,ωρ=limsupc→0supx∈Bcτ,ωxlρ2×lρ2⩽I1/21,τ,ω. Moreover, B0(τ,ω)={(u0,v0)∈lρ2×lρ2:u0ρ2+v0ρ2⩽I(1,τ,ω)} is a closed tempered random absorbing set for the continuous cocycle Φ0 associated with the limiting system:(58)du0dt=-σu0+v0,dv0dt=σα-σ-λu0+σ-αv0+Bθtωu0-βsin⁡u0+gt,u0τ=uτ0,v0τ=vτ0=u1τ0+σuτ0,(59)⋃c⩽1Acτ,ω⊆⋃c⩽1Bcτ,ω⊆B0τ,ω.
(iii) Givenc<1. Let us prove the precompactness of ⋃c⩽1‍Ac(τ,ω) for every τ∈R and ω∈Ω. For one thing, by Theorem 5, for every ε>0, c>0, τ∈R, and ω∈Ω, there exist T=T(τ,ω,B,c,ε)>0 and R=R(τ,ω,c,ε)>1 such that, for all t⩾T and (uτ-tc,vτ-tc)∈B0(τ-t,θ-tω), the solution (uc,vc) of (8) satisfies(60)∑i>Rρiuicτ,τ-t,θ-τω,uτ-tc,vτ-tc2+vicτ,τ-t,θ-τω,uτ-tc,vτ-tc2⩽ε,from which, along with (59) and the invariance of Ac(τ,ω), we have, for every τ∈R, ω∈Ω, and t⩾T,(61)supuc,vc∈⋃c⩽1Acτ,ω⁡∑i>Rρiuic2+vic2⩽ε.For another, by (59) we find that the set {(uic,vic)i⩽R:(uc,vc)∈⋃c⩽1‍Ac(τ,ω)} is bounded in a finite-dimensional space and hence ⋃c⩽1‍Ac(τ,ω) is precompact in lρ2×lρ2.
(iv) Letφc=(uc,vc) and φ0=(u0,v0) be mild solutions of (8) and (58) with initial data (uτc,vτc) and (uτ0,vτ0), respectively, and set U=uc-u0, V=vc-v0, (U,V)=φc-φ0.
Let(uc,m(t,τ,ω,uτc,vτc),vc,m(t,τ,ω,uτc,vτc)) and (u0,m(t,τ,ω,uτ0,vτ0),v0,m(t,τ,ω,uτ0,vτ0)) be the solutions of the following random differential equations with initial data:(62)duc,mdt=-σuc,m+vc,m+cuc,mzθtω,dvc,mdt=σα-σ-λuc,m+σ-αvc,m+Bmt,ωuc,m+2cσzθtω-c2z2θtωuc,m-czθtωvc,m-βsin⁡uc,m+gt,uc,mτ=uτc,vc,mτ=vτc=u1τc+σuτc-cuτczω,(63)du0,mdt=-σu0,m+v0,m,dv0,mdt=σα-σ-λu0,m+σ-αv0,m+Bmt,ωu0,m-βsin⁡u0,m+gt,u0,mτ=uτ0,v0,mτ=vτ0=u1τ0+σuτ0,respectively. Then, φc,m(·,τ,ω,uτc,vτc), φ0,m(·,τ,ω,uτ0,vτ0)∈C([τ,+∞),lρ2×lρ2) and satisfy the differential equations (62) and (63), respectively. Moreover, φc(t,τ,ω,uτc,vτc) and φ0(t,τ,ω,uτ0,vτ0) are limit functions of subsequences of {φc,m(t,τ,ω,uτc,vτc)} and {φ0,m(t,τ,ω,uτ0,vτ0)}∈lρ2×lρ2. So φc(t,τ,ω,uτc,vτc)-φ0(t,τ,ω,uτ0,vτ0) is a limit function of a subsequence of {φc,m(t,τ,ω,uτc,vτc)-φ0,m(t,τ,ω,uτ0,vτ0)} in lρ2×lρ2, and (Um(t,τ,ω,uτc,vτc,uτ0,vτ0),Vm(t,τ,ω,uτc,vτc,uτ0,vτ0)) satisfies(64)dUmdt=-σUm+Vm+czθtωuc,m,dVmdt=σα-σ-λUm+σ-αVm+Bmt,ωUm+2cσzθtω-c2z2θtωuc,m-czθtωvc,m-βsin⁡uc,m+βsin⁡u0,m,Umτ=uτc-uτ0,Vmτ=vτc-vτ0.Now let us estimate Umρ2+Vmρ2=uc,m-u0,mρ2+vc,m-v0,mρ2. Taking the inner product of (64) with Um(t) and Vm(t) in lλ,ρ2 and lρ2, respectively, then we get(65)12ddtUmλ,ρ2+Vmρ2=-σUmλ,ρ2+cuc,mzθtω,Umλ,ρ+σα-σUm,Vmρ+σ-αVmρ2+Bmt,ωUm,Vmρ+2cσzθtω-c2z2θtωuc,m,Vmρ-czθtωvc,m,Vmρ+βsin⁡u0,m-sin⁡uc,m,Vmρ.We now estimate the terms in (65):(66)czθtωuc,m,Umλ,ρ⩽c2z2θtω2σuc,mλ,ρ2+σ2Umλ,ρ2,σα-σUm,Vmρ⩽σα-σλ+12λUmλ,ρ2+Vmρ2,Bmt,ωUm,Vmρ⩽ηθtω1+q+q~λ+1λUmλ,ρ2+Vmρ2,2cσzθtω-c2z2θtωuc,m,Vmρ⩽α4Vmρ2+2cσzθtω-c2z2θtω2αuc,mρ2,czθtωvc,m,Vmρ⩽α4Vmρ2+c2z2θtωαvc,mρ2,βsin⁡u0,m-sin⁡uc,m,Vmρ⩽βλ+12λVmρ2+Umλ,ρ2.Then it follows from (66) that(67)12ddtUmλ,ρ2+Vmρ2⩽λ+1λσα-σ2+ηθtω1+q+q~+β2Umλ,ρ2+Vmρ2+c2z2θtω2σuc,mλ,ρ2-σ2Umλ,ρ2+σ-α2Vmρ2+2cσzθtω-c2z2θtω2αuc,mρ2+c2z2θtωαvc,mρ2⩽-λ+λ+1λσα-σ2+ηθtω1+q+q~+β2Umλ,ρ2+Vmρ2+λc2z2θtω2σ+2cσzθtω-c2z2θtω2αuc,mρ2+c2z2θtωαvc,mρ2=-ϵ+λ+1ηθtω1+q+q~λUmλ,ρ2+Vmρ2+H2c,t,ωuc,mρ2+vc,mρ2,where λ=min{α/2-σ,σ/2}, ϵ is as in (10) and H2(c,t,ω)=λc2z2(θtω)/2σ+(2cσz(θtω)-c2z2(θtω))2/α+c2z2(θtω)/α. Applying Gronwall’s inequality to (67) from τ to t+τ, we have(68)Umt+τ,τ,ω,uτc,vτc,uτ0,vτ0λ,ρ2+Vmt+τ,τ,ω,uτc,vτc,uτ0,vτ0ρ2⩽e2∫τt+τ-ϵ+λ+1ηθrω1+q+q~/λdruτc-uτ0λ,ρ2+vτc-vτ0ρ2+2∫τt+τe2∫st+τ-ϵ+λ+1ηθrω1+q+q~/λdrH2c,s,ω·uc,ms,τ,ω,uτc,vτcρ2+vc,ms,τ,ω,uτc,vτcρ2ds.We now replace ω in the above by θ-τω to yield (69)Umt+τ,τ,θ-τω,uτc,vτc,uτ0,vτ0λ,ρ2+Vmt+τ,τ,θ-τω,uτc,vτc,uτ0,vτ0ρ2⩽e2∫0t-ϵ+λ+1ηθrω1+q+q~/λdruτc-uτ0λ,ρ2+vτc-vτ0ρ2+2∫0te2∫st-ϵ+λ+1ηθrω1+q+q~/λdrH2c,s,ω·uc,ms,τ,θ-τω,uτc,vτcρ2+vc,ms,τ,θ-τω,uτc,vτcρ2ds.From (69), we see that, for τ∈R, t∈R+, ω∈Ω, c→0, and (uτc,vτc), (uτ0,vτ0)∈lρ2×lρ2 with (uτc,vτc)→(uτ0,vτ0),(70)limc→0⁡φc,mt+τ,τ,θ-τω,uτc,vτc=φ0,mt+τ,τ,θ-τω,uτ0,vτ0inlρ2×lρ2.Let {cn}⊂[-1,1] be a sequence of numbers with cn→0 when n→+∞. Then φc(t+τ,τ,θ-τω,uτc,vτc) and φ0(t+τ,τ,θ-τω,uτ0,vτ0) being limit functions of subsequences of {φc,m(t+τ,τ,θ-τω,uτc,vτc)} and {φ0,m(t+τ,τ,θ-τω,uτ0,vτ0)} in lρ2×lρ2 imply that, for τ∈R, t∈R+, ω∈Ω, cn→0, and (uτcn,vτcn), (uτ,vτ)∈lρ2×lρ2 with (uτcn,vτcn)→(uτ0,vτ0); the following holds:(71)limn→∞⁡φcnt+τ,τ,θ-τω,uτcn,vτcn=φ0t+τ,τ,θ-τω,uτ0,vτ0inlρ2×lρ2.The proof is completed.



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*Source: 7986203-2016-08-10.xml*