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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains

Xiaoyao Jia
• Corresponding author
• Mathematics and Statistics School, Henan University of Science and Technology, No.263 Kai-Yuan Road, Luo-Long District, Luoyang, Henan Province, 471023, China
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• Other articles by this author:
/ Xiaoquan Ding
• Mathematics and Statistics School, Henan University of Science and Technology, No.263 Kai-Yuan Road, Luo-Long District, Luoyang, Henan Province, 471023, China
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• Other articles by this author:
/ Juanjuan Gao
• Mathematics and Statistics School, Henan University of Science and Technology, No.263 Kai-Yuan Road, Luo-Long District, Luoyang, Henan Province, 471023, China
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Published Online: 2018-08-03 | DOI: https://doi.org/10.1515/math-2018-0076

## Abstract

In this paper we investigate the stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domain ℝn (n ≥ 2). We first transform the retarded reaction-diffusion equations into the deterministic reaction-diffusion equations with random parameter by Ornstein-Uhlenbeck process. Next, we show the original equations generate the random dynamical systems, and prove the existence of random attractors by conjugation relation between two random dynamical systems. In this process, we use the cut-off technique to obtain the pullback asymptotic compactness.

MSC 2010: 35B40; 35B41; 35K45; 35K57

## 1 Introduction

In this paper we investigate a class of stochastic partial differential equations, which are widely used in quantum field theory, statistical mechanics and financial mathematics. It is known that there are many dynamical systems, depending on both current and historical states. They are referred to as time-decay dynamical systems and can be described by retarded partial differential equations. Hence, it is very important to study the properties of retarded partial differential equations.

In this paper, we consider the asymptotic behavior of the solutions to the following stochastic retarded reaction-diffusion equation with multiplicative noise in the whole space ℝn (n ≥ 2):

$du+(λu−Δu)dt=(F(x,u(t,x))+G(x,u(t−h,x))+g(x))dt+ϵu∘dw.$(1)

Here λ is a positive constant; h > 0 is the delay time of the system; F and G are given functions satisfying certain conditions which will be given in Section 2; g is a given function defined on ℝn; w is a two-sided real-valued Wiener process on a probability space (Ω, 𝓕, ℙ), with

$Ω={ω∈C(R,R):ω(0)=0},$

the Borel σ-algebra 𝓕 on Ω is generated by the compact open topology [2] and ℙ is the corresponding Wiener measure on 𝓕; ϵ is a positive parameter; ∘ denotes the Stratonovich sense in the stochastic term. We identify ω(t) with wt(ω), i.e. wt(ω) = w(t, ω) = ω (t), t ∈ ℝ.

One of the most important problems for stochastic differential equations is to study the asymptotic behavior of the solutions. The asymptotic behavior of random dynamical systems can be described by its random attractors. The existence of attractors for partial differential equations has been studied by many authors, see [3, 12, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 32, 34] and the references therein. However, to the best of our knowledge, the studies of attractors for retarded partial differential equations are very few, especially for the stochastic retarded partial differential equations. The attractors for the deterministic retarded partial differential equations were investigated in [6, 7, 9, 33, 35], and random attractors for stochastic retarded partial differential equations were studied in [13, 14, 30, 36]. In [13], the authors studied the existence of random attractor for stochastic retarded reaction-diffusion equations with additive noise, while in [14, 30, 36], the authors studied the random attractors for stochastic retarded lattice dynamical systems. In this paper, we will study the existence of the random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domain ℝn. In the bounded domain case, the compactness of random absorbing set can be obtained by Sobolev embeddings theory. However, in the unbounded domain case, we can not get compactness in this way, because Sobolev embeddings are not compact. Hence, the cut-off technique will be used to obtain the compactness. This technique has been used by many authors (see [13, 21, 26, 27]).

The remainder of the paper is organized as follows. In section 2, we recall an important theorem which will be used to obtain the existence of random attractor, and transform the stochastic retarded partial differential equations to random ones by Ornstein-Uhlenbeck process. We will show that for each ωΩ, the random equation has a unique solution. In Section 3, we get some useful estimates for the solution of the random equation. At last, we get the existence of the random attractor.

## 2 Preliminaries and Random Dynamical Systems

In this section we introduce some notations and recall some basic knowledge about random attractors for random dynamical systems (RDS). The reader can refer to [2, 3, 10, 11, 17] for more details.

First, we introduce some notations which will be used in the following. Let ∥⋅∥and (⋅, ⋅) be the norm and inner product in L2(ℝn). For fixed h > 0, let 𝔖 = C ([−h, 0], L2(ℝn)), and let the norm of 𝔖 be ∥f𝔖 = supt∈[−h,0]f(t)∥. Then, it is easy to see that 𝔖 is a Banach space. Let (X, ∥⋅∥X) be a Banach space with Borel σ−algebra 𝓑(X), and let (Ω, 𝓕,ℙ) be a probability space. For ab, t ∈ [a, b]and continuous function uC([ah, b], L2(ℝd)), define ut(s) = u(t + s) for s ∈ [−h, 0]. From the definition, one can see that, for t ∈ [a, b], ut ∈ 𝔖.

Let (X, ∥⋅∥X) be a Banach space with Borel σ−algebra 𝓑(X), and let (Ω, 𝓕,ℙ) be a probability space.

#### Definition 2.1

(Ω, 𝓕, ℙ, (θt)t∈ℝ) is called a metric dynamical system, if θ:ℝ × ΩΩ is (𝓑(ℝ) × 𝓕, 𝓕)− measurable, θ0 is the identity on Ω, θs+t = θtθs for all s, t ∈ ℝ and θt ℙ = ℙ for all t ∈ ℝ.

#### Definition 2.2

A stochastic process ϕ is called a continuous random dynamical system (RDS) over (Ω, 𝓕, ℙ, (θt)t∈ℝ), if ϕ is (𝓑([0, +∞)) × 𝓕 × 𝓑(X), 𝓑(X))−neasurable, and for all ωΩ,

1. the mapping ϕ(t, ω, ⋅) : XX, xϕ(t, ω, x) is continuous for every t ≥ 0;

2. ϕ(0, ω, ⋅) is the identity on X;

3. ϕ(s + t, ω, ⋅) = ϕ(t, θs ω, ⋅) ∘ ϕ(s, ω, ⋅) for all s, t > 0.

#### Definition 2.3

1. A random set {B(ω)}ωΩ of X is called bounded if there exists x0X and a random variable r(ω) > 0, such that

$B(ω)⊂{x∈X:||x−x0||X≤r(ω),x0∈X},forallω∈Ω.$

2. A random set {B(ω)}ωΩ is called a compact random set if B(ω) is compact for all ωΩ.

3. A random set {B(ω)}ωΩX is called tempered with respect to (θt)t∈ℝ, if for ℙ − a.e.ωΩ,

$limt→+∞e−ytsupx∈B(θ−tω)||x||X=0forally>0.$(2)

4. A random variable r(ω) is said to be tempered with respect to (θt)t∈ℝ, if for ℙ − a.e.ωΩ,

$limt→+∞e−yt|r(θ−tω)|=0forally>0.$(3)

In the following, we assume that ϕ is a continuous RDS on X over (Ω, 𝓕, ℙ, (θt)t∈ℝ), and 𝒟 is a collection of random subsets of X.

#### Definition 2.4

Let {K(ω)}ωΩ ∈ 𝒟. Then {K(ω)}ωΩ is called a random absorbing set for ϕ in 𝒟 if for every {B(ω)} ∈ 𝒟 and ℙ − a.e. ωΩ, there exists T(B, ω) > 0, such that

$ϕ(t,θ−tω,B(θ−tω))⊆K(ω)forallt≥T(B,ω).$(4)

#### Definition 2.5

A random set {A(ω)}ωΩ of X is called a 𝒟-random attractor (or 𝒟-pullback attractor) for ϕ, if the following conditions are satisfied:

1. {A(ω)}ωΩ is compact, and ωd(x, A(ω)) is measurable for every xX;

2. {A(ω)}ωΩ is invariant, that is,

$ϕ(t,ω,A(ω))=A(θtω)∀t≥0.$(5)

3. {A(ω)}ωΩ attracts every set in 𝒟, that is, for every {B(ω)}ωΩ ∈ 𝒟, ωΩ,

$limt→+∞dist(ϕ(t,θ−tω,B(θ−tω)),A(ω))=0,$(6)

where dist$\begin{array}{}\left(Y,Z\right)=\underset{y\in Y}{sup}\underset{z\in Z}{inf}||y-z|{|}_{X}\end{array}$ is the Hausdorff semi-metrix (YX, ZX).

#### Definition 2.6

ϕ is said to be 𝒟-pullback asymptotically compact in X, if for all B ∈ 𝒟 and ℙ − a.e. ωΩ, $\begin{array}{}{\left\{\varphi \left({t}_{n},{\theta }_{-{t}_{n}}\omega ,{x}_{n}\right)\right\}}_{i=1}^{\mathrm{\infty }}\end{array}$ has a convergent subsequence in X, as tn → ∞, and xnB(θtn ω).

At the end of this section, we refer to [3, 17] for the existence of random attractor for continuous RDS.

#### Proposition 2.7

Let {K(ω)}ωΩ ∈ 𝒟 be a random absorbing set for the continuous RDS ϕ in 𝒟 and ϕ is 𝒟-pullback asymptotically compact in X. Then ϕ has a unique 𝒟-random attractor {A(ω)}ωΩ which is given by

$A(ω)=∩τ≥0∪t≥τϕ(t,θ−tω,K(θ−tω))¯.$(7)

In the rest of this paper we will assume that 𝒟 is the collection of all tempered random subsets of 𝔖 and we will prove that the stochastic retarded reaction-diffusion equation has a 𝒟-pullback random attractor.

In the remaining part of this section we show that there is a continuous random dynamical system generated by the following stochastic retarded reaction-diffusion equation on ℝn with the multiplicative noise:

$du+(λu−Δu)dt=(F(x,u(t,x))+G(x,u(t−h,x))+g(x))dt+ϵu∘dw,x∈Rn,t>0,$(8)

with the initial condition

$u(t,x)=u0(t,x),x∈Rn,t∈[−h,0].$(9)

Here g is a given function in L2 (ℝn), and F, G are continuous functions satisfying the following conditions:

• (A1)

F:ℝn × ℝ → ℝ is a continuous function such that for all x ∈ ℝn and s ∈ ℝ,

$F(x,s)s≤−α1|s|p+β1(x);$(10)

$|F(x,s)|≤α2|s|p−1+β2(x);$(11)

$∂∂sF(x,s)≤α3;$(12)

$|∂∂xF(x,s)|≤β3(x).$(13)

Here α1, α2, α3 are positive constants, p > 2, β1(x), β2(x), β3(x) are nonnegative functions on ℝn, such that β1(x) ∈ L1(ℝn), and β2(x), β3(x) ∈ L2(ℝn).

• (A2)

G : ℝn × ℝ → ℝ is a continuous function such that for all x ∈ ℝn and s1, s2 ∈ ℝ,

$|G(x,s1)−G(x,s2)|≤α4|s1−s2|;$(14)

$|G(x,s)|≤β4(x)|s|+β5(x).$(15)

Here α4 is a positive constant, β4(x), β5(x) are nonnegative functions on ℝn such that β4(x) ∈ $\begin{array}{}{L}^{\frac{2p}{p-2}}\end{array}$(ℝn) ∩ L(ℝn), β5(x) ∈ L2(ℝn).

#### Example 2.8

For x ∈ ℝn, s, s1, s2 ∈ ℝ and p > 2, let

$F(x,s)=−α1|s|p−1sgn(s);G(x,s)=α4e−x2s.$

It is easy to check that the functions F and G in Example 2.8 satisfy Condition (A1) and (A2).

In what follows we consider the probability space (Ω, 𝔉, ℙ) which is defined in Section 1. Let

$θtω(⋅)=ω(⋅+t)−ω(t),t∈R.$(16)

Then (Ω, 𝔉, ℙ, (θt)t∈ℝ) is an ergodic metric dynamical system. Since the probability space (Ω,𝔉,ℙ) is canonical, one has

$w(t,ω)=ω(t),w(t,θsω)=w(t+s,ω)−w(s,ω).$(17)

To study the random attractor for problem (8)-(9), we first transform that system into a deterministic system with random parameter. Let

$z(θtω)≡−μ∫−∞0eμs(θtω)(s)ds,t∈R.$

Then, one has that z(θt ω) is the Ornstein-Uhlenbeck process and solves the following equation (see [15] for details):

$dz+μzdt=dw(t).$(18)

Moreover, the random variable z(θt ω) is tempered, and z(θt ω) is ℙ − a.e. continuous. It follows from Proposition 4.3.3 [2] that there exists a tempered function r(ω) > 0 such that

$|z(ω)|+|z(ω)|2≤r(ω),$(19)

where r(ω) satisfies, for α > 0 and for ℙ − a.e. ωΩ,

$r(θtω)≤eα2|t|r(ω),t∈R.$(20)

Then it follows that for α > 0 and for ℙ − a.e. ωΩ,

$|z(θtω)|+|z(θtω)|2≤eα2|t|r(ω),t∈R,$(21)

$|z(θt−hω)|+|z(θt−hω)|2≤eα2|t−h|r(ω)≤eα2heα2|t|r(ω),t∈R.$(22)

By [9], z(θt ω) has the following properties:

$limt→±∞z(θtω)t=0,$(23)

$limt→±∞1|t|∫−t0|z(θsω)|ds=E|z|,$(24)

$E|z|≤E|z|2=12μ.$(25)

It follows from (23) and (25) that there exists a positive constant μ > 0 small enough, such that for t > 0 large enough,

$2ϵμ∫−t0|z(θsω)|ds≤λ4t;2ϵ|z(θtω)|≤λ8t.$(26)

In this paper we consider the weak solutions of (8)-(9) and (30)-(31).

#### Definition 2.9

For any u0 ∈ 𝔖, let u : [0,∞) × ΩL2 (ℝn). Suppose that u(⋅, ω, u0) : [0, ∞) → L2 (ℝn) is continuous and $\begin{array}{}\frac{\mathrm{\partial }u}{\mathrm{\partial }t}\end{array}$ is measurable. Then we say that u is a weak solution of (8)-(9), if u(t, ω, u0) = u0(t, ω), t ∈ [−h, 0], ωΩ, and uL2 ([0, T], H1(ℝn)) ∩ Lp([0, T], Lp(ℝn)), for any T > 0, and for any ϕ$\begin{array}{}{C}_{0}^{\mathrm{\infty }}\end{array}$ (ℝn)

$(u(T),ϕ)−(u0(0),ϕ)+∫0T(u(t),λϕ−Δϕ)dt=∫0T(F(⋅,u(t))+G(⋅,u(t−h))+g,ϕ)dt+ϵ∫0T(ϕ,u∘dw),$(27)

ℙ − a.e. ωΩ.

Suppose that u is a weak solution of (8)-(9). Let v(t) = eϵz(θtω) u(t), v0(t) = eϵz(θtω) u0(t) and ψ = eϵz(θtω) ϕ. Then one has that

$(u(T),ϕ)−(u0(0),ϕ)=∫0T(∂u∂t,ϕ)dt=∫0T(∂∂t(eϵz(θtω)v(t)),ϕ)dt=∫0T(∂∂tv(t),ψ)dt+ϵ∫0T(ψ,v∘dz)=(v(T),ψ)−(v0(0),ψ)+ϵ∫0T(ψ,v∘dw)−μϵ∫0T(ψ,z(θtω)v)dt.$(28)

In the last step of (28) we use (18). Hence, it follows from (28) and the definition of weak solution that

$(v(T),ψ)−(v0(0),ψ)+∫0T(v(t),λψ−Δψ)dt=∫0Te−ϵz(θtω)(F(⋅,eϵz(θtω)v(t))+G(⋅,eϵz(θt−hω)v(t−h))+g,ψ)dt+μϵ∫0T(ψ,z(θtω)v)dt.$(29)

Then function v satisfying (29) is said to be the weak solution of the following equation:

$∂v∂t+λv−Δv=e−ϵz(θtω)F(x,eϵz(θtω)v(t,x))+e−ϵz(θtω)G(x,eϵz(θt−hω)v(t−h,x))+e−ϵz(θtω)g(x)+ϵμz(θtω)v,$(30)

with the initial condition

$v0(t,x)=e−ϵz(θtω)u0(t,x),x∈Rn,t∈[−h,0].$(31)

Equation (30)-(31) can be seen as a deterministic partial differential equation with random coefficients.

We use a similar method as in [4] to obtain the existence of weak solution to equation (30)-(31). First, using the Galerkin method as in [28], we can get that, under the condition (A1) and (A2), for any bounded domain O ⊂ ℝn, (30)-(31) has a unique weak solution v(⋅, ω, v0) ∈ C([0, ∞), L2(O)) ∩ $\begin{array}{}{L}_{loc}^{2}\end{array}$ ([0, ∞), $\begin{array}{}{H}_{0}^{1}\end{array}$ (O)) ∩ $\begin{array}{}{L}_{loc}^{p}\end{array}$ ([0, ∞), Lp(O)), for ℙ − a.e. ωΩ. We can take the domain to be a family of balls, and the radius of balls tend to ∞. Then, one can get that vL2(ℝn) is the unique weak solution of (30)-(31) on ℝn. Then, u(t) = eϵz(θtω) v(t) is a unique weak solution to (8)-(9). Similar as Theorem 12 [13], we can show that (30)-(31) generates a continuous random dynamical system (Φ(t))t≥0 over (Ω, 𝓕, ℙ, (θt)t∈ℝ), with

$Φ(t,θ−tω,B(θ−tω))=vt(⋅,ω,v0),ω∈Ω,v0∈S,$(32)

and (8)-(9) generates a continuous random dynamical system (Ψ(t))t≥0 over (Ω, 𝓕, ℙ, (θt)t∈ℝ), with

$Ψ(t,θ−tω,B(θ−tω))=ut(⋅,ω,u0),ω∈Ω,u0∈S.$(33)

Notice that two dynamical systems are conjugate to each other. Therefore, in the following sections, we only consider the existence of random attractor of (Φ(t))t≥0.

## 3 Uniform estimates of solutions

In this section we prove the existence of the random attractor for random dynamical system (Φ(t))t≥0. We first give some useful estimates on the mild solutions of equation (30)-(31).

#### Lemma 3.1

Random dynamical system Φ has a random absorbing set {K(ω)}ωΩ in 𝒟, that is, for any {B(ω)}ωΩ ∈ 𝒟 and ℙ − a.e.ωΩ, there is TB(ω) > 0, such that Φ(t, θtω, B(θt ω)) ⊂ K(ω) for all tTB(ω).

#### Proof

Taking the inner product of (30) with v, we get that

$12ddt||v||2+λ||v||2+||∇v||2=e−ϵz(θtω)∫RnF(x,eϵz(θtω)v(t,x))vdx+e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h,x))vdx+e−ϵz(θtω)∫Rng(x)vdx+ϵμz(θtω)||v||2.$(34)

We now estimate each term on the right hand side of (34). For the first term, by condition (A1) and Young inequality we have that

$e−ϵz(θtω)∫RnF(x,eϵz(θtω)v(t,x))vdx≤e−2ϵz(θtω)∫Rn−α1|eϵz(θtω)v|p+β1(x)dx=−α1e(p−2)ϵz(θtω)||v||Lpp+e−2ϵz(θtω)||β1||L1.$(35)

Note that by Young inequality, we have that for all a, b, c > 0, α, β, y > 1, and $\begin{array}{}\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{y}=1,\end{array}$

$abc≤aαα+bββ+cyy.$(36)

For the second term on the right hand side of (34), by condition (A2) and (36), we obtain that

$e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h,x))vdx≤e−ϵz(θtω)∫Rnβ4(x)|eϵz(θt−hω)v(t−h,x)v|dx+e−ϵz(θtω)∫Rnβ5(x)|v|dx=∫Rn(ep−2pϵz(θtω)|v|)|v(t−h,x)|(e2−2ppϵz(θtω)+ϵz(θt−hω)β4(x))dx+e−ϵz(θtω)∫Rnβ5(x)|v|dx≤α12e(p−2)ϵz(θtω)||v||Lpp+λ4e−λ2h||v(t−h,x)||2+c1e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω)||β4||2pp−2+λ4||v||2+1λe−2ϵz(θtω)||β5||2,$(37)

where c1 is a positive constant depending on α, λ and h. For the third term on the right hand side of (34), by Young inequality,

$e−ϵz(θtω)∫Rng(x)vdx≤1λe−2ϵz(θtω)||g||2+λ4||v||2.$(38)

Then it follows from (34) - (38) that, for all t ≥ 0

$ddt||v||2+2||∇v||2+α1e(p−2)ϵz(θtω)||v||Lpp≤(2ϵμ|z(θtω)|−λ)||v||2+λ2e−λ2h||v(t−h,x)||2+c2e−2ϵz(θtω)+c3e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω)=(2ϵμ|z(θtω)|−λ2)||v||2+λ2(e−λ2h||v(t−h,x)||2−||v||2)+c2e−2ϵz(θtω)+c3e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω),$(39)

with $\begin{array}{}{c}_{2}=2||{\beta }_{1}|{|}_{{L}^{1}}+\frac{2}{\lambda }||{\beta }_{5}|{|}^{2}+\frac{2}{\lambda }||g|{|}^{2},{c}_{3}=2{c}_{1}||{\beta }_{4}|{|}^{\frac{2p}{p-2}}.\end{array}$ Applying Gronwall inequality, we obtain that, for all t ≥ 0,

$||v(t)||2+2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ||∇v||2ds+α1∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+(p−2)ϵz(θsω)||v||Lppds≤e−λ2t+2ϵμ∫0t|z(θτω)|dτ||v0||2+λ2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ(e−λ2h||v(s−h)||2−||v||2)ds+c2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θtω)ds+c3∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.$(40)

Notice that

$∫0teλ2s−2ϵμ∫0s|z(θτω)|dτe−λ2h||v(s−h)||2ds=∫−ht−heλ2(s+h)−2ϵμ∫0s+h|z(θτω)|dτe−λ2h||v(s)||2ds=∫−h0eλ2s−2ϵμ∫0s+h|z(θτω)|dτ||v(s)||2ds+∫0t−heλ2s−2ϵμ∫0s+h|z(θτω)|dτ||v(s)||2ds≤h||v0||S+∫0t−heλ2s−2ϵμ∫0s|z(θτω)|dτ||v(s)||2ds≤h||v0||S+∫0teλ2s−2ϵμ∫0s|z(θτω)|dτ||v(s)||2ds.$(41)

Hence, it follows from (41) that

$λ2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ(e−λ2h||v(s−h)||2−||v||2)ds=λ2e−λ2t+2ϵμ∫0t|z(θtω)|dτ∫0teλ2s−2ϵμ∫0s|z(θtω)|dτe−λ2h||v(s−h)||2ds−λ2e−λ2t+2ϵμ∫0t|z(θtω)|dτ∫0teλ2s−2ϵμ∫0s|z(θtω)|dτ||v||2ds≤λh2e−λ2t+2ϵμ∫0t|z(θtω)|dτ||v0||S.$(42)

Then, (40) and (42) imply that

$||v(t)||2+2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ||∇v||2ds+α1∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ+(p−2)ϵz(θsω)||v||Lppds≤(1+λh2)e−λ2t+2ϵμ∫0t|z(θtω)|dτ||v0||S2+c2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ−2ϵz(θtω)ds+c3∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.$(43)

For fixed σ ∈ [−h, 0], we have that, for t ≥ −σ,

$||v(t+σ)||2≤(1+λh2)e−λ2(t+σ)+2ϵμ∫0t+σ|z(θτω)|dτ||v0||S2+c2∫0t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ−2ϵz(θsω)ds+c3∫0t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds≤(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτω)|dτ||v0||S2+c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)ds+c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.$(44)

and for t ∈ [0, −σ],

$||v(t+σ)||2≤||v0||S2≤(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτω)|dτ||v0||S2+c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)ds+c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.$(45)

Due to (44) and (45), we find that for all t ≥ 0

$||vt||S2≤(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτω)|dτ||v0||S2+c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)ds+c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.$(46)

Replacing ω by θtω in (46), we get that for all t ≥ 0

$||vt(θ−tω,v0(θ−tω))||S2≤(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτ−tω)|dτ||v0(θ−tω)||S2+c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds+c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds.$(47)

By (26), one has that for any v0(θt ω) ∈ B(θt ω),

$limt→+∞(1+λh2)eλ2(h−t)+2ϵμ∫0t|z(θτ−tω)|dτ||v0||S2=limt→+∞(1+λh2)eλ2(h−t)+2ϵμ∫−t0|z(θτω)|dτ||v0||S2≤limt→+∞(1+λh2)eλ2(h−t)+λ4t||v0||S2=0.$(48)

It follows from (26) that for any v0(θt ω) ∈ B(θt ω),

$c2eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds=c2eλh2∫−t0eλ2s+2ϵμ∫s0|z(θτω)|dτ−2ϵz(θsω)ds≤c2eλh2∫−∞0eλ4s−2ϵz(θtω)ds≤c2eλh2∫−∞0eλ2s−λ4s−λ8sds<+∞.$(49)

Similarly, we can get that

$c3eλh2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds=c3eλh2∫−t0eλ2s+2ϵμ∫s0|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds≤c3eλh2∫−∞0eλ4s+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds<+∞.$(50)

Set

$ρ12(ω)=1+c2eλh2∫−∞0eλ4s−2ϵz(θsω)ds+c3eλh2∫−∞0eλ4s+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.$(51)

Then, there exists a TB(ω) > 0, such that for all t > TB(ω),

$||vt(θ−tω,v0(θ−tω))||S2≤ρ12(ω).$(52)

To show $\begin{array}{}{\rho }_{1}^{2}\end{array}$ (ω) is tempered, we need only to prove that for any y > 0 small enough, the following holds

$limt→+∞e−ytρ12(θ−tω)=0.$(53)

Using (26) again, one has that for y < λ small enough,

$limt→+∞e−yt∫−∞0eλ4s−2ϵz(θs−tω)ds≤limt→+∞e−yt∫−∞0ey4s−2ϵz(θs−tω)ds=limt→+∞e−yt∫−∞tey4(s+t)−2ϵz(θsω)ds=limt→+∞e−34yt(∫−∞0ey4s−2ϵz(θsω)ds+∫0tey4s−2ϵz(θsω)ds)≤limt→+∞e−34yt(∫−∞0ey4s−y8sds+∫0tey4s+y8sds)=0.$(54)

Similarly, we can get that

$limt→+∞e−yt∫−∞0eλ4s+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds=0.$(55)

In view of (54) and (55), we find that $\begin{array}{}{\rho }_{1}^{2}\end{array}$ (ω) is tempered. This ends the proof.

In Lemma 3.1, we show that {K(ω)}ωΩ is a random absorbing set for Φ. To prove Φ has a random attractor, we need to show that Φ is 𝒟-pullback asymptotically compact. Therefore, we should obtain some estimates for ∇ v.

#### Lemma 3.2

There exists a tempered random variable ρ2(ω) > 0 such that for any {B(ω)}ωΩ ∈ 𝒟 and v0(ω) ∈ B(ω), there exists a TB(ω) > 0 such that the solution v of (30)-(31) satisfies, for ℙ-a.e. ωΩ, for all tTB(Ω),

$∫tt+1||∇v(s,θ−tω,v0(θ−tω))||2ds≤ρ2(ω).$(56)

#### Proof

By (43), we have that for all t ≥ 0

$2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ||∇v||2ds≤(1+λh2)e−λ2t+2ϵμ∫0t|z(θtω)|dτ||v0||S2+ c2∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ−2ϵz(θtω)ds+c3∫0teλ2(s−t)+2ϵμ∫st|z(θtω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)ds.$(57)

Let TB(ω) be the positive constant defined in Lemma 4.1. Replacing ω by θt ω in (57), by (48) - (50), one has that for all tTB(ω),

$2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ||∇v(s,θ−tω,v0(θ−tω))||2ds≤(1+λh2)e−λ2t+2ϵμ∫0t|z(θτ−tω)|dτ||v0(θ−tω)||S2+c2∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds +c3∫0teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds ≤ρ12(ω).$(58)

In another aspect, for s ∈ [t, t + 1],

$∫0t+1eλ2(s−t−1)+2ϵμ∫st+1|z(θτ−t−1ω)|dτ||∇v(s,θ−t−1ω,v0(θ−t−1ω))||2ds≥∫tt+1eλ2(s−t−1)+2ϵμ∫st+1|z(θτ−t−1ω)|dτ||∇v(s,θ−t−1ω,v0(θ−t−1ω))||2ds≥e−λ2−2ϵμmax−1≤τ≤0|z(θτω)|∫tt+1||∇v(s,θ−t−1ω,v0(θ−t−1ω))||2ds.$(59)

It follows from (58) and (59) that

$∫tt+1||∇v(s,θ−t−1ω,v0(θ−t−1ω))||2ds≤eλ2+2ϵμmax−1≤τ≤0|z(θτω)|ρ12(ω).$(60)

Replacing ω by θ1 ω in the last inequality we obtain that

$∫tt+1||∇v(s,θ−tω,v0(θ−tω))||2ds≤eλ2+2ϵμmax−1≤τ≤0|z(θτω)|ρ12(ω)≡ρ2(ω).$(61)

This ends the proof. □

#### Lemma 3.3

There exists a tempered random variable ρ3(ω) > 0 such that for any {B(ω)}ωΩ ∈ 𝒟 and v0(ω) ∈ B(ω), there exists a TB(ω) > 0 such that the solution v of (30)-(31) satisfies, for ℙ-a.e. ωΩ, for all tTB(ω) + h + 1, and σ1, σ2 ∈ [−h, 0],

$∇v(t,θ−tω,v0(θ−tω))2≤ρ3(ω),$(62)

$∫t+σ1t+σ2||Δv(s,θ−tω,v0(θ−tω))||2ds≤ρ4(ω).$(63)

#### Proof

Taking the inner product of (30) with −Δ v, we get that

$12ddt||∇v||2+λ||∇v||2+||Δv||2=−e−ϵz(θtω)∫RnF(x,eϵz(θtω)v(t,x))Δvdx− e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h,x))Δvdx−e−ϵz(θtω)∫Rng(x)Δvdx+ϵμz(θtω)||∇v||2.$(64)

We now estimate each term on the right-hand side of (64). For the first term, by Young inequality and condition (A1), we have

$−e−ϵz(θtω)∫RnF(x,eϵz(θtω)v(t,x))Δvdx=e−ϵz(θtω)∫Rn∂F∂x(x,eϵz(θtω)v)⋅∇vdx+∫Rn∂F∂s(x,eϵz(θtω)v)|∇v|2dx≤e−ϵz(θtω)||β3|| ||∇v||+α3||∇v||2≤(1+α3)||∇v||2+14e−2ϵz(θtω)||β3||2.$(65)

For the second term, it follows from condition (A2) and Young inequality that

$−e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h,x))Δvdx≤e−ϵz(θtω)∫Rneϵz(θt−hω)β4(x)|v(t−h,x)|+β5(x)Δvdx≤18||Δv||2+2e−2ϵz(θtω)+2ϵz(θt−hω)∫Rnβ42(x)|v(t−h,x)|2dx+18||Δv||2+2e−2ϵz(θtω)∫Rnβ52(x)dx≤14||Δv||2+2e−2ϵz(θtω)+2ϵz(θt−hω)||β4||L∞2||v(t−h)||2+2e−2ϵz(θtω)||β5||2.$(66)

The third term is bounded by

$−e−ϵz(θtω)∫Rng(x)Δvdx≤14||Δv||2+e−2ϵz(θtω)||g||2.$(67)

By (64) - (67), we find that for all t ≥ 0,

$ddt||∇v||2+||Δv||2≤2(1+α3)||∇v||2+c5e−2ϵz(θtω)+4e−2ϵz(θtω)+2ϵz(θt−hω)||β4||L∞2||v(t−h)||2,$(68)

with $\begin{array}{}{c}_{5}=4||{\beta }_{5}|{|}^{2}+2||g|{|}^{2}+\frac{1}{2}||{\beta }_{3}|{|}^{2}.\end{array}$ Let TB(ω) be the positive constant defined in Lemma 3.1, take tTB(ω) and s ∈ [t, t + 1]. Integrating (68) over [s, t + 1], we get that

$||∇v(t+1,ω,v0(ω))||2≤||∇v(s,ω,v0(ω))||2+2(1+α3)∫st+1||∇v(τ,ω,v0(ω))||2dτ +c5∫st+1e−2ϵz(θτω)dτ+4||β4||L∞2∫st+1e−2ϵz(θτω)+2ϵz(θτ−hω)||v(τ−h,ω,v0(ω))||2dτ.$(69)

Integrating the above inequality with respect to s over [t, t + 1], we get that

$||∇v(t+1,ω,v0(ω))||2≤2(2+α3)∫tt+1||∇v(τ,ω,v0(ω))||2dτ +c5∫tt+1e−2ϵz(θτω)dτ+4||β4||L∞2∫tt+1e−2ϵz(θτω)+2ϵz(θτ−hω)||v(τ−h,ω,v0(ω))||2dτ.$(70)

Replacing ω by θt−1 ω in (70), we obtain that

$||∇v(t+1,θ−t−1ω,v0(θ−t−1ω))||2≤2(2+α3)∫tt+1||∇v(τ,θ−t−1ω,v0(θ−t−1ω))||2dτ +c5∫tt+1e−2ϵz(θτ−t−1ω)dτ+4||β4||L∞2∫tt+1e−2ϵz(θτ−t−1ω)+2ϵz(θτ−t−h−1ω)||v(τ−h,θ−t−1ω,v0(θ−t−1ω))||2dτ.$(71)

It follows from (56) that for all tTB(ω) + h,

$2(2+α3)∫tt+1||∇v(τ,θ−t−1ω,v0(θ−t−1ω))||2dτ≤2(2+α3)ρ2(θ−1ω),$(72)

$∫tt+1e−2ϵz(θτ−t−1ω)dτ≤e2ϵmax−1≤τ≤0|z(θτω)|,$(73)

and

$∫tt+1e−2ϵz(θτ−t−1ω)+2ϵz(θτ−t−h−1ω)||∇v(τ−h,θ−t−1ω,v0(θ−t−1ω))||2dτ≤e2ϵmax−1≤τ≤0|z(θτω)|+|z(θτ−hω)|∫t−ht−h+1||∇v(τ,θ−t−1ω,v0(θ−t−1ω))||2dτ=e2ϵmax−1≤τ≤0|z(θτω)|+|z(θτ−hω)|∫t−ht−h+1||∇v(τ,θ−t+h−1(θ−hω),v0(θ−t+h−1(θ−hω))||2dτ≤e2ϵmax−1≤τ≤0|z(θτω)|+|z(θτ−hω)|ρ2(θ−hω).$(74)

It follows from (71) - (74) that, for all tTB(ω) + h

$||∇v(t+1,θ−t−1ω,v0(θ−t−1ω))||2≤2(2+α3)ρ2(θ−1ω)+c5e2ϵmax−1≤τ≤0|z(θτω)| +4||β4||L∞2e2ϵmax−1≤τ≤0|z(θτω)|+|z(θτ−hω)|ρ2(θ−hω)≡ρ3(ω).$(75)

Then we have that for all tTB(ω) + h +1,

$||∇v(t,θ−tω,v0(θ−tω))||2≤ρ3(ω).$(76)

Let th, −hσ1σ2 ≤ 0. Integrating (68) over [t + σ1, t + σ2], we obtain that

$||∇v(t+σ2,ω,v0(ω))||2+∫t+σ1t+σ2||Δv(τ,ω,v0(ω))||2dτ ≤||∇v(t+σ1,ω,v0(ω))||2+2(1+α3)∫t+σ1t+σ2||∇v(τ,ω,v0(ω))||2dτ +c5∫t+σ1t+σ2e−2ϵz(θτω)dτ+4||β4||L∞2∫t+σ1t+σ2e−2ϵz(θτω)+2ϵz(θτ−hω)||v(τ−h,ω,v0(ω))||2dτ.$(77)

Replacing ω by θt ω in the last inequality, we have that

$||∇v(t+σ2,θ−tω,v0(θ−tω))||2+∫t+σ1t+σ2||Δv(τ,θ−tω,v0(θ−tω))||2dτ ≤||∇v(t+σ1,θ−tω,v0(θ−tω))||2+2(1+α3)∫t+σ1t+σ2||∇v(τ,θ−tω,v0(θ−tω))||2dτ +c5∫t+σ1t+σ2e−2ϵz(θτ−tω)dτ+4||β4||L∞2∫t+σ1t+σ2e−2ϵz(θτ−tω)+2ϵz(θτ−t−hω)||v(τ−h,θ−tω,v0(θ−tω))||2dτ.$(78)

Using (76), we get that

$∫t+σ1t+σ2||∇v(τ,θ−tω,v0(θ−tω))||2dτ=∫t+σ1t+σ2||∇v(τ,θ−τ(θτ−tω),v0(θ−τ(θτ−tω)))||2dτ≤hmax−h≤τ≤0ρ3(θτω)).$(79)

By (52), we can get that

$4||β4||L∞2∫t+σ1t+σ2e−2ϵz(θτ−tω)+2ϵz(θτ−t−hω)||v(τ−h,θ−tω,v0(θ−tω))||2dτ≤4||β4||L∞2max−2h≤τ≤−hρ12(θτω)∫t+σ1t+σ2e−2ϵz(θτ−tω)+2ϵz(θτ−t−hω)dτ=4h||β4||L∞2max−2h≤τ≤−hρ12(θτω)e2ϵmax−h≤τ≤0|z(θτω)|+|z(θτ−hω)|.$(80)

Notice that

$∫t+σ1t+σ2e−2ϵz(θτ−tω)dτ=∫σ1σ2e−2ϵz(θτω)dτ≤he2ϵmax−h≤τ≤0|z(θτω)|.$(81)

It follows from (76), (78)-(81) that

$∫t+σ1t+σ2||Δv(τ,θ−tω,v0(θ−tω))||2dτ≤(1+2h(1+α3))max−h≤τ≤0ρ3(θτω)+c5he2ϵmax−h≤τ≤0|z(θτω)|+ 4h||β4||L∞2max−2h≤τ≤−hρ12(θτω)e2ϵmax−h≤τ≤0|z(θτω)|+|z(θτ−hω)|≡ρ4(ω).$(82)

This ends the proof. □

#### Lemma 3.4

Let B(ω) ∈ 𝒟 and v0(ω) ∈ B(ω). Then for any ϵ1 > 0 and ℙ-a.e. ωΩ, there exists a T = T(B, ω, ϵ1) > 0, and R = R (ω, ϵ1) > 0 such that the solution (30)-(31), satisfies for all tT,

$sups∈[−h,0]∫|x|≥R∗|vt(s,θ−tω,v0(θ−tω))|dx≤ϵ1.$(83)

#### Proof

Let ρ be a smooth function defined on ℝ+ such that 0 ≤ ρ(s) ≤ 1 for all s ∈ ℝ+, and

$ρ(s)=0, 0≤s≤1,1, s≥2.$(84)

Then there exists a positive constant c0 such that |ρ′(s)| ≤ c0 for all s ∈ ℝ+. Taking the inner product of (30) with $\begin{array}{}\rho \left(\frac{|x{|}^{2}}{{k}^{2}}\right)v,\end{array}$ we get that

$12ddt∫Rnρ|x|2k2|v|2dx+λ∫Rnρ|x|2k2|v|2dx−∫Rnρ|x|2k2vΔvdx=e−ϵz(θtω)∫RnF(x,eϵz(θtω)v)ρ|x|2k2vdx+e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h))ρ|x|2k2vdx +e−ϵz(θtω)∫Rng(x)ρ|x|2k2vdx+ϵμz(θtω)∫Rnρ|x|2k2|v|2dx.$(85)

We now estimate each term in (85). For the third term on the left hand-side of (85), we have that

$−∫Rnρ|x|2k2vΔvdx=∫Rnρ|x|2k2|∇v|2dx+∫Rnρ′|x|2k2v2xk2⋅∇vdx =∫Rnρ|x|2k2|∇v|2dx+∫k≤|x|≤2kρ′|x|2k2v2xk2⋅∇vdx,$(86)

and

$∫k≤|x|≤2kρ′|x|2k2v2xk2⋅∇vdx≤22k∫k≤|x|≤2k|ρ′|x|2k2| |v| |∇v|dx≤4c0k∫Rn|v| |∇v|dx≤2c0k||v||2+||∇v||2.$(87)

By (86) and (87), one has that

$−∫Rnρ|x|2k2vΔvdx≥∫Rnρ|x|2k2|∇v|2dx−2c0k||v||2+||∇v||2.$(88)

For the first term on the right-hand side of (85) by condition (A1), we have

$e−ϵz(θtω)∫RnF(x,eϵz(θtω)v)ρ|x|2k2vdx=e−2ϵz(θtω)∫RnF(x,eϵz(θtω)v)ρ|x|2k2eϵz(θtω)vdx≤−α1e(p−2)ϵz(θtω)∫Rnρ|x|2k2|v|pdx+e−2ϵz(θtω)∫Rnρ|x|2k2β1(x)dx.$(89)

For the second term on the right-hand side of (85) we have that

$e−ϵz(θtω)∫RnG(x,eϵz(θt−hω)v(t−h))ρ|x|2k2vdx≤e−ϵz(θtω)∫Rnβ4(x)eϵz(θt−hω)v(t−h)ρ|x|2k2|v|dx+e−ϵz(θtω)∫Rnβ5(x)ρ|x|2k2|v|dx.$(90)

By (36), the first term on the right-hand side of (90) is bounded by

$e−ϵz(θtω)∫Rnβ4(x)eϵz(θt−hω)v(t−h)ρ|x|2k2|v|dx=∫Rne−p−2pϵz(θtω)|v|ρ1p|x|2k2 ρ12|x|2k2|v(t−h)|ρp−22p|x|2k2e2−2ppϵz(θtω)+ϵz(θt−hω)β4(x)dx≤α1∫Rne(p−2)ϵz(θtω)ρ|x|2k2|v|pdx+λ4e−λ2h∫Rnρ|x|2k2|v(t−h)|2dx +c6∫Rnρ|x|2k2e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω)β4(x)2pp−2dx,$(91)

and the second term on the right-hand side of (90) is bounded by

$e−ϵz(θtω)∫Rnβ5(x)ρ|x|2k2|v|dx≤λ4∫Rnρ|x|2k2|v|2dx+1λe−2ϵz(θtω)∫Rnρ|x|2k2β52(x)dx.$(92)

For the third term on the right-hand side of (85) we have that

$e−ϵz(θtω)∫Rng(x)ρ|x|2k2vdx≤λ4∫Rnρ|x|2k2|v|2dx+1λe−2ϵz(θtω)∫Rnρ|x|2k2g2(x)dx.$(93)

It follows from (85) - (93) that

$ddt∫Rnρ|x|2k2|v|2dx≤2ϵμ|z(θtω)|−λ2∫Rnρ|x|2k2|v|2dx +λ2∫Rnρ|x|2k2e−λ2h|v(t−h)|2−|v|2dx+4c0k||v||2+||∇v||2 +2e−2ϵz(θtω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx +2c6∫Rnρ|x|2k2e4−4pp−2ϵz(θtω)+2pp−2ϵz(θt−hω)β42pp−2(x)dx.$(94)

Let TB(ω) be the positive constant defined in Lemma 3.1. Set T1(B, ω) ≥ TB(ω) + h +1. Applying Gronwall inequality to (94), we obtain that for tT1,

$∫Rnρ|x|2k2|v|2dx≤eλ2(T1−t)+2ϵμ∫T1t|z(θτω)|dτ∫Rnρ|x|2k2|v(T1)|2dx +λ2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ∫Rnρ|x|2k2e−λ2h|v(t−h)|2−|v|2dx ds +4c0k∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ||v||2+||∇v||2ds +2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds +2c6∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds.$(95)

If we take tT1 + h, then, by (95), we can get that for all σ ∈ [−h, 0],

$∫Rnρ|x|2k2|v(t+σ)|2dx≤eλ2(T1−t−σ)+2ϵμ∫T1t+σ|z(θτω)|dτ∫Rnρ|x|2k2|v(T1)|2dx +λ2∫T1t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ∫Rnρ|x|2k2e−λ2h|v(s−h)|2−|v|2dx ds +4c0k∫T1t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ||v||2+||∇v||2ds +2∫T1t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ−2ϵz(θsω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds +2c6∫T1t+σeλ2(s−t−σ)+2ϵμ∫st+σ|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds.$(96)

It follows that for all tT1 + h,

$sup−h≤σ≤0∫Rnρ|x|2k2|v(t+σ)|2dx≤eλ2(T1+h−t)+2ϵμ∫T1t|z(θτω)|dτ||v(T1)||S2 +λ2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ∫Rnρ|x|2k2e−λ2h|v(s−h)|2−|v|2dx ds +4c0keλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ||v||2+||∇v||2ds +2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ−2ϵz(θsω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds +2c1eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτω)|dτ+4−4pp−2ϵz(θsω)+2pp−2ϵz(θs−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds.$(97)

Replacing ω by θtω, we obtain from (97) that for all tT1(B, ω),

$sup−h≤σ≤0∫Rnρ|x|2k2|vt(σ,θ−tω,v0(θ−tω))|2dx≤eλh2eλ2(T1−t)+2ϵμ∫T1t|z(θτ−tω)|dτ||vT1(θ−t(ω),v0(θ−t(ω)))||S2 +λ2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ∫Rnρ|x|2k2e−λ2h|v(s−h,θ−tω,v0(θ−tω))|2−|v(s,θ−tω,v0(θ−tω))|2dx ds +4c0keλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ||v(s,θ−tω,v0(θ−tω))||2+||∇v(s,θ−tω,v0(θ−tω))||2ds +2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds +2c1eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds.$(98)

Now we estimate each term on the right hand side of (98). We first replace t by T1 and replace ω by θtω in (46), we have that

$||v(T1,θ−tω,θ−tv0(ω))||S2≤(1+λh2) eλ2(h−T1)+2ϵμ∫0T1|z(θτ−tω)|dτ||v0(θ−tω)||S2+c2eλ2h∫0T1eλ2(s−T1)+2ϵμ∫sT1|z(θτ−tω)|dτ−2ϵz(θs−tω)ds+c3eλ2h∫0T1eλ2(s−T1)+2ϵμ∫sT1|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds.$(99)

Therefore, for the first term on the right hand side of (98),

$eλh2eλ2(T1−t)+2ϵμ∫T1t|z(θτ−tω)|dτ||vT1(θ−t(ω),v0(θ−t(ω)))||S2≤(1+λh2)eλh e−λ2t+2ϵμ∫0t|z(θτ−tω)|dτ||v0(θ−tω)||S2+c2eλh∫0T1eλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds+c3eλh∫0T1eλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds.$(100)

We now estimate the terms in (100) as follows. For the first term, by (26), we find that for tT1 and v0(θtω) ∈ B(θtω),

$limt→+∞(1+λh2)eλh e−λ2t+2ϵμ∫0t|z(θτ−tω)|dτ||v0(θ−tω)||S2=limt→+∞(1+λh2)eλh e−λ2t+2ϵμ∫−t0|z(θτω)|dτ||v0(θ−tω)||S2≤limt→+∞(1+λh2)eλh e−λ2t+λ4t||v0(θ−tω)||S2=0.$(101)

For the second term, by (26)

$limt→+∞c2eλh∫0T1eλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds=limt→+∞c2eλh∫0T1eλ2(s−t)+2ϵμ∫s−t0|z(θτω)|dτ−2ϵz(θs−tω)ds≤limt→+∞c2eλh∫0T1eλ2(s−t)−λ4(s−t)−λ8(s−t)ds=0.$(102)

Similarly, we can get the following estimate for the third term on the right-hand side of (100):

$limt→+∞c3eλh∫0T1eλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)ds=0.$(103)

It follows from (100) - (103) that

$limt→+∞eλh2eλ2(T1−t)+2ϵμ∫T1t|z(θτ−tω)|dτ||vT1(θ−t(ω),v0(θ−t(ω)))||S2=0,$(104)

which implies that for any ϵ1 > 0, there is a T2 = T2(B, ω, ϵ) > T1 such that

$eλh2eλ2(T1−t)+2ϵμ∫T1t|z(θτ−tω)|dτ||vT1(θ−t(ω),v0(θ−t(ω)))||S2≤ϵ1.$(105)

Next, we estimate the second term on the right hand side of (98). Similarly as (42), we can obtain that

$λ2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ∫Rnρ|x|2k2e−λ2h|v(s−h,θ−tω,v0(θ−tω))|2−|v(s,θ−tω,v0(θ−tω))|2dx ds≤λ2eλh2eλ2(T1−t)+2ϵμ∫0t|z(θτ−tω)|dτ∫Rnρ|x|2k2sup−h≤σ≤0|vT1(σ,θ−tω,v0(θ−tω))|2dx ds≤λ2eλh2eλ2(T1−t)+2ϵμ∫0t|z(θτ−tω)|dτ||vT1(θ−tω,v0(θ−tω))||S.$(106)

By (104) and (106), we know that for any ϵ1 > 0, there exists a T3 = T3(B, ω, ϵ) > T1 such that

$λ2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ××∫Rnρ|x|2k2e−λ2h|v(s−h,θ−tω,v0(θ−tω))|2−|v(s,θ−tω,v0(θ−tω))|2dx ds≤ϵ1.$(107)

For the third term on the right hand side of (98), by Lemma 3.1 and Lemma 3.3, we obtain that

$∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ||v(s,θ−tω,v0(θ−tω))||2+||∇v(s,θ−tω,v0(θ−tω))||2ds≤∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτρ12(θs−tω)+ρ3(θs−tω)ds=∫T1−t0eλ2s+2ϵμ∫s0|z(θτω)|dτρ12(θsω)+ρ3(θsω)ds≤∫−∞0eλ2s−λ4sρ12(θsω)+ρ3(θsω)ds<∞.$(108)

This implies that there exists a T4 = T4(B, ω, ϵ) > T1 and R2 = R2(ω, ϵ1) such that for all tT4 and kR2,

$4c0keλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ||v(s,θ−tω,v0(θ−tω))||2+||∇v(s,θ−tω,v0(θ−tω))||2ds≤ϵ1.$(109)

Note that β1L1(ℝn), β5L2(ℝn), gL2(ℝn). Thus, there is R3 = R3(ω, ϵ1) such that for all kR3

$∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx≤∫|x|≥kβ1(x)+1λβ52(x)+1λg2(x)dx≤ϵ1.$(110)

Then, for the fourth term on the right hand side of (98) we have that

$2eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)∫Rnρ|x|2k2β1(x)+1λβ52(x)+1λg2(x)dx ds≤2ϵ1eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ−2ϵz(θs−tω)ds=2ϵ1eλh2∫T1−t0eλ2s+2ϵμ∫s0|z(θτω)|dτ−2ϵz(θsω)ds≤2ϵ1eλh2∫−∞0eλ2s−λ4s−λ8sds=16λeλh2ϵ1.$(111)

Notice that $\begin{array}{}{\beta }_{4}\in {L}^{\frac{2p}{p-2}}\left({\mathbb{R}}^{n}\right).\end{array}$ Thus, there is R4 = R4(ϵ1, ω) such that for all kR4,

$∫Rnρ|x|2k2β42pp−2(x)dx≤∫|x|≥kβ42pp−2(x)dx≤ϵ1.$(112)

Then, similarly as (111), there exists a constant c > 0, such that

$2c1eλh2∫T1teλ2(s−t)+2ϵμ∫st|z(θτ−tω)|dτ+4−4pp−2ϵz(θs−tω)+2pp−2ϵz(θs−t−hω)∫Rnρ|x|2k2β42pp−2(x)dx ds≤cϵ1.$(113)

Let T = max {T1, T2, T3. T4} and R = max {R1, R2, R3}. Then, it follows from (98), (105), (107), (109), (111) and (113) that

$sup−h≤σ≤0∫Rnρ|x|2k2|vt(σ,θ−tω,v0(θ−tω))|2dx≤(c+16λeλh2+3)ϵ1,$(114)

This ends the proof.□

#### Lemma 3.5

Let B(ω) ∈ 𝒟 and v0(ω) ∈ B(ω). Then there are tempered random variables ρ5(ω), ρ6(ω), such that the solution of (30)-(31) satisfies for all tTB(ω) + 2h + 1, and σ, σ1, σ2 ∈ [–h, 0],

$||vt(σ,θ−tω,v0(θ−tω))||H1(Rn)2≤ρ5(ω);$(115)

$||vt(σ1,θ−tω,v0(θ−tω))−vt(σ2,θ−tω,v0(θ−tω))||≤ρ6(ω)|σ1−σ2|1/2.$(116)

#### Proof

By Lemma 3.1 and Lemma 3.3, we get that for all tTB(ω) + h + 1, and σ ∈ [–h,0 ],

$||vt(σ,θ−tω,v0(θ−tω))||H1(Rn)2=||v(t+σ,θ−tω,v0(θ−tω))||2+||∇v(t+σ,θ−tω,v0(θ−tω))||2≤max−h≤σ≤0ρ12(θσω)+ρ3(θσω)≡ρ5(ω).$(117)

For σ1, σ2 ∈ [–h, 0] (assuming σ1σ2 for simplicity), one has that,

$||vt(σ1,θ−tω,v0(θ−tω))−vt(σ2,θ−tω,v0(θ−tω))||2=||v(t+σ1,θ−tω,v0(θ−tω))−v(t+σ2,θ−tω,v0(θ−tω))||2=||∫t+σ1t+σ2ddtv(s,θ−tω,v0(θ−tω))||2ds≤λ∫t+σ1t+σ2||v(s,θ−tω,v0(θ−tω))||ds+∫t+σ1t+σ2||Δv(s,θ−tω,v0(θ−tω))||ds +∫t+σ1t+σ2||e−ϵz(θs−tω)F(x,eϵz(θs−tω)v(s,θ−tω,v0(θ−tω)))||ds +∫t+σ1t+σ2||e−ϵz(θs−tω)G(x,eϵz(θs−t−hω)v(s−h,θ−tω,v0(θ−tω)))||ds +∫t+σ1t+σ2||e−ϵz(θs−tω)g||ds+ϵμ∫t+σ1t+σ2||z(θs−tω)v(s,θ−tω,v0(θ−tω))||ds.$(118)

Now, we estimate each term on the right hand side of (118). By Lemma 3.1, and Lemma 3.3 we find that for all tTB(ω) + 2h + 1,

$∫t+σ1t+σ2||v(s,θ−tω,v0(θ−tω))||ds=∫t+σ1t+σ2||v(s,θ−s(θs−tω),v0(θ−s(θs−tω)))||ds≤max−h≤τ≤0ρ1(θτω)|σ1−σ2|;$(119)

$∫t+σ1t+σ2||z(θs−tω)v(s,θ−tω,v0(θ−tω))||ds≤max−h≤τ≤0z(θτω)ρ1(θτω)|σ1−σ2|;$(120)

$∫t+σ1t+σ2||Δv(s,θ−tω,v0(θ−tω))||ds≤∫t+σ1t+σ2||Δv(s,θ−tω,v0(θ−tω))||2ds1/2|σ1−σ2|1/2≤ρ4(ω)|σ1−σ2|1/2;$(121)

and

$∫t+σ1t+σ2||e−ϵz(θs−tω)g||ds≤max−h≤τ≤0eϵ|z(θτω)|||g|| |σ1−σ2|.$(122)

Using conditions (A1) and (A2), we obtain

$∫t+σ1t+σ2||e−ϵz(θs−tω)F(x,eϵz(θs−tω)v(s,θ−tω,v0(θ−tω)))||ds≤∫t+σ1t+σ2α2e(p−2)ϵz(θs−tω)||v(s,θ−tω,v0(θ−tω))||p−1+e−ϵz(θs−tω)||β2||ds≤α2max−h≤τ≤0e(p+2)ϵ|z(θτω)|ρ1p−1(θτω)+max−h≤τ≤0eϵ|z(θτω)|||β2|||σ1−σ2|,$(123)

and

$∫t+σ1t+σ2||e−ϵz(θs−tω)G(x,eϵz(θs−t−hω)v(s−h,θ−tω,v0(θ−tω)))||ds≤max−h≤τ≤0eϵ|z(θτω)|+ϵ|z(θτ−hω)|ρ1(θτ−hω)+||β5|||σ1−σ2|.$(124)

The lemma follows from (119) to (124). This ends the proof.□

Next, we use Ascoli theorem to show that Φ is 𝒟 pullback asymptotically compact.

#### Lemma 3.6

The random dynamical system Φ is 𝒟 pullback asymptotically compact in 𝔖; that is, for ℙ-a.e. ωΩ, the sequence $\begin{array}{}\mathit{\Phi }\left({t}_{n},{\theta }_{-{t}_{n}},{v}_{0}\left({\theta }_{-{t}_{n}}\right)\right){\right\}}_{n=1}^{\mathrm{\infty }}\end{array}$ has a convergent subsequence in 𝔖 provided tn → ∞ and v0(θtn) ∈ B(θtnω).

#### Proof

Let Qk be the set of {x ∈ ℝn : |x| < k}. Since tn → ∞, there exists N = N(B, ω) such that tn > TB(ω) + 2h + 1, for all n > N. Using Lemma 3.5, we obtain that for n > N and σ ∈ [–h, 0],

$||vtn(σ,θ−tnω,v0(θ−tnω))||H1(Qk)≤ρ5(ω).$(125)

This implies that $\begin{array}{}\left\{\mathit{\Phi }\left({t}_{n},{\theta }_{-{t}_{n}}\omega ,{v}_{0}\left({\theta }_{-{t}_{n}}\omega \right)\right){\right\}}_{n=N}^{\mathrm{\infty }}\end{array}$ is relatively compact in L2(Qk), since the embedding from H1(Qk) to L2(Qk) is compact. From Lemma 3.5, we can also obtain that for all n > N, and σ1, σ2 ∈ [–h, 0],

$||vtn(σ1,θ−tnω,v0(θ−tnω))−vtn(σ1,θ−tnω,v0(θ−tnω))||L2(Qk)≤ρ6(ω)|σ1−σ2|1/2.$(126)

This means that $\begin{array}{}\left\{\mathit{\Phi }\left({t}_{n},{\theta }_{-{t}_{n}}\omega ,{v}_{0}\left({\theta }_{-{t}_{n}}\omega \right)\right){\right\}}_{n=N}^{\mathrm{\infty }}\end{array}$ is equicontinuous. By Ascoli Theorem we have that for fixed $\begin{array}{}k\in \mathbb{N},\left\{{v}^{{t}_{n}}\left(\sigma ,{\theta }_{-{t}_{n}}\omega ,{v}_{0}\left({\theta }_{-{t}_{n}}\omega \right)\right){\right\}}_{n=1}^{\mathrm{\infty }}\end{array}$ is relatively compact in C([–h, 0];L2(Qk)). Therefore, for each k ∈ ℕ, there exists a subsequence {Φ(tni, θtni ω, v0(θtni ω))} converges to ηk(⋅, ω) in C([–h, 0]; L2(Qk)). Notice that, for fixed σ ∈ [–h, 0] and ωΩ, ηk+1(σ, ω) coincides with ηk(σ, ω) on Qk. Hence, for any ϵ > 0, there exists N1 = N1(ω, ϵ) ∈ ℕ, such that for all i > N1,

$supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))−η(σ,ω)||L2(Qk)≤ϵ.$(127)

By Lemma 3.1, we have that for all σ ∈ [–h, 0], and i > N,

$||vtni(σ,θ−tniω,v0(θ−tniω))||L2(Qk)≤supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))||L2(Rn)≤ρ1(ω).$(128)

Hence, for all k ∈ ℕ,

$||η(σ,ω)||L2(Qk)≤limi→∞||vtni(σ,θ−tniω,v0(θ−tniω))||L2(Qk)≤ρ1(ω).$(129)

It follows that,

$||η(σ,ω)||L2(Rn)≤supk∈N||η(σ,ω)||L2(Qk)≤ρ1(ω).$(130)

Therefore, η(σ, ω) ∈ L2(ℝn). By Lemma 3.4, for every B ∈ 𝒟 and ϵ1 > 0, there exist K = K(ω, ϵ1) and N2 = N2(ω, ϵ1), such that for all i > N2 and k > K,

$supσ∈[−h,0]||vtni(σ,θ−tni,v0(θ−tni))||L2(|x|≥k)≤ϵ1.$(131)

Notice that for all l > k > K and σ ∈ [–h, 0],

$||ξ(σ,ω)||L2(k≤|x|N2||vtni(σ,θ−tniω,v0(θ−tniω))||L2(|x|≥k)≤ϵ1.$(132)

It follows that

$supσ∈[−h,0]||ξ(σ,ω)||L2(|x|≥k)≤supσ∈[−h,0],l>k||ξ(σ,ω)||L2(k≤|x|(133)

Then, (113), (127) and (131) imply that for all i > max N1, N2,

$supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))−ξ(σ,ω)||≤supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))−ξ(σ,ω)||L2(Qk)+2supσ∈[−h,0]||vtni(σ,θ−tniω,v0(θ−tniω))||L2(|x|≥k) +2supσ∈[−h,0]||ξ(σ,ω)||L2(|x|≥k)≤5ϵ1.$(134)

This ends the proof.□

#### Theorem 3.7

The random dynamical system Φ has a unique 𝒟-random attractor in 𝔖.

#### Proof

By Lemma 3.1, we know that Φ has a random absorbing set, and by Lemma 3.6, we obtain that Φ is 𝒟-pullback asymptotically compact in 𝔖. Therefore, by Proposition 2.1, Φ has an unique 𝒟-pullback attractor. This ends the proof.□

## Acknowledgement

This work is partially supported by the National Natural Science Foundation of China under Grants 11301153 and 11271110, the Key Programs for Science and Technology of the Education Department of Henan Province under Grand 12A110007, and the Scientific Research Funds of Henan University of Science and Technology.

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Accepted: 2018-06-12

Published Online: 2018-08-03

Competing interests: The authors declare that there is no conflict of interest regarding the publication of this paper.

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 862–884, ISSN (Online) 2391-5455,

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