# Pullback Attractor for Nonautonomous Ginzburg-Landau Equation with Additive Noise

Authors: Yangrong Li; Hongyong Cui
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Long time behavior of stochastic Ginzburg-Landau equations with nonautonomous deterministic external forces, dispersion coefficients, and nonautonomous perturbations is studied. The domain is taken as a bounded intervalI in R. By making use of Sobolev embeddings and Gialiardo-Nirenberg inequality we obtain the existence and upper semicontinuity of the pullback attractor in L2(I) for the equation. The upper semicontinuity shows the stability of attractors under perturbations.


---

## Body

## 1. Introduction

Consider the following stochastic complex Ginzburg-Landau equation with nonautonomous deterministic external force and dispersion coefficients:(GL)du=[(λ+iα(t))Δu-(κ+iβ(t))|u|2u+δu+g(x,t)]dt+ɛh(x)dW(t),
where u(x,t) is unknown andcomplex-valued; x∈I=]0,1[⊆R; dispersion coefficients α(t), β(t) and external force g(x,t) are all time-dependent; λ, κ, δ, and ɛ are positive real constants; h(x) is a given complex-valued function; W(t) is supposed to be a real-valued Wiener process on a complete probability space. (For detail of conditions see Lemma 12.)As a kind of Schrödinger type equation, complex Ginzburg-Landau (G.-L.) equations arise in various areas of physics and chemistry as pointed out by Temam [1]. Due to their rich mathematical properties, Ginzburg-Landau equations have drawn much attention of mathematicians and many related articles appear in literature, such as [1–7]. In [1–5], existence and attractors fordeterministic G.-L. equations are studied and in [6, 7], the authors take perturbations into consideration and have investigated the existence of attractors of solutions.This paper deals with anonautonomous case, equation (GL). Note that, except for perturbations, external force g(t) and dispersions α(t) and β(t) are all time-dependent. To investigate the long time behavior of solutions for such stochastic equations with nonautonomous deterministic terms, Wang [8, 9] generalized the theory on existence and upper semicontinuity of attractors for random dynamical systems (RDS), which provides us with abstract criteria (see Lemmas 10 and 11) to investigate system (GL).Since the spatial domainI is bounded and the G.-L. equation has smoothing effect, Sobolev embeddings are applied to obtain the asymptotic compactness of the system, as well as constructing a compact random absorbing set, which is needed to investigate not only the existence, but also the upper semicontinuity (or the stability at ɛ=0) of pullback attractors for G.-L. system (GL). Gialiardo-Nirenberg inequality also contributes to estimation of the nonlinear term.The main result of this paper contains Theorems20 and 22.This paper is organized as follows. In Section2 we recall some concepts and abstract results on attractors of RDS and define a cocycle for nonautonomous system (GL). Uniform estimates of solutions are established in Section 3. The main result of this paper is concluded in the last section.Notations. In this paper, any function space should be considered complex-valued except that there are particular explanations. We denote by ∥·∥X the norm of any (complex- or real-valued) Banach space X. The norm of a q-times Lebesgue integrable space Lq(I), 0<q⩽∞, is denoted by ∥·∥q for short and ∥·∥2≕∥·∥. We often write H=L2(I) and V=H01(I) for convenience and denote the inner product of H by (·,·) with (u,v)=∫I‍uv¯dx, for all u,v∈H. To investigate the stability of attractors, we use subscript “ɛ” or superscript “ɛ” to indicate the dependence on ɛ when necessary. Letter c stands for positive constants independent of ɛ and may change its value from line to line; c(⋯) indicates the exclusive dependence on the parameters in the bracket.


## 2. Preliminary Result and the Cocycle

### 2.1. Pullback Attractors for Cocycles

In this part we recall some theory of pullback attractors for RDS with nonautonomous deterministic terms. The reader is referred to [8–10] for more details.Suppose we are given a nonempty unbounded subsetT of R, a probability space (Ω,F,P) and a Banach space (X,∥·∥X) with Borel σ-algebra B(X).Definition 1.
A nonautonomous random setB={B(τ,ω)}τ,ω≔{B(τ,ω):τ∈T,ω∈Ω} of X is a set-valued mapping: T×Ω→2X which is measurable with respect to F in Ω. That is, the value B(τ,ω) of every (τ,ω)∈T×Ω is a closed nonempty subset of X and the mapping ω∈Ω→d(x,B(τ,ω)) is (F,B(R))-measurable for each fixed x∈X and τ∈T.Definition 2.
Suppose there are two groups{Θs}s∈R and {θs}s∈R acting on T and Ω, respectively. (T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R) are called two parametric dynamical systems if (i)
Θ:R×T→T is a mapping such that Θ0 is the identity operator on T;(ii)
Θs+t=Θt∘Θs for all t,s∈R;(i′)
θ:R×Ω→Ω is a (B(R)×F,F)-measurable mapping such that θ0 is the identity operator on Ω;(ii′)
θs+t=θs∘θt for all s,t∈R and θsP=P for all s∈R.Definition 3.
ClassDγ is defined for any fixed γ∈R+ as the collection of all nonempty and bounded nonautonomous random sets of X, of which every member B={B(τ,ω)}τ,ω satisfies
(1)lim⁡t→+∞e-γt∥B(Θ-tτ,θ-tω)∥X2=0foreachτ∈TandP-a.e.ω∈Ω,
where ∥B∥X=sup⁡x∈B∥x∥X.Definition 4.
Let(T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R) be two parametric dynamical systems. A mapping ϕ:R+×T×Ω×X→X is called a cocycle on X over (T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R) if for all τ∈T, ω∈Ω, and s,t∈R+, it satisfies that (i)
ϕ(·,τ,·)·:R+×Ω×X→X is (B(R+)×F×B(X),B(X))-measurable;(ii)
ϕ(0,τ,ω) is the identity operator on X;(iii)
ϕ(t+s,τ,ω)=ϕ(t,Θsτ,θsω)∘ϕ(s,τ,ω). If, in addition, ϕ(t,τ,ω):X→X is always a continuous mapping on X, ϕ is called a continuous cocycle.Let in the sequelϕ be a continuous cocycle on a Banach space X over (T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R).Definition 5.
SupposeE={E(τ,ω)}τ,ω∈Dγ. Then E is called a Dγ-pullback absorbing set for ϕ if for all τ∈T, P-a.e. ω∈Ω and, for every B∈Dγ, there exists a T=T(B,τ,ω)>0 such that
(2)ϕ(t,Θ-tτ,θ-tω)B(Θ-tτ,θ-tω)⊆E(τ,ω)∀t⩾T.
If, in addition, for each τ∈T and P-a.e. ω∈Ω, E(τ,ω) is a closed subset of X and E is measurable with respect to F in Ω, then we say E is a closed measurable Dγ-pullback absorbing set for ϕ.Definition 6.
ϕ is said to be Dγ-pullback asymptotically compact in X if for all τ∈T and P-a.e. ω∈Ω, sequence {ϕ(tn,Θ-tnτ,θ-tnω,xn)}n=1∞ has a convergent subsequence in X whenever xn∈B(Θ-tnτ,θ-tnω) with {B(τ,ω)}τ,ω∈Dγ and tn→∞.Definition 7.
A random variableR(ω):Ω→(0,∞) is called tempered with respect to {θt}t∈R if
(3)lim⁡t→±∞1tlog⁡R(θtω)=0P-a.s.,
where t→-∞ applies only to two-sided time.Proposition 8.
By Definition7 the following properties hold true. (i)
If a random variableR(ω) is tempered, then
(4)lim⁡t→+∞e-ϵtR(θ-tω)=0,lim⁡t→+∞e-ϵtR(θtω)=0∀ϵ>0.(ii)
If random variablesR1(ω) and R2(ω) are both tempered, so are R1(ω)+R2(ω) and R1(ω)R2(ω).Definition 9.
A nonautonomous random setA={A(τ,ω)}τ,ω of X is called a Dγ-pullback attractor for ϕ if the following conditions are satisfied for every τ∈T and P-a.e. ω∈Ω. Consider (i)
A∈Dγ and A(τ,ω) is compact in X;(ii)
A is invariant; that is,
(5)ϕ(t,τ,ω)A(τ,ω)=A(Θtτ,θtω),∀t⩾0;(iii)
A attracts every set in Dγ; that is, for every B={B(τ,ω)}τ,ω∈Dγ,
(6)lim⁡t→∞dist⁡X(ϕ(t,Θ-tτ,θ-tω)B(Θ-tτ,θ-tω),A(τ,ω))=0,wheredist⁡X(·,·) denotes the Hausdorff semimetric in X defined as
(7)dist⁡X(Y,Z)=sup⁡y∈Yinf⁡z∈Z∥y-z∥X,∀Y,Z⊆X.The following results are borrowed from Wang [8–11].Lemma 10.
Letϕ be a continuous cocycle on X over (Ω,F,P,{θs}s∈R) and (T,{Θs}s∈R). Then ϕ has a unique Dγ-pullback attractor A in Dγ if and only if ϕ is Dγ-pullback asymptotically compact in X and ϕ has a closed measurable Dγ-pullback absorbing set E in Dγ.Given a metric spaceΛ. For every λ∈Λ, let ϕλ be a continuous cocycle over (T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R).Lemma 11.
Suppose for eachλ∈Λ, ϕλ has a Dγ-pullback attractor Aλ={Aλ(τ,ω)}τ,ω, particularly, Aλ0={Aλ0(τ)}τ; then for every τ∈R and ω∈Ω,
(8)distX(Aλ(τ,ω),Aλ0(τ,ω))⟶0asλ⟶λ0inΛ,
if provided (i)
for everyt∈R+, τ∈R, ω∈Ω, and λn∈Λ with λn→λ0, xn, x∈X with xn→x, it holds that
(9)lim⁡n→∞ϕλn(t,τ,ω)xn=ϕλ0(t,τ,ω)x;(ii)
there exists a mappingR0(·,·):R×Ω→R such that the family
(10)B={B(τ,ω)={x∈X:∥x∥X⩽R0(τ,ω)}:τ∈R,ω∈Ω}belongstoDγ;(iii)
λ∈Λ, ϕλ has a Dγ-pullback absorbing set Eλ={Eλ(τ,ω)}τ,ω such that for all τ∈R and ω∈Ω,
(11)limsup⁡λ→λ0∥Eλ(τ,ω)∥X⩽R0(τ,ω),
where R0 is as in (ii) and ∥S∥X=sup⁡x∈S∥x∥X, for all S⊆X;(iv)
for everyτ∈R and ω∈Ω,
(12)⋃λ∈Λ‍Aλ(τ,ω)isprecompactinX.


### 2.2. Nonautonomous Stochastic G.-L. Equations

Givenτ∈R and t⩾τ, consider equations defined for x∈I=]0,1[⊆R,
(13)du=[(λ+iα(t))Δu-(κ+iβ(t))|u|2u+δu+|u|2g(x,t)]dt+ɛh(x)dW(t),
with initial-boundary value conditions
(14)u(x,τ)=u0(x),u(x,·)|∂I≡0,
where the unknown u(x,t) is a complex-valued function; dispersion coefficients α(t), β(t), and external force g(x,t) are all time-dependent and real-valued functions; λ, κ, δ, and ɛ∈]0,1] are positive constants; h(x)∈H01(I)∩H2(I); W(t) is supposed to be a two-sided real-valued Wiener process on a complete probability space (Ω,F,P), where Ω={ω∈C(R,R):ω(0)=0},F is Borel σ-algebra induced by the compact open topology of Ω, and P is the corresponding Wiener measure on (Ω,F).To define a proper cocycle for system (13), we first define the related parametric dynamic systems as follows. Let {Θs}s∈R be the group acting on R satisfying
(15)Θsτ=τ+s∀s,τ∈R,
and {θs}s∈R the group acting on (Ω,F,P) such that
(16)θsω(·)=ω(·+s)-ω(s)∀ω∈Ω,s∈R,
where we have identified W(t) with
(17)W(t,ω)=ω(t),t⩾τ.
Then by Definition 2 it is evident that (R,{Θs}s∈R) and (Ω,F,P,(θs)s∈R) are two parametric dynamical systems.We now define a continuous cocycleϕ over (R,{Θs}s∈R) and (Ω,F,P,{θs}s∈R) for system (13). Consider the one-dimensional Ornstein-Uhlenbeck equation
(18)dy-δydt=dW,
of which a stationary solution is provided by
(19)y(t)=y(θtω)≡-δ∫0∞e-δτ(θtω)(τ)dτ,t∈R.
It is known that there exists a θt-invariant set Ω~⊆Ω with P(Ω~)=1 such that y(θtω) is continuous in t for every ω∈Ω~, and the random variable |y(θtω)| is tempered (see, e.g., [8, 10, 12, 13], and hereafter we will not distinguish Ω~ from Ω). Therefore, by Proposition 8 and [12, Proposition 4.3.3] (see also [14–16]), there exists a tempered variable r(ω)>0 such that
(20)|y(ω)|6⩽r(ω),
where r(ω) satisfies
(21)r(θtω)⩽e(γ/2)|t|r(ω),t∈R.
Let z(θtω)=hy(θtω), f(s)=|s|2s, and
(22)v(t,τ,ω,v0)=u(t,τ,ω,u0)-ɛz(θtω)withv0=u0-ɛz(θτω).
Then if u(t) solves (13)-(14), v(t) should satisfy, by (18), (22), and h(x)∈H01(I)∩H2(I),
(23)dvdt=(λ+iα(t))Δv-(κ+iβ(t))f(v+ɛz(θtω))+δv+g(x,t)+ɛ(λ+iα(t))Δz(θtω),
with conditions
(24)v(τ,τ,ω,v0)=v0=u0-ɛz(θτω),v(t,τ,ω,v0)|∂I=0,
for all t⩾τ∈R and x∈I=]0,1[. Since (23)-(24) is a deterministic problem, by the “standard” Galërkin method as in [17] or similar arguments of [2] (see also [1, 5, 6] for autonomous G.-L. equations), we have the following well-possessedness result.Lemma 12.
Assume that(i)
λ∈R+, κ∈R+, δ∈R+, ɛ∈]0,1], h(x)∈H01(I;C)∩H2(I;C);(ii)
β(t)∈Cloc(R;R) and sup⁡t∈R|β(t)|⩽3κ;(iii)
α(t)∈Cloc(R;R) and g(x,t)∈Lloc2(R;H) are such that
(25)∫-∞τe(γ/2)s(|α(t)|4+∥g(x,s)∥2)ds<+∞foreveryτ∈R.Then, for each v0∈H, the initial-boundary value problem (23)-(24) has a unique weak solution
(26)v(t,τ,ω,v0)∈Cloc(τ,∞;H)∩Lloc2(τ,∞;V)∩Lloc4(τ,∞;L4(I)).
Besides, v(t,τ,ω,v0) is (F,B(H))-measurable in ω∈Ω and continuous in v0 with respect to the norm of H for each t⩾τ.Let(27)u(t,τ,ω,u0)=v(t,τ,ω,v0)+ɛz(θtω)withu0=v0+ɛz(θτω).
Then under assumptions of Lemma 12 it is evident that u solves problem (13)-(14) and is (F,B(H))-measurable in ω∈Ω, continuous in both t⩾τ, and u0∈H. Consider the mapping ϕ:R+×R×Ω×H→H with
(28)ϕ(t,τ,ω,u0)=u(t+τ,τ,θ-τω,u0)=v(t+τ,τ,θ-τω,v0)+ɛz(θtω),
where v0=u0-ɛz(ω). By the property of solution trajectories of well-possessed nonautonomous dynamical systems one can readily check according to Definition 4 that (28) defines a continuous cocycle ϕ for problem (13)-(14) on H over (R,{Θs}) and (Ω,F,P,{θs}), where {Θs}s∈R and {θs}s∈R are given by (15) and (16), respectively.To investigate the (Dγ-) pullback attractor in H for system (13), hereafter in this paper, we let, for an arbitrarily fixed γ>0, Dγ be the class satisfying Definition 3 with X=H.


## 2.1. Pullback Attractors for Cocycles

In this part we recall some theory of pullback attractors for RDS with nonautonomous deterministic terms. The reader is referred to [8–10] for more details.Suppose we are given a nonempty unbounded subsetT of R, a probability space (Ω,F,P) and a Banach space (X,∥·∥X) with Borel σ-algebra B(X).Definition 1.
A nonautonomous random setB={B(τ,ω)}τ,ω≔{B(τ,ω):τ∈T,ω∈Ω} of X is a set-valued mapping: T×Ω→2X which is measurable with respect to F in Ω. That is, the value B(τ,ω) of every (τ,ω)∈T×Ω is a closed nonempty subset of X and the mapping ω∈Ω→d(x,B(τ,ω)) is (F,B(R))-measurable for each fixed x∈X and τ∈T.Definition 2.
Suppose there are two groups{Θs}s∈R and {θs}s∈R acting on T and Ω, respectively. (T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R) are called two parametric dynamical systems if (i)
Θ:R×T→T is a mapping such that Θ0 is the identity operator on T;(ii)
Θs+t=Θt∘Θs for all t,s∈R;(i′)
θ:R×Ω→Ω is a (B(R)×F,F)-measurable mapping such that θ0 is the identity operator on Ω;(ii′)
θs+t=θs∘θt for all s,t∈R and θsP=P for all s∈R.Definition 3.
ClassDγ is defined for any fixed γ∈R+ as the collection of all nonempty and bounded nonautonomous random sets of X, of which every member B={B(τ,ω)}τ,ω satisfies
(1)lim⁡t→+∞e-γt∥B(Θ-tτ,θ-tω)∥X2=0foreachτ∈TandP-a.e.ω∈Ω,
where ∥B∥X=sup⁡x∈B∥x∥X.Definition 4.
Let(T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R) be two parametric dynamical systems. A mapping ϕ:R+×T×Ω×X→X is called a cocycle on X over (T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R) if for all τ∈T, ω∈Ω, and s,t∈R+, it satisfies that (i)
ϕ(·,τ,·)·:R+×Ω×X→X is (B(R+)×F×B(X),B(X))-measurable;(ii)
ϕ(0,τ,ω) is the identity operator on X;(iii)
ϕ(t+s,τ,ω)=ϕ(t,Θsτ,θsω)∘ϕ(s,τ,ω). If, in addition, ϕ(t,τ,ω):X→X is always a continuous mapping on X, ϕ is called a continuous cocycle.Let in the sequelϕ be a continuous cocycle on a Banach space X over (T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R).Definition 5.
SupposeE={E(τ,ω)}τ,ω∈Dγ. Then E is called a Dγ-pullback absorbing set for ϕ if for all τ∈T, P-a.e. ω∈Ω and, for every B∈Dγ, there exists a T=T(B,τ,ω)>0 such that
(2)ϕ(t,Θ-tτ,θ-tω)B(Θ-tτ,θ-tω)⊆E(τ,ω)∀t⩾T.
If, in addition, for each τ∈T and P-a.e. ω∈Ω, E(τ,ω) is a closed subset of X and E is measurable with respect to F in Ω, then we say E is a closed measurable Dγ-pullback absorbing set for ϕ.Definition 6.
ϕ is said to be Dγ-pullback asymptotically compact in X if for all τ∈T and P-a.e. ω∈Ω, sequence {ϕ(tn,Θ-tnτ,θ-tnω,xn)}n=1∞ has a convergent subsequence in X whenever xn∈B(Θ-tnτ,θ-tnω) with {B(τ,ω)}τ,ω∈Dγ and tn→∞.Definition 7.
A random variableR(ω):Ω→(0,∞) is called tempered with respect to {θt}t∈R if
(3)lim⁡t→±∞1tlog⁡R(θtω)=0P-a.s.,
where t→-∞ applies only to two-sided time.Proposition 8.
By Definition7 the following properties hold true. (i)
If a random variableR(ω) is tempered, then
(4)lim⁡t→+∞e-ϵtR(θ-tω)=0,lim⁡t→+∞e-ϵtR(θtω)=0∀ϵ>0.(ii)
If random variablesR1(ω) and R2(ω) are both tempered, so are R1(ω)+R2(ω) and R1(ω)R2(ω).Definition 9.
A nonautonomous random setA={A(τ,ω)}τ,ω of X is called a Dγ-pullback attractor for ϕ if the following conditions are satisfied for every τ∈T and P-a.e. ω∈Ω. Consider (i)
A∈Dγ and A(τ,ω) is compact in X;(ii)
A is invariant; that is,
(5)ϕ(t,τ,ω)A(τ,ω)=A(Θtτ,θtω),∀t⩾0;(iii)
A attracts every set in Dγ; that is, for every B={B(τ,ω)}τ,ω∈Dγ,
(6)lim⁡t→∞dist⁡X(ϕ(t,Θ-tτ,θ-tω)B(Θ-tτ,θ-tω),A(τ,ω))=0,wheredist⁡X(·,·) denotes the Hausdorff semimetric in X defined as
(7)dist⁡X(Y,Z)=sup⁡y∈Yinf⁡z∈Z∥y-z∥X,∀Y,Z⊆X.The following results are borrowed from Wang [8–11].Lemma 10.
Letϕ be a continuous cocycle on X over (Ω,F,P,{θs}s∈R) and (T,{Θs}s∈R). Then ϕ has a unique Dγ-pullback attractor A in Dγ if and only if ϕ is Dγ-pullback asymptotically compact in X and ϕ has a closed measurable Dγ-pullback absorbing set E in Dγ.Given a metric spaceΛ. For every λ∈Λ, let ϕλ be a continuous cocycle over (T,{Θs}s∈R) and (Ω,F,P,{θs}s∈R).Lemma 11.
Suppose for eachλ∈Λ, ϕλ has a Dγ-pullback attractor Aλ={Aλ(τ,ω)}τ,ω, particularly, Aλ0={Aλ0(τ)}τ; then for every τ∈R and ω∈Ω,
(8)distX(Aλ(τ,ω),Aλ0(τ,ω))⟶0asλ⟶λ0inΛ,
if provided (i)
for everyt∈R+, τ∈R, ω∈Ω, and λn∈Λ with λn→λ0, xn, x∈X with xn→x, it holds that
(9)lim⁡n→∞ϕλn(t,τ,ω)xn=ϕλ0(t,τ,ω)x;(ii)
there exists a mappingR0(·,·):R×Ω→R such that the family
(10)B={B(τ,ω)={x∈X:∥x∥X⩽R0(τ,ω)}:τ∈R,ω∈Ω}belongstoDγ;(iii)
λ∈Λ, ϕλ has a Dγ-pullback absorbing set Eλ={Eλ(τ,ω)}τ,ω such that for all τ∈R and ω∈Ω,
(11)limsup⁡λ→λ0∥Eλ(τ,ω)∥X⩽R0(τ,ω),
where R0 is as in (ii) and ∥S∥X=sup⁡x∈S∥x∥X, for all S⊆X;(iv)
for everyτ∈R and ω∈Ω,
(12)⋃λ∈Λ‍Aλ(τ,ω)isprecompactinX.


## 2.2. Nonautonomous Stochastic G.-L. Equations

Givenτ∈R and t⩾τ, consider equations defined for x∈I=]0,1[⊆R,
(13)du=[(λ+iα(t))Δu-(κ+iβ(t))|u|2u+δu+|u|2g(x,t)]dt+ɛh(x)dW(t),
with initial-boundary value conditions
(14)u(x,τ)=u0(x),u(x,·)|∂I≡0,
where the unknown u(x,t) is a complex-valued function; dispersion coefficients α(t), β(t), and external force g(x,t) are all time-dependent and real-valued functions; λ, κ, δ, and ɛ∈]0,1] are positive constants; h(x)∈H01(I)∩H2(I); W(t) is supposed to be a two-sided real-valued Wiener process on a complete probability space (Ω,F,P), where Ω={ω∈C(R,R):ω(0)=0},F is Borel σ-algebra induced by the compact open topology of Ω, and P is the corresponding Wiener measure on (Ω,F).To define a proper cocycle for system (13), we first define the related parametric dynamic systems as follows. Let {Θs}s∈R be the group acting on R satisfying
(15)Θsτ=τ+s∀s,τ∈R,
and {θs}s∈R the group acting on (Ω,F,P) such that
(16)θsω(·)=ω(·+s)-ω(s)∀ω∈Ω,s∈R,
where we have identified W(t) with
(17)W(t,ω)=ω(t),t⩾τ.
Then by Definition 2 it is evident that (R,{Θs}s∈R) and (Ω,F,P,(θs)s∈R) are two parametric dynamical systems.We now define a continuous cocycleϕ over (R,{Θs}s∈R) and (Ω,F,P,{θs}s∈R) for system (13). Consider the one-dimensional Ornstein-Uhlenbeck equation
(18)dy-δydt=dW,
of which a stationary solution is provided by
(19)y(t)=y(θtω)≡-δ∫0∞e-δτ(θtω)(τ)dτ,t∈R.
It is known that there exists a θt-invariant set Ω~⊆Ω with P(Ω~)=1 such that y(θtω) is continuous in t for every ω∈Ω~, and the random variable |y(θtω)| is tempered (see, e.g., [8, 10, 12, 13], and hereafter we will not distinguish Ω~ from Ω). Therefore, by Proposition 8 and [12, Proposition 4.3.3] (see also [14–16]), there exists a tempered variable r(ω)>0 such that
(20)|y(ω)|6⩽r(ω),
where r(ω) satisfies
(21)r(θtω)⩽e(γ/2)|t|r(ω),t∈R.
Let z(θtω)=hy(θtω), f(s)=|s|2s, and
(22)v(t,τ,ω,v0)=u(t,τ,ω,u0)-ɛz(θtω)withv0=u0-ɛz(θτω).
Then if u(t) solves (13)-(14), v(t) should satisfy, by (18), (22), and h(x)∈H01(I)∩H2(I),
(23)dvdt=(λ+iα(t))Δv-(κ+iβ(t))f(v+ɛz(θtω))+δv+g(x,t)+ɛ(λ+iα(t))Δz(θtω),
with conditions
(24)v(τ,τ,ω,v0)=v0=u0-ɛz(θτω),v(t,τ,ω,v0)|∂I=0,
for all t⩾τ∈R and x∈I=]0,1[. Since (23)-(24) is a deterministic problem, by the “standard” Galërkin method as in [17] or similar arguments of [2] (see also [1, 5, 6] for autonomous G.-L. equations), we have the following well-possessedness result.Lemma 12.
Assume that(i)
λ∈R+, κ∈R+, δ∈R+, ɛ∈]0,1], h(x)∈H01(I;C)∩H2(I;C);(ii)
β(t)∈Cloc(R;R) and sup⁡t∈R|β(t)|⩽3κ;(iii)
α(t)∈Cloc(R;R) and g(x,t)∈Lloc2(R;H) are such that
(25)∫-∞τe(γ/2)s(|α(t)|4+∥g(x,s)∥2)ds<+∞foreveryτ∈R.Then, for each v0∈H, the initial-boundary value problem (23)-(24) has a unique weak solution
(26)v(t,τ,ω,v0)∈Cloc(τ,∞;H)∩Lloc2(τ,∞;V)∩Lloc4(τ,∞;L4(I)).
Besides, v(t,τ,ω,v0) is (F,B(H))-measurable in ω∈Ω and continuous in v0 with respect to the norm of H for each t⩾τ.Let(27)u(t,τ,ω,u0)=v(t,τ,ω,v0)+ɛz(θtω)withu0=v0+ɛz(θτω).
Then under assumptions of Lemma 12 it is evident that u solves problem (13)-(14) and is (F,B(H))-measurable in ω∈Ω, continuous in both t⩾τ, and u0∈H. Consider the mapping ϕ:R+×R×Ω×H→H with
(28)ϕ(t,τ,ω,u0)=u(t+τ,τ,θ-τω,u0)=v(t+τ,τ,θ-τω,v0)+ɛz(θtω),
where v0=u0-ɛz(ω). By the property of solution trajectories of well-possessed nonautonomous dynamical systems one can readily check according to Definition 4 that (28) defines a continuous cocycle ϕ for problem (13)-(14) on H over (R,{Θs}) and (Ω,F,P,{θs}), where {Θs}s∈R and {θs}s∈R are given by (15) and (16), respectively.To investigate the (Dγ-) pullback attractor in H for system (13), hereafter in this paper, we let, for an arbitrarily fixed γ>0, Dγ be the class satisfying Definition 3 with X=H.


## 3. Uniform Estimates of Solutions

In this section we estimate the solution of problem (13)-(14) to establish a Dγ-random absorbing set, as well as to obtain the Dγ-pullback asymptotic compactness for the cocycle ϕ under assumptions of Lemma 12. We begin with two useful lemmas.Lemma 13 (Young’s inequality).
Leta, b>0. Then for every p, q satisfying 1<p, q<∞, (1/p)+(1/q)=1, it holds that
(29)a·b⩽ϵapp+ϵ-q/pbqq∀ϵ>0.Lemma 14 (Gagliardo-Nirenberg’s inequality).
Letu∈Lq(Rn), Dmu∈Lr(Rn), 1⩽q, r⩽∞. Then for 0⩽j<m, (j/m)⩽a<1, there exists a constant C such that
(30)∥Dju∥p⩽C∥Dmu∥ra·∥u∥q1-a,
where 1/p=(j/n)+a((1/r)-(m/n))+(1-a)(1/q).By Lemma14 we derive the following estimates for later convenience.Corollary 15.
Letx∈I⊆R. Then it holds true for every well-defined n(x) that
(31)∥n∥44⩽C∥n∥3∥∇n∥,∥n∥66⩽C∥n∥4∥∇n∥2,∥∇n∥2⩽C∥Δn∥6/7∥n∥48/7.Lemma 16.
Let assumptions of Lemma12 hold. Then for every τ∈R, ω∈Ω, and D∈Dγ, there exists a T=T(τ,ω,D)⩾1 and a positive constant L, which depends on γ but is independent of ɛ, τ, ω, and D, such that the solution v(t,τ,ω,v0) with v0∈D(τ-t,θ-τω) of (23)-(24) satisfies, for all t⩾T, that
(32)∥v(τ,τ-t,θ-tω,v0)∥2⩽Le-γτ∫-∞τeγs(|α(t)|4+∥g(x,s)∥2)ds+ɛLr(ω)+L,(33)∫τ-tτeγs∥∇v(s,τ-t,θ-τω,v0)∥2ds+∫τ-tτeγs∥v(s,τ-t,θ-τω,v0)∥44ds⩽L∫-∞τeγs(|α(t)|4+∥g(x,s)∥2)ds+ɛLeγτr(ω)+L,
where r(ω) is the tempered random variable given by (20) and (21).Proof.
Taking the inner product of (23) with v in H and taking the real part, we get
(34)12ddt∥v∥2=-λ∥∇v∥2-Re((κ+iβ(t))f(v+ɛz(θtω)),v)+δ∥v∥2+ɛRe(g(t),v)+Re((λ+iα)Δz(θtω),v).
By conditions f(s)=|s|2s, |β(t)|⩽3κ, and z(θtω)=hy(θtω) we derive that
(35)-Re((κ+iβ(t))f(v+ɛz(θtω)),v)=-Re((κ+iβ(t))|v+ɛz(θtω)|2(v+ɛz(θtω)),v)=-Re(22(κ+iβ(t))(|v|2v+2v|ɛz|2+ɛv2z¯=-Re(κ+iβ(t))+|ɛz|2z+ɛ2v¯z2+2ɛ|v|2z),v22)⩽-κ∥v∥44+ɛ|∫I(κ+iβ(t))(v|v|2z¯+|z|2zv¯+v¯2z2+2|v|2zv¯)dx|⩽-κ∥v∥44+2ɛκ∫I(3|v|3|z|+|z|3|v|+|v|2|z|2)dx⩽-κ∥v∥44+κ2∥v∥44+ɛc∥z∥44(by  Lemma13)⩽-κ2∥v∥44+ɛc|y(θtω)|4(by(31)andh∈H01(I)),
where c=c(κ,∥h∥,∥∇h∥). Since similarly we have
(36)Re(g(t),v)+ɛRe((λ+iα(t))Δz(θtω),v)⩽12∥g(x,t)∥2+12∥v∥2+λ2∥∇v∥2+ɛc|λ+iα(t)|2|y(θtω)|2⩽12∥g(x,t)∥2+12∥v∥2+λ2∥∇v∥2+ɛc(|α(t)|4+|y(θtω)|4+|y(θtω)|2),
where c=c(λ,∥∇h∥); from (34)–(36) it follows that
(37)ddt∥v∥2+λ∥∇v∥2+κ∥v∥44⩽(2δ+1)∥v∥2+∥g(x,t)∥2+ɛc(|α(t)|4+|y(θtω)|4+|y(θtω)|2),
which implies that
(38)ddt∥v∥2+γ∥v∥2+λ∥∇v∥2+κ2∥v∥44⩽-κ2∫I(|v|2-2δ+1+γκ)2dx+κ2∫I(2δ+1+γκ)2dx+∥g(x,t)∥2+ɛc(|α(t)|4+|y(θtω)|4+|y(θtω)|2)⩽∥g(x,t)∥2+c|α(t)|4+ɛc|y(θtω)|4+ɛc,
where c=c(γ,κ,δ). Multiply (38) by eγt and integrate over ]τ-t,τ[, t∈R+, to get, for each ω∈Ω,
(39)∥v(τ,τ-t,ω,v0)∥2+λ∫τ-tτeγ(s-τ)∥∇v(s,τ-t,ω,v0)∥2ds+κ2∫τ-tτeγ(s-τ)∥v(s,τ-t,ω,v0)∥44ds⩽∫τ-tτeγ(s-τ)(c|α(s)|4+∥g(x,s)∥2)ds+ɛc∫τ-tτeγ(s-τ)(|y(θsω)|4+1)ds+e-γt∥v0∥2.
Notice that |y|4⩽|y|6+c. Therefore, replacing ω in (39) with θ-τω and by (20)-(21) we obtain
(40)∥v(τ,τ-t,θ-τω,v0)∥2+λe-γτ∫τ-tτeγs∥∇v(s,τ-t,θ-τω,v0)∥2ds+κ2e-γτ∫τ-tτeγs∥v(s,τ-t,θ-τω,v0)∥44ds⩽∫τ-tτeγ(s-τ)(c|α(s)|4+∥g(x,s)∥2)ds+ɛc∫τ-tτeγ(s-τ)(|y(θs-τω)|6+1)ds+e-γt∥v0∥2⩽ce-γτ∫-∞τeγs(|α(s)|4+∥g(x,s)∥2)ds+ɛc∫-∞0eγsr(θsω)ds+c+e-γt∥D(τ-t,θ-tω)∥2,
where c is a positive constant depending on ϵ but independent of ɛ, τ, ω, and D. Since D∈Dγ, by Definition 3 there exists a T=T(τ,ω,D)⩾1 such that
(41)e-γt∥D(τ-t,θ-tω)∥2⩽1∀t⩾T,
which along with (40) and (20)-(21) completes the proof.Lemma 17.
Let assumptions of Lemma12 hold. Then for every τ∈R, ω∈Ω, and D∈Dγ, there exists a T=T(τ,ω,D)⩾1 and a positive constant C, which depends on γ but is independent of ɛ, τ, ω, and D, such that the solution v(t,τ,ω,v0) with v0∈D(τ-t,θ-τω) of (23)-(24) satisfies, for all t⩾T,
(42)∫τ-1τ∥∇v(s,τ-t,θ-τω,v0)∥2ds+∫τ-1τ∥v(s,τ-t,θ-τω,v0)∥44ds⩽Ce-γτ∫-∞τeγs(|α(s)|4+∥g(x,s)∥2)ds+ɛCr(ω)+C,
where r(ω) is the tempered random variable given by (20) and (21).Proof.
Notice thateγs⩾eγ(τ-1) for all s∈]τ-1,τ[. Hence, by (33) we have
(43)∫τ-1τ∥∇v(s,τ-t,θ-τω,v0)∥2ds+∫τ-1τ∥v(s,τ-t,θ-τω,v0)∥44ds⩽eγ(1-τ)∫τ-tτeγs∥∇v(s,τ-t,θ-τω,v0)∥2ds+eγ(1-τ)∫τ-tτeγs∥v(s,τ-t,θ-τω,v0)∥44ds⩽ce-γτ∫-∞τeγs(|α(s)|4+∥g(x,s)∥2)ds+ɛcr(ω)+c,
for all t⩾T⩾1, which concludes the lemma.Lemma 18.
Let assumptions of Lemma12 hold. Then for every τ∈R, ω∈Ω, and D∈Dγ, there exists a T=T(τ,ω,D)⩾1 and a positive constant K, which depends on γ but is independent of ɛ, τ, ω, and D, such that the solution v(t,τ,ω,v0) with v0∈D(τ-t,θ-τω) of (23)-(24) satisfies, for all t⩾T,
(44)∥∇v(τ,τ-t,θ-τω,v0)∥2⩽K(ɛr(ω)+1)e-γτ∫-∞τeγs(|α(s)|4+∥g(x,s)∥2)ds+K(ɛr(ω)+1)2,
where r(ω) is the tempered random variable given by (20) and (21).Proof.
Taking the inner product of (23) with -Δv in H and taking the real part, we have
(45)12ddt∥∇v∥2+λ∥Δv∥2=Re((κ+iβ(t))f(v+ɛz(θtω)),Δv)+δ∥∇v∥2-Re(g(t),Δv)-ɛRe((λ+iα(t))Δz(θtω),Δv).
Estimate the first term in the right hand side of (45) to get (52). Since f(s)=|s|2s, we have
(46)Re((κ+iβ(t))f(v+ɛz(θtω)),Δv)=Re((κ+iβ(t))(|v|2v,Δv))+ɛRe((κ+iβ(t))(v2z¯+2|v|2z),Δv)+Re((κ+iβ(t))(2ɛ2v|z|2+ɛ3|z|2z+ɛ2v¯z2),Δv).
By the condition |β(t)|⩽3κ, for all t∈R, we have
(47)Re((κ+iβ(t))(|v|2v,Δv))=-Re((κ+iβ(t))∫I(|v|2|∇v|2+v∇|v|2∇v¯)dx)=-κ∫I|v|2|∇v|2dx-κ∫IRe(v∇v¯)∇|v|2dx+β(t)∫I‍Im⁡(v∇v¯)∇|v|2dx=-κ∫I|v|2|∇v|2dx-κ2∫I(∇|v|2)2dx-iβ(t)2∫I(v∇v¯-v¯∇v)∇|v|2dx=-14∫I(3κ(∇|v|2)2+2iβ(t)(v∇v¯-v¯∇v)∇|v|2+(∇|v|2)2κ|v∇v¯-v¯∇v|2)dx=-14∫IηMη⊤dx⩽0,∀t∈R,
where η⊤ denotes the conjugate transpose of matrix η and
(48)η=[∇|v|2,v∇v¯-v¯∇v],M=[3κ-iβiβκ].
Recall the Agmon inequality that
(49)∥z(x)∥∞⩽c∥z∥1/2∥∇z∥1/2,x∈I⊆R.
By Lemma 13, |β(t)|⩽3κ, and (49) we estimate the second term in the right hand side of (46) to obtain
(50)ɛRe((κ+iβ(t))(v2z¯+2|v|2z),Δv)⩽ɛ|κ+iβ(t)|∫I3|v|2|z||Δv|dx⩽λ4∥Δv∥2+ɛc∥z(θtω)∥∞2∥v∥44⩽λ4∥Δv∥2+ɛc|y(θtω)|2∥v∥44,
where c=c(λ,κ,∥h∥,∥∇h∥). Similarly, for the last term of (46) we have
(51)Re((κ+iβ(t))(2ɛ2v|z|2+ɛ3|z|2z+ɛ2v¯z2),Δv)⩽2ɛκ∫I(3|v||z|2|Δv|+|z|3|Δv|)dx⩽λ4∥Δv∥2+ɛc(∥z∥∞4∥v∥44+∥z∥∞4+∥z(θtω)∥6)⩽λ4∥Δv∥2+ɛc(|y(θtω)|4∥v∥44+|y(θtω)|4+|y(θtω)|6).
From (46)–(51) and Lemma 13 it follows that
(52)Re((κ+iβ(t))f(v+ɛz(θtω)),Δv)⩽λ2∥Δv∥2+ɛc(|y(θtω)|6+1)∥v∥44+ɛc|y(θtω)|6+c.
For the last term of (45), by Lemma 13 again we get
(53)-ɛRe((λ+iα(t))Δz(θtω),Δv)⩽λ4∥Δv∥2+ɛc|λ+iα(t)|2∥Δz(θtω)∥2⩽λ4∥Δv∥2+ɛc|α(t)|2∥Δz(θtω)∥2+ɛcλ2∥Δz(θtω)∥2⩽λ4∥Δv∥2+c|α(t)|4+ɛc|y(θtω)|6+c,
where c=c(λ,∥Δh∥). Since, by (31) and Lemma 13 again,
(54)δ∥∇v∥2⩽λ8∥Δv∥2+c∥v∥44+c,-Re(g(t),Δv)⩽4λ∥g(x,t)∥2+λ8∥Δv∥2,
from (45), (52), and (53) we conclude that
(55)ddt∥∇v∥2⩽c(ɛ|y(θtω)|6+1)(∥v∥44+1)+c(∥g(x,t)∥2+|α(t)|4),
where c is a positive constant independent of τ, ω, and D. Given t⩾0, τ∈R, ω∈Ω, and s∈]τ-1,τ[, integrating (55) over ]s,τ[ and by (20) we find that
(56)∥∇v(τ,τ-t,ω,v0)∥2⩽∥∇v(s,τ-t,ω,v0)∥2+c∫sτ(ɛ|y(θξω)|6+1)(∥v(ξ)∥44+1)dξ+c∫sτ(∥g(x,ξ)∥2+|α(ξ)|4)dξ⩽∥∇v(s,τ-t,ω,v0)∥2+c∫τ-1τ(ɛr(θξω)+1)∥v(ξ)∥44dξ+c∫τ-1τ(ɛr(θξω)+1)dξ+c∫τ-1τ(∥g(x,ξ)∥2+|α(ξ)|4)dξ,
where c is a positive constant independent of τ, ω, and D. Integrating (56) with respect to s over ]τ-1,τ[ and replacing ω with θ-τω, by (21) we derive that
(57)∥∇v(τ,τ-t,θ-τω,v0)∥2⩽∫τ-1τ∥∇v(ξ,τ-t,θ-τω,v0)∥2dξ+(ɛcr(ω)+c)∫τ-1τ∥v(ξ,τ-t,θ-τω,v0)∥44dξ+ɛcr(ω)+c+c∫τ-1τ(∥g(x,ξ)∥2+|α(ξ)|4)dξ,
where c depends on γ but is independent of ɛ, τ, ω, and D. Let T=T(τ,ω,D)⩾1 be the same as in Lemma 17. Then from (57) and (42) it follows that
(58)∥∇v(τ,τ-t,θ-τω,v0)∥2⩽(ɛcr(ω)+c)e-γτ∫-∞τeγs(|α(s)|4+∥g(x,s)∥2)ds+(ɛcr(ω)+c)2+c∫τ-1τ(∥g(x,ξ)∥2+|α(ξ)|4)dξ⩽(ɛcr(ω)+c)e-γτ∫-∞τeγs(|α(s)|4+∥g(x,s)∥2)ds+(ɛcr(ω)+c)2,
which completes the proof.To derive uniform estimates on the solutionsu of (13)-(14), recall from (28) that
(59)u(τ,τ-t,θ-τω,u0)=v(τ,τ-t,θ-τω,v0)+z(ω),
where v0=u0-z(θ-tω). Hence, for X=H or V, we have
(60)∥u(τ,τ-t,θ-τω,u0)∥X2⩽2∥v(τ,τ-t,θ-τω,v0)∥X2+2∥z(ω)∥X2⩽2∥v(τ,τ-t,θ-τω,v0)∥X2+2∥h(x)∥X2(r(ω)+1).
Moreover, by the temperance of r(ω) it is evident that v0 comes from a nonautonomous random set in Dγ provided, and so does u0. Therefore, Lemmas 16 and 18 imply the following lemma.Lemma 19.
Let assumptions of Lemma12 hold. Then for every τ∈R, ω∈Ω, and D∈Dγ, there exists a T=T(τ,ω,D)⩾1 and a positive constant M, which depends on γ but is independent of ɛ, τ, ω, and D, such that the solution u(t,τ,ω,u0) with u0∈D(τ-t,θ-τω) of (13)-(14) satisfies, for all t⩾T, that
(61)∥u(τ,τ-t,θ-tω,u0)∥2⩽Me-γτ∫-∞τeγs(|α(t)|4+∥g(x,s)∥2)ds+ɛMr(ω)+M,
and that
(62)∥u(τ,τ-t,θ-τω,u0)∥V2⩽M(ɛr(ω)+1)e-γτ∫-∞τeγs(|α(s)|4+∥g∥2)ds+M(ɛr(ω)+1)2,
where r(ω) is the tempered random variable given by (20) and (21).


## 4. Pullback Attractors for Stochastic G.-L. Equations

### 4.1. Existence

In this part, for eachɛ∈]0,1] we establish the existence of the Dγ-pullback attractor Aɛ for system (13).Consider the nonautonomous random setEɛ={Eɛ(τ,ω)}τ,ω with
(63)Eɛ(τ,ω)≔{u∈H:∥u∥V2⩽Jɛ(τ,ω)}
for each τ∈R and ω∈Ω, where Jɛ(τ,ω) is given by
(64)Jɛ(τ,ω)=M(ɛr(ω)+1)e-γτ∫-∞τeγs(|α(s)|4+∥g∥2)ds+M(ɛr(ω)+1)2,
where M is the constant found out by Lemma 19. It is evident that Jϵ(τ,·):ω→R+ is (F,B(R))-measurable for each τ∈R. Moreover, by the temperance of r(ω) and assumption (iii) of Lemma 12 one can readily verify that
(65)lim⁡t→+∞e-γt∥Eϵ(Θ-tτ,θ-tω)∥2=lim⁡t→+∞e-γtJϵ(Θ-tτ,θ-tω)=0,
which indicates by Definition 3 that Eϵ∈Dγ. Therefore, by Definition 5, definition (28), and Lemma 19 (62), Eϵ is a closed (and moreover, compact by Sobolev compactness embeddings) measurable Dγ-pullback absorbing set for ϕ. In addition, it is evident that ϕ is Dγ-pullback asymptotically compact in H. Hence, by Lemma 10 we obtain the following existence result.Theorem 20.
Let assumptions of Lemma12 hold. Then for each ɛ∈ ]0,1] the cocycle associated with problem (13)-(14) has a unique Dγ-pullback attractor Aɛ={Aɛ(τ,ω)}τ,ω in L2(I).


### 4.2. Upper Semicontinuity

In the sequel we denote byϕɛ(t,τ,ω) the cocycle corresponding to the solution uɛ(t,τ,ω,u0ɛ) with uɛ(τ)=u0ɛ of (13) (to indicate the dependence of ɛ), and let ϕ0(t,τ) be the solution operator generated by the deterministic nonautonomous system
(66)dudt=(λ+iα(t))Δu-(κ+iβ(t))|u|2u+δu+g(x,t),(67)u(x,τ)=u0(x),u(x,·)|∂I≡0,
and given by
(68)ϕ0(t,τ)u0=u(t+τ,τ,u0),
where u(t,τ,u0) with u(τ)=u0 solves the problem (66)-(67). Analogously to cocycles ϕɛ(t,τ,ω), under assumptions of Lemma 12  ϕ0(t,τ) is a continuous operator from H to H, and moreover, it has a Dγ-pullback attractor A0={A0(τ)}τ.Lemma 21.
Let assumptions of Lemma12 hold. Then for every T∈R+, τ∈R, ω∈Ω, and u0ɛ, u0∈X with u0ɛ→u0 as ɛ↓0, it holds that
(69)lim⁡ɛ↓0ϕɛ(T,τ,ω)u0ɛ=ϕ0(T,τ)u0.Proof.
Letvɛ(t,τ,ω,v0ɛ)=uɛ(t,τ,ω,u0ɛ)-ɛz(θtω) with v0ɛ(ω)=u0ɛ-ɛz(ω) and write W(t,τ,ω)=vɛ(t,τ,ω,v0ɛ)-u(t,τ,u0). Then by (23) and (66), W satisfies
(70)dWdt=(λ+iα(t))ΔW-(κ+iβ(t))(f(v+ɛz(θtω))-f(u))+δW+ɛ(λ+iα(t))Δz(θtω),
where f(s)=|s|2s. Since, for each fixed ω∈Ω,
(71)∥W(τ)∥=∥v0ɛ-u0∥=∥u0ɛ-ɛz(ω)-u0∥⟶0asɛ↓0,
there exists a ɛ0∈]0,1] such that, for the fixed ω,
(72)∥v0ɛ∥2⩽∥u0∥2+1∀ɛ∈]0,ɛ0].
Taking the inner product of (70) with W in H and taking the real part we have
(73)12ddt∥W∥2+λ∥∇W∥2=δ∥W∥2-Re((κ+iβ(t))(f(uɛ)-f(u)),W)+ɛRe((λ+iα(t))Δz(θtω),W).
In the sequel we consider ɛ∈]0,ɛ0]. By the condition β(t)⩽3κ, for all t∈R, we have
(74)-Re((κ+iβ(t))(f(uɛ)-f(u)),W)=-Re((κ+iβ(t))(|uɛ|2(uɛ-u)+(|uɛ|2-|u|2)u),W)⩽2κ∫I(|uɛ|2|W+ɛz(θtω)|+|uɛ+u∥W+ɛz(θtω)∥u|)×|W|dx⩽c∥uɛ∥∞2(ɛ∥z(θtω)∥2+∥W∥2)+c(∥uɛ∥∞2+∥u∥∞2)(ɛ∥z(θtω)∥2+∥W∥2)⩽c(∥uɛ∥∞2+∥u∥∞2)∥W∥2+ɛc∥z(θtω)∥2×(∥uɛ∥∞2+∥u∥∞2),
where c depends on κ but is independent of ɛ. Similarly by Lemma 13 we have
(75)ɛRe((λ+iα(t))Δz(θtω),W)⩽ɛc∥Δz(θtω)∥2(1+|α(t)|2)+ɛ∥W∥2.
Then from (73)–(75) it follows that
(76)ddt∥W(t,τ,ω)∥2⩽c(∥uɛ(t,τ,ω,u0ɛ)∥∞2+∥u∥∞2+1)∥W(t,τ,ω)∥2+ɛc∥z(θtω)∥2(∥uɛ(t,τ,ω,u0ɛ)∥∞2+∥u∥∞2+|α(t)|2+1)⩽cN(t,τ,ω)∥W(t,τ,ω)∥2+ɛc∥z(θtω)∥2N(t,τ,ω),
where c is a positive constant independent of ɛ and N(t,τ,ω) is given by
(77)N(t,τ,ω)≔sup⁡0<0⩽ɛ0∥uɛ(t,τ,ω,u0ɛ)∥∞2+∥u∥∞2+|α(t)|2+1.
Let T>τ be an arbitrarily fixed time. By Agmon inequality (49) and (62) we have
(78)∫ττ+TN(s,τ,θ-Tω)ds<+∞.
Therefore, replacing ω with θ-Tω in (76) and by Gronwall lemma we have
(79)∥W(τ+T,τ,θ-Tω)∥2⩽ec∫ττ+T‍N(s,τ,θ-Tω)ds∥W(τ)∥2+ɛc∫ττ+Te∫sτ+T‍N(υ,τ,θ-Tω)dυ∥z(θs-Tω)∥2N(s,τ,θ-Tω)ds⩽ec∫ττ+T‍N(s,τ,θ-Tω)ds∥W(τ)∥2+ɛc(sup⁡τ-T<s<τ|y(θsω)|2)e∫ττ+T‍N(υ,τ,θ-Tω)dυ·∫ττ+TN(s,τ,θ-Tω)ds.
Since W(t,τ,ω)=uɛ(t,τ,ω,u0ɛ)-ɛz(θtω)-u(t,τ,u0), from (71) and (79) it follows that
(80)∥uɛ(τ+T,τ,θ-Tω,u0ɛ)-u(τ+T,τ,u0)∥2⩽2∥W(τ+T,τ,θ-Tω)∥2+2ɛ2∥z(θτω)∥2⟶0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiasɛ↓0,
which completes the proof.Theorem 22.
Let assumptions of Lemma12 hold. Then for every τ∈R and ω∈Ω,
(81)lim⁡ɛ↓0distH(Aɛ(τ,ω),A0(τ))=0.Proof.
We prove that the result verifies the four conditions of Lemma11. First, it is evident that condition (i) is actually proved by Lemma 21. To verify the rest, let
(82)R0(τ)=Jɛ(τ,ω)|ɛ=0=Me-γτ∫-∞τeγs(|α(s)|4+∥g∥2)ds+M,
where M is the constant as in (63). Since Eɛ={Eɛ(τ,ω)}τ,ω for each ɛ∈]0,1] defined by (63) is a compact Dγ-pullback absorbing set for ϕɛ in H, conditions (ii) and (iii) of Lemma 11 hold and condition (iv) follows from the fact that
(83)⋃ɛ∈]0,1]‍Aɛ(τ,ω)⊆⋃ɛ∈]0,1]‍Eɛ(τ,ω)⊆E1(τ,ω).
Then the theorem is concluded.


## 4.1. Existence

In this part, for eachɛ∈]0,1] we establish the existence of the Dγ-pullback attractor Aɛ for system (13).Consider the nonautonomous random setEɛ={Eɛ(τ,ω)}τ,ω with
(63)Eɛ(τ,ω)≔{u∈H:∥u∥V2⩽Jɛ(τ,ω)}
for each τ∈R and ω∈Ω, where Jɛ(τ,ω) is given by
(64)Jɛ(τ,ω)=M(ɛr(ω)+1)e-γτ∫-∞τeγs(|α(s)|4+∥g∥2)ds+M(ɛr(ω)+1)2,
where M is the constant found out by Lemma 19. It is evident that Jϵ(τ,·):ω→R+ is (F,B(R))-measurable for each τ∈R. Moreover, by the temperance of r(ω) and assumption (iii) of Lemma 12 one can readily verify that
(65)lim⁡t→+∞e-γt∥Eϵ(Θ-tτ,θ-tω)∥2=lim⁡t→+∞e-γtJϵ(Θ-tτ,θ-tω)=0,
which indicates by Definition 3 that Eϵ∈Dγ. Therefore, by Definition 5, definition (28), and Lemma 19 (62), Eϵ is a closed (and moreover, compact by Sobolev compactness embeddings) measurable Dγ-pullback absorbing set for ϕ. In addition, it is evident that ϕ is Dγ-pullback asymptotically compact in H. Hence, by Lemma 10 we obtain the following existence result.Theorem 20.
Let assumptions of Lemma12 hold. Then for each ɛ∈ ]0,1] the cocycle associated with problem (13)-(14) has a unique Dγ-pullback attractor Aɛ={Aɛ(τ,ω)}τ,ω in L2(I).


## 4.2. Upper Semicontinuity

In the sequel we denote byϕɛ(t,τ,ω) the cocycle corresponding to the solution uɛ(t,τ,ω,u0ɛ) with uɛ(τ)=u0ɛ of (13) (to indicate the dependence of ɛ), and let ϕ0(t,τ) be the solution operator generated by the deterministic nonautonomous system
(66)dudt=(λ+iα(t))Δu-(κ+iβ(t))|u|2u+δu+g(x,t),(67)u(x,τ)=u0(x),u(x,·)|∂I≡0,
and given by
(68)ϕ0(t,τ)u0=u(t+τ,τ,u0),
where u(t,τ,u0) with u(τ)=u0 solves the problem (66)-(67). Analogously to cocycles ϕɛ(t,τ,ω), under assumptions of Lemma 12  ϕ0(t,τ) is a continuous operator from H to H, and moreover, it has a Dγ-pullback attractor A0={A0(τ)}τ.Lemma 21.
Let assumptions of Lemma12 hold. Then for every T∈R+, τ∈R, ω∈Ω, and u0ɛ, u0∈X with u0ɛ→u0 as ɛ↓0, it holds that
(69)lim⁡ɛ↓0ϕɛ(T,τ,ω)u0ɛ=ϕ0(T,τ)u0.Proof.
Letvɛ(t,τ,ω,v0ɛ)=uɛ(t,τ,ω,u0ɛ)-ɛz(θtω) with v0ɛ(ω)=u0ɛ-ɛz(ω) and write W(t,τ,ω)=vɛ(t,τ,ω,v0ɛ)-u(t,τ,u0). Then by (23) and (66), W satisfies
(70)dWdt=(λ+iα(t))ΔW-(κ+iβ(t))(f(v+ɛz(θtω))-f(u))+δW+ɛ(λ+iα(t))Δz(θtω),
where f(s)=|s|2s. Since, for each fixed ω∈Ω,
(71)∥W(τ)∥=∥v0ɛ-u0∥=∥u0ɛ-ɛz(ω)-u0∥⟶0asɛ↓0,
there exists a ɛ0∈]0,1] such that, for the fixed ω,
(72)∥v0ɛ∥2⩽∥u0∥2+1∀ɛ∈]0,ɛ0].
Taking the inner product of (70) with W in H and taking the real part we have
(73)12ddt∥W∥2+λ∥∇W∥2=δ∥W∥2-Re((κ+iβ(t))(f(uɛ)-f(u)),W)+ɛRe((λ+iα(t))Δz(θtω),W).
In the sequel we consider ɛ∈]0,ɛ0]. By the condition β(t)⩽3κ, for all t∈R, we have
(74)-Re((κ+iβ(t))(f(uɛ)-f(u)),W)=-Re((κ+iβ(t))(|uɛ|2(uɛ-u)+(|uɛ|2-|u|2)u),W)⩽2κ∫I(|uɛ|2|W+ɛz(θtω)|+|uɛ+u∥W+ɛz(θtω)∥u|)×|W|dx⩽c∥uɛ∥∞2(ɛ∥z(θtω)∥2+∥W∥2)+c(∥uɛ∥∞2+∥u∥∞2)(ɛ∥z(θtω)∥2+∥W∥2)⩽c(∥uɛ∥∞2+∥u∥∞2)∥W∥2+ɛc∥z(θtω)∥2×(∥uɛ∥∞2+∥u∥∞2),
where c depends on κ but is independent of ɛ. Similarly by Lemma 13 we have
(75)ɛRe((λ+iα(t))Δz(θtω),W)⩽ɛc∥Δz(θtω)∥2(1+|α(t)|2)+ɛ∥W∥2.
Then from (73)–(75) it follows that
(76)ddt∥W(t,τ,ω)∥2⩽c(∥uɛ(t,τ,ω,u0ɛ)∥∞2+∥u∥∞2+1)∥W(t,τ,ω)∥2+ɛc∥z(θtω)∥2(∥uɛ(t,τ,ω,u0ɛ)∥∞2+∥u∥∞2+|α(t)|2+1)⩽cN(t,τ,ω)∥W(t,τ,ω)∥2+ɛc∥z(θtω)∥2N(t,τ,ω),
where c is a positive constant independent of ɛ and N(t,τ,ω) is given by
(77)N(t,τ,ω)≔sup⁡0<0⩽ɛ0∥uɛ(t,τ,ω,u0ɛ)∥∞2+∥u∥∞2+|α(t)|2+1.
Let T>τ be an arbitrarily fixed time. By Agmon inequality (49) and (62) we have
(78)∫ττ+TN(s,τ,θ-Tω)ds<+∞.
Therefore, replacing ω with θ-Tω in (76) and by Gronwall lemma we have
(79)∥W(τ+T,τ,θ-Tω)∥2⩽ec∫ττ+T‍N(s,τ,θ-Tω)ds∥W(τ)∥2+ɛc∫ττ+Te∫sτ+T‍N(υ,τ,θ-Tω)dυ∥z(θs-Tω)∥2N(s,τ,θ-Tω)ds⩽ec∫ττ+T‍N(s,τ,θ-Tω)ds∥W(τ)∥2+ɛc(sup⁡τ-T<s<τ|y(θsω)|2)e∫ττ+T‍N(υ,τ,θ-Tω)dυ·∫ττ+TN(s,τ,θ-Tω)ds.
Since W(t,τ,ω)=uɛ(t,τ,ω,u0ɛ)-ɛz(θtω)-u(t,τ,u0), from (71) and (79) it follows that
(80)∥uɛ(τ+T,τ,θ-Tω,u0ɛ)-u(τ+T,τ,u0)∥2⩽2∥W(τ+T,τ,θ-Tω)∥2+2ɛ2∥z(θτω)∥2⟶0,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiasɛ↓0,
which completes the proof.Theorem 22.
Let assumptions of Lemma12 hold. Then for every τ∈R and ω∈Ω,
(81)lim⁡ɛ↓0distH(Aɛ(τ,ω),A0(τ))=0.Proof.
We prove that the result verifies the four conditions of Lemma11. First, it is evident that condition (i) is actually proved by Lemma 21. To verify the rest, let
(82)R0(τ)=Jɛ(τ,ω)|ɛ=0=Me-γτ∫-∞τeγs(|α(s)|4+∥g∥2)ds+M,
where M is the constant as in (63). Since Eɛ={Eɛ(τ,ω)}τ,ω for each ɛ∈]0,1] defined by (63) is a compact Dγ-pullback absorbing set for ϕɛ in H, conditions (ii) and (iii) of Lemma 11 hold and condition (iv) follows from the fact that
(83)⋃ɛ∈]0,1]‍Aɛ(τ,ω)⊆⋃ɛ∈]0,1]‍Eɛ(τ,ω)⊆E1(τ,ω).
Then the theorem is concluded.



---

*Source: 921750-2014-10-19.xml*