# Random Attractors for Stochastic Retarded Reaction-Diffusion Equations on Unbounded Domains

Authors: Xiaoquan Ding; Jifa Jiang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is devoted to a stochastic retarded reaction-diffusion equation on alld-dimensional space with additive white noise. We first show that the stochastic retarded reaction-diffusion equation generates a random dynamical system by transforming this stochastic equation into a random one through a tempered stationary random homeomorphism. Then, we establish the existence of a random attractor for the random equation. And the existence of a random attractor for the stochastic equation follows from the conjugation relation between two random dynamical systems. The pullback asymptotic compactness is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique.


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## Body

## 1.  Introduction

The study of stochastic functional differential equations is motivated by the fact that, when one wants to model some evolution phenomena arising in physics, chemistry, biology, and other sciences, some hereditary characteristics such as aftereffect, time-lag, and time delay can appear in the variables. On the other hand, one of the most interesting problems concerning stochastic functional differential equations is to understood the asymptotic behavior of the solutions when time grows to infinite, since it can provide useful information about the future of the phenomenon described in the model.It is known that the asymptotic behavior of random systems can be captured by a random attractor, which was introduced in [1, 2] as an extension of the attractor theory of deterministic systems in [3–5]. For stochastic PDEs without any hereditary features, the existence of random attractors has been investigated by many authors; see, for example, [6–20] and the references therein. However, this problem is not well studied in the case of stochastic retarded PDEs.In this paper, we investigate the asymptotic behavior of solutions to the following stochastic retarded reaction-diffusion equation with additive noise defined in the entire spaceℝd:
(1)du+(λu-Δu)dt=(f(ut)(x)+g(x))dt+dW,
where λ is a positive constant, g is a given function defined on ℝd, f is a nonlinear functional satisfying certain conditions, and W is a two-sided infinite dimensional Wiener process on a probability space which will be specified later.We note that the asymptotic behavior of several deterministic retarded PDEs on bounded domains was studied in [21–25], and the case of retarded Navier-Stokes equations on some unbounded domains was treated in [26]. The random attractor for retarded stochastic differential equations was considered in [27] by monotone methods. Recently, in the case of stochastic retarded lattice dynamical systems defined on the entire integer set, the existence of a random attractor was proved in [28, 29]. Here we prove the existence of a random attractor for the stochastic retarded reaction-diffusion equation defined in ℝd. It is worth mentioning that the asymptotic behavior of the nonretarded version of (1) was investigated recently in [9].Notice that Sobolev embeddings are not compact when domains are unbounded. This introduces a major obstacle for proving existence of attractors for PDEs on unbounded domains. Under certain circumstances, the tail-estimates method can be used to deal with the problem caused by the unboundedness of domains. This approach was developed in [30, 31] for deterministic nonretarded PDEs and used in [9–11, 13, 17, 18] for stochastic systems. At the same time, the present of delays is another obstacle, which makes phase spaces not reflexive and increases the difficulty of uniform estimates. In this paper, we will develop a tail-estimates approach for stochastic retarded PDEs on unbounded domains and prove the existence of a compact random attractor for the stochastic retarded reaction-diffusion equation (1), in particular, defined on the unbounded domain ℝd. The idea is based on the observation that the solutions of the equation are uniformly small when space and time variables are sufficiently large. It is clear that our method can be used for a variety of other equations, as it was for the nonretarded case.For convenience, hereafter we adopt the following notations. We denote by∥·∥ and (·,·) the norm and the inner product in L2(ℝd). Otherwise, the norm of a general Banach space X is written as ∥·∥X. For ν>0, let 𝒞 denote the Banach space of all continuous functions ξ:[-ν,0]→L2(ℝd) endowed with the supremum norm ∥ξ∥𝒞=sups∈[-ν,0]∥ξ(s)∥. For any real number a≤b, t∈[a,b], and any continuous function u:[a-ν,b]→L2(ℝd), ut denotes the element of 𝒞 given by ut(s)=u(t+s) for s∈[-ν,0].The rest of the paper is organized as follows. In the next section, we introduce basic concepts concerning random dynamical systems and random attractors. In Section3, we define a continuous random dynamical system for the stochastic retarded reaction-diffusion equation on ℝd. The existence of the random attractor is given in Section 4.


## 2. Preliminaries

In this section, we introduce some basic concepts related to random attractors for random dynamical systems. The reader is referred to [1, 2, 6, 32–34] for more details.Let(X,∥·∥X) be a separable Banach space with Borel σ-algebra ℬ(X) and (Ω,ℱ,ℙ) a probability space.Definition 1.
(
Ω
,
ℱ
,
ℙ
,
(
ϑ
t
)
t
∈
ℝ
) is called a metric dynamical system if ϑ:ℝ×Ω→Ω is (ℬ(ℝ)⊗ℱ,ℱ)-measurable, and ϑ0 is the identity on Ω, ϑs+t=ϑt∘ϑs for all s,t∈ℝ and ϑtℙ=ℙ for all t∈ℝ.Definition 2.
A setA⊂Ω is called invariant with respect to (ϑt)t∈ℝ, if, for all t∈ℝ, it holds that
(2)ϑt-1A=A.Definition 3.
A continuous random dynamical system onX over a metric dynamical system (Ω,ℱ,ℙ,(ϑt)t∈ℝ) is a mapping
(3)φ:ℝ+×Ω×X⟶X,(t,ω,x)⟶φ(t,ω,x),
which is (ℬ(ℝ+)⊗ℱ⊗ℬ(X),ℬ(X))-measurable, and, for all ω∈Ω,(i)
φ(t,ω,·):X→X is continuous for all t∈ℝ+;(ii)
φ(0,ω,·) is the identity on X;(iii)
φ(t+s,ω,·)=φ(t,ϑsω,·)∘φ(s,ω,·) for all s,t∈ℝ+.Definition 4.
A random setD is a multivalued mapping D:Ω→ℬ(X) such that, for every x∈X, the mapping ω→d(x,D(ω)) is measurable, where d(x,B) is the distance between the element x and the set B⊂X. It is said that the random set is bounded (resp., closed or compact) if D(ω) is bounded (resp., closed or compact) for ℙ a.e. ω∈Ω.Definition 5.
A random variabler:Ω→(0,∞) is called tempered with respect to (ϑt)t∈ℝ, if for ℙ a.e. ω∈Ω(4)limt→∞e-βtr(ϑ-tω)=0∀β>0.
A random set D is called tempered if D(ω) is contained in a ball with center zero and tempered radius r(ω) for all ω∈Ω.Remark 6.
Ifr>0 is tempered, then(1)
for anyτ∈ℝ, r(ϑτ·) is tempered;(2)
for anyα>0 and ℙ a.e. ω∈Ω(5)R(ω)=∫-∞0‍eαsr(ϑsω)ds<∞,andR is tempered.Moreover, if, forℙ a.e. ω∈Ω, r(ϑtω) is continuous in t, then R(ϑtω) is continuous in t for such ω, and, for any ν>0, supσ∈[-ν,0]r(ϑσ·) is tempered.Hereafter, we always assume thatφ is a continuous random dynamical system over (Ω,ℱ,ℙ,(ϑt)t∈ℝ), and 𝒟 is a collection of random subsets of X.Definition 7.
A random setK is called a random absorbing set in 𝒟 if, for every B∈𝒟 and ℙ a.e. ω∈Ω, there exists tB(ω)>0 such that
(6)φ(t,ϑ-tω,B(ϑ-tω))⊆K(ω)∀t≥tB(ω).Definition 8.
A random set𝒜 is called a 𝒟-random attractor (𝒟-pullback attractor) for φ if the following hold: (i)
𝒜 is a random compact set;(ii)
𝒜 is strictly invariant; that is, for ℙ a.e. ω∈Ω and all t≥0,
(7)φ(t,ω,𝒜(ω))=𝒜(ϑtω);(iii)
𝒜 attracts all sets in 𝒟; that is, for all B∈𝒟 and ℙ a.e. ω∈Ω,
(8)limt→∞d(φ(t,ϑ-tω,B(ϑ-tω)),𝒜(ω))=0,whered is the Hausdorff semimetric given by d(E,F)=supx∈Einfy∈F∥x-y∥X for any E⊆X and F⊆X.
We remark that if𝒜∈𝒟, then this attractor is unique [33].Definition 9.
φ is said to be 𝒟-pullback asymptotically compact in X if, for all B∈𝒟 and ℙ a.e. ω∈Ω, {φ(tn,ϑ-tnω,xn)}n=1∞ has a convergent subsequence in X whenever tn→∞, and xn∈B(ϑ-tnω).The following existence result on a random attractor for a continuous random dynamical system can be found in [2, 34]. First, recall that a collection 𝒟 of random subsets of X is called inclusion closed if whenever E is an arbitrary random set and F is in 𝒟 with E(ω)⊂F(ω) for ℙ a.e. ω∈Ω, then E must belong to 𝒟.Proposition 10.
Let𝒟 be an inclusion-closed collection of random subsets of X and φ a continuous random dynamical system on X over (Ω,ℱ,ℙ,(ϑt)t∈ℝ). Suppose that K∈𝒟 is a closed random absorbing set for φ in 𝒟 and φ is 𝒟-pullback asymptotically compact in X. Then φ has a unique 𝒟-random attractor 𝒜 which is given by
(9)𝒜(ω)=⋂τ≥0‍⋃t≥τ‍φ(t,ϑ-tω,K(ϑ-t))¯.In this paper, we will take𝒟 as the collection of all tempered random subsets of 𝒞 and prove the stochastic retarded reaction-diffusion equation on ℝd has a 𝒟-random attractor.


## 3. Stochastic Retarded Reaction-Diffusion Equations onℝd with Additive Noise

In this section, we show that there is a continuous random dynamical system generated by the stochastic retarded reaction-diffusion equation onℝd with additive white noise:
(10)du+(λu-Δu)dt=(f(ut)(x)+g(x))dt+dW,du+(λu-Δu)dt=dt=dt=x∈ℝd,t>0,
with the initial condition
(11)u(t,x)=u0(t,x),x∈ℝd,t∈[-ν,0].
Here λ is a positive constant, g is a given function in L2(ℝd), W is an L2(ℝd)-valued two-sided Wiener process with a symmetric nonnegative finite trace covariance operator Q defined on a probability space which will be specified below, and f:𝒞→L2(ℝd) is a continuous mapping satisfying the following conditions: (A1)
f(0)=0;(A2)
there exists a positive continuous functionlf(r) with
(12)limr→∞lf(r)rk0=0for some positive integerk0 such that, for all ξ, η∈𝒞 with ∥ξ∥≤r and ∥η∥≤r,
(13)∥f(ξ)-f(η)∥≤lf(r)∥ξ-η∥𝒞;(A3)
there exist positive constantsα0 and cf such that, for all α∈(0,α0),  t>0,  u∈C([-ν,t];L2(ℝd)), and x∈ℝd,
(14)∫0t‍eαs|f(us)(x)|2ds≤cf2∫-νt‍eαs|u(s)(x)|2ds;(A4)
λ>cf.In the sequel, we consider the probability space(Ω,ℱ,ℙ) where
(15)Ω={ω∈C(ℝ,L2(ℝd)):ω(0)=0},ℱ is the Borel σ-algebra induced by the compact-open topology of Ω, and ℙ the corresponding Wiener measure on (Ω,ℱ) with respect to the covariance operator Q. Let
(16)ϑtω(·)=ω(·+t)-ω(t),t∈ℝ.
Then (Ω,ℱ,ℙ,(ϑt)t∈ℝ) is an ergodic metric dynamical system. Since the above probability space is canonical, we have
(17)W(t,ω)=ω(t),W(t,ϑsω)=W(t+s,ω)-W(s,ω).
Similar to Proposition A.1 in [34], we can find that there exists a full ℙ-measure {ϑt}t∈ℝ-invariant set Ω~∈ℱ such that for each ω∈Ω~(18)limt→±∞∥W(t,ω)∥t=0.Letℱ¯ be the ℙ-completion of ℱ, and let
(19)ℱt=⋁s≤tℱst,t∈ℝ,
with
(20)ℱst=σ{W(τ2)-W(τ1):s≤τ1≤τ2≤t}∨𝒩,
where σ{W(τ2)-W(τ1):s≤τ1≤τ2≤t} is the smallest σ-algebra generated by the random variable W(τ2)-W(τ1) for all τ1, τ2 such that s≤τ1≤τ2≤t and 𝒩 is the collection of ℙ-null sets of ℱ¯.Note that(21)ϑτ-1ℱst=ℱs+τt+τ,
so (Ω,ℱ,ℙ,(ϑt)t∈ℝ,(ℱst)s≤t) is a filtered metric dynamical system (see [32, pages 72 and 91] for more details). In addition, it is important to note that the measurability of ϑ is not true if we replace ℱ by its completion; see [32, page 547] for details.In this paper, the solution of problem (10)-(11) is interpreted in a mild sense:
(22)u(t)=S(t)u0(0)+∫0t‍S(t-s)(f(us)+g)ds+∫0t‍S(t-s)dW(s),t>0,u(t)=u0(t),t∈[-ν,0],ℙ a.s. for any u0∈𝒞, where S(t) is the analytic semigroup on L2(ℝd) generated by Δ-λI. By the theory in [35], we deal with (22) on the complete probability space (Ω,ℱ¯,ℙ).We now associate a continuous random dynamical system with the stochastic retarded reaction-diffusion equation (10)-(11) over (Ω,ℱ,ℙ,(ϑt)t∈ℝ). To this end, we introduce an auxiliary Ornstein-Uhlenbeck process on (Ω,ℱ,ℙ,(ϑt)t∈ℝ) and transform the stochastic retarded reaction-diffusion equation into a random one. Let
(23)z(t,ω)={∫-∞t‍(Δ-λI)S(t-s)(W(s,ω)-W(t,ω))ds,ω∈Ω~,0,ω∉Ω~.
Then by (18), (23) is well defined. The process z(t), t∈ℝ, is a stationary, Gaussian process. By Lemma 5.13 in [35], we can see that it is a mild solution of the linear equation
(24)dz(t)=(Δ-λI)z(t)dt+dW(t).
That is, for all t∈ℝ and ℙ a.s. 						(25)z(t)=∫-∞t‍S(t-s)dW(s).
Moreover, the random variable ∥z(0,ω)∥ is tempered, and, for each ω∈Ω, the mapping t→z(t,ω) is continuous.Settingv(t)=u(t)-z(t) for t≥-ν in (22), then by (25), we obtain a deterministic equation, ℙ a.s. in Ω,
(26)v(t)=S(t)v0(0)+∫0t‍S(t-s)(f(vs+zs)+g)ds,t>0,v(t)=v0(t),t∈[-ν,0],
which is the mild form of the evolution equation
(27)dvdt=Δv-λv+f(vt+zt)+g,
with the initial condition
(28)v(t)=v0(t),t∈[-ν,0].
Here v0(t)=u0(t)-z0(t,ω), t∈[-ν,0].Problem (27)-(28) is a deterministic partial functional differential equation with random coefficients, which can be solved pathwise. We now establish the following result for problem (27)-(28).Theorem 11.
LetT>0 and ω∈Ω be fixed. Then the following properties hold:(1)
for eachv0∈𝒞, problem (27)-(28) has a unique mild solution v(·,ω,v0) that belongs to
(29)C([-ν,T];L2(ℝd))∩W1,2(ɛ,T;L2(ℝd))∩L2(ɛ,T;H2(ℝd)),for anyɛ∈(0,T), and for a.e. t∈[0,T](30)dvdt=Δv-λv+f(vt+zt)+g.(2)
Letv1(·,ω,v10) and v2(·,ω,v20) be the mild solutions of problem (27)-(28) for the initial data v10 and v20, respectively. Then there exists a constant c(T)>0 such that for all t∈[0,T](31)∥v1t(·,ω,v10)-v2t(·,ω,v20)∥𝒞≤∥v10-v20∥𝒞e(c(T)-λ)t+λν.Proof.
(1) By(A1)-(A2), following the same lines of Theorem 6.1.4 in [36], one can show that, for each v0∈𝒞, there exists a Tmax≤∞ such that (26) has a unique solution v on [0,Tmax). Moreover, if Tmax<∞, then
(32)limsupt↑Tmax∥vt∥𝒞=∞.
We prove now that this local solution is a global one. For fixed T∈(0,Tmax), by regularity of mild solutions for an analytic semigroup [37, page 145], we inform that
(33)v∈C([-ν,T];L2(ℝd))∩W1,2(ɛ,T;L2(ℝd))∩L2(ɛ,T;H2(ℝd)),
for any ɛ∈(0,T), and (27) holds for a.e. t∈[0,T]. Then, taking the inner product of (27) with v in L2(ℝd), we get that(34)12ddt∥v∥2+λ∥v∥2+∥∇v∥2=(f(vt+zt),v)+(g,v).
By (A4), we can choose β>0 small enough such that 2λ>2cf+β. Using the Young inequality, we find that
(35)(f(vt+zt),v)≤∥f(vt+zt)∥∥v∥≤cf2∥v∥2+12cf∥f(vt+zt)∥2,(g,v)≤∥g∥∥v∥≤β2∥v∥2+12β∥g∥2.
Then it follows from (34) and (35) that
(36)ddt∥v∥2+2∥∇v∥2≤-(2λ-cf-β)∥v∥2+1cf∥f(vt+zt)∥2+1β∥g∥2.
Choose α∈(0,α0) small enough such that 2λ>2cf+α+β. Then by (36), we obtain
(37)ddt(eαt∥v∥2)+2eαt∥∇v∥2≤-(2λ-cf-α-β)eαt∥v∥2+eαtcf∥f(vt+zt)∥2+eαtβ∥g∥2.
Now, we can also choose γ>0 such that 2λ>(2+γ)cf+α+β. Integrating (37) over [0,t]  (t∈[0,T]) leads to
(38)eαt∥v(t)∥2≤∥v(0)∥2-(2λ-cf-α-β)∫0t‍eαs∥v(s)∥2ds+1cf∫0t‍eαs∥f(vs+zs)∥2ds+∥g∥2β∫0t‍eαsds.
Using the Young inequality and (A3), we find that
(39)1cf∫0t‍eαs∥f(vs+zs)∥2ds≤cf∫-νt‍eαs∥v(s)+z(s)∥2ds≤cf∫0t‍eαs[(1+γ)∥v(s)∥2+(1+γ-1)∥z(s)∥2]ds+cf∫-ν0‍eαs∥v(s)+z(s)∥2ds.
Then by (38) and (39), we obtain
(40)eαt∥v(t)∥2≤-(2λ-(2+γ)cf-α-β)×∫0t‍eαs∥v(s)∥2ds+∥v(0)∥2+∥g∥2αβeαt+cf∫-ν0‍eαs∥v(s)+z(s)∥2ds+c1∫0t‍eαs∥z(s)∥2ds≤∥v(0)∥2+∥g∥2αβeαt+2νcf[∥v0∥𝒞2+∥z0∥𝒞2]+c1∫0t‍eαs∥z(s)∥2ds,
where c1=cf(1+γ-1). Consequently,
(41)∥v(t)∥2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]e-αt+c1∫0t‍eα(s-t)∥z(s)∥2ds+∥g∥2αβ.
Hence, for fixed σ∈[-ν,0], we get that, for t∈(-σ,T],
(42)∥v(t+σ)∥2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]e-α(t+σ)+c1∫0t+σ‍eα(s-t-σ)∥z(s)∥2ds+∥g∥2αβ≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eα(ν-t)+c1eαν∫0t‍eα(s-t)∥z(s)∥2ds+∥g∥2αβ,
and, for t∈[0,-σ],
(43)∥v(t+σ)∥2≤∥v0∥𝒞2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eα(ν-t).
In view of (42) and (43), we find that, for all t∈[0,T],
(44)∥vt∥𝒞2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eα(ν-t)+c1eαν∫0t‍eα(s-t)∥z(s)∥2ds+∥g∥2αβ.
Therefore, for all t∈[0,T],
(45)∥vt∥𝒞2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eαν+c1eαν∫0T‍∥z(s)∥2ds+∥g∥2αβ,
which, together with (32), implies that Tmax=∞. This proves property (1).
(2) By (44), there exists a constant r(T)>0 such that
(46)∥v1t∥𝒞≤r(T),∥v2t∥𝒞≤r(T).
Then from (A2) and (26), we have that for t∈[0,T](47)∥v1(t)-v2(t)∥≤e-λt∥v10(0)-v20(0)∥+lf(r(T))∫0t‍e-λ(t-s)∥v1s-v2s∥𝒞ds.
Hence, for fixed σ∈[-ν,0], we get that, for t∈(-σ,T],
(48)∥v1(t+σ)-v2(t+σ)∥≤e-λ(t+σ)∥v10(0)-v20(0)∥+lf(r(T))∫0t+σ‍e-λ(t+σ-s)∥v1s-v2s∥𝒞ds≤eλ(ν-t)∥v10-v20∥𝒞+lf(r(T))eλν∫0t‍e-λ(t-s)∥v1s-v2s∥𝒞ds,
and, for t∈[0,-σ],
(49)∥v1(t+σ)-v2(t+σ)∥≤∥v10-v20∥𝒞≤eλ(ν-t)∥v10-v20∥𝒞.
In view of (48) and (49), we find that, for all t∈[0,T],
(50)∥v1t-v2t∥≤eλ(ν-t)∥v10-v20∥𝒞+lf(r(T))eλν∫0t‍e-λ(t-s)∥v1s-v2s∥𝒞ds.
The Gronwall inequality implies that, for all t∈[0,T],
(51)∥v1t-v2t∥𝒞≤∥v10-v20∥𝒞e(lf(r(T))eλν-λ)t+λν.
This prove property (2). The proof is complete.Conversely, if, for eachω∈Ω, v(t,ω,v0)∈C([-ν,∞);L2(ℝd)) is a mild solution of problem (27)-(28) with v0(·)=u0(·)-z0(·,ω), then by (25) the continuous process
(52)u(t,ω,u0)=v(t,ω,v0)+z(t,ω)
is a mild solution of problem (10)-(11).Theorem 12.
Problem (27)-(28) generates a continuous random dynamical system ϕ over (Ω,ℱ,ℙ,(ϑ)t∈ℝ), where
(53)ϕ(t,ω,v0)=vt(·,ω,v0),fort≥0,ω∈Ω,v0∈𝒞.
Moreover, if one defines ψ by
(54)ψ(t,ω,u0)=ut(·,ω,u0),fort≥0,ω∈Ω,u0∈𝒞,
then ψ is another continuous random dynamical system associated with problem (10)-(11).Proof.
By a classical successive approximation argument, one can easily show that, for fixedv0∈𝒞, v(t,ω,v0) is an ℱt-adapted continuous process. Hence, for fixed v0∈𝒞, ϕ(t,ω,v0) is also an ℱt-adapted continuous process. On the other hand, from property (2) of Theorem 11, it follows that, for fixed t≥0 and ω∈Ω, ϕ(t,ω,·):[0,∞)×𝒞→𝒞 is continuous. Consequently, ϕ(t,ω,v0) is (ℬ(ℝ+)⊗ℱ⊗ℬ(𝒞),ℬ(𝒞))-measurable.
By (26) we have that, for s, t≥0 and σ∈[-ν,0],
(55)ϕ(t,ϑsω,ϕ(s,ω,v0))(σ)=S(t+σ)ϕ(s,ω,v0)(0)+∫0t+σ‍S(t+σ-τ)×(f(ϕ(τ,ϑsω,ϕ(s,ω,v0))+zτ(ϑsω))+g)dτ.
Then again by (26) we get
(56)ϕ(t,ϑsω,ϕ(s,ω,v0))(σ)=S(t+s+σ)v0(0)+∫0s‍S(s-τ)×(f(ϕ(τ,ω,v0)+zτ(ω))+g)dτ+∫st+s+σ‍S(t+s+σ-τ)×(f(ϕ(τ-s,ϑsω,ϕ(s,ω,v0))+zτ(ω))+g)dτ.
For each ω∈Ω consider
(57)Φ(τ,ω,v0)={ϕ(τ,ω,v0),if0≤τ≤s,ϕ(τ-s,ϑsω,ϕ(s,ω,v0)),ifs<τ≤t+s.
Then for τ=t+s we have
(58)Φ(t+s,ω,v0)=ϕ(t,ϑsω,ϕ(s,ω,v0))fors,t≥0.
It follows from (56) that
(59)Φ(t+s,ω,v0)(σ)=S(t+s+σ)v0(0)+∫0t+s+σ‍S(t+s+σ-τ)×(f(Φ(τ,ω,v0)+zτ(ω))+g)dτ,
for all σ∈[-ν,0]. By the uniqueness of the solution of (26) we find that
(60)Φ(t+s,ω,v0)=ϕ(t+s,ω,v0),
while (58) implies
(61)ϕ(t+s,ω,v0)=ϕ(t,ϑsω,ϕ(s,ω,v0))fors,t≥0.
Therefore, ϕ is a continuous random dynamical system.
As forψ, noticing that
(62)ψ(t,ω,u0)=ϕ(t,ω,u0-z0(ω))+zt(ω),(t,ω,u0u0u0)fort≥0,ω∈Ω,u0∈𝒞,
we get from (61) that, for s,t≥0,
(63)ψ(t,ϑsω,ψ(s,ω,u0))=ϕ(t,ϑsω,ϕ(s,ω,u0-z0(ω)))+zt(ϑsω)=ϕ(t+s,ω,u0-z0(ω))+zt+s(ω)=ψ(t+s,ω,u0).
Therefore, ψ is also a continuous random dynamical system. Furthermore, ϕ and ψ are conjugated random dynamical systems; that is,
(64)ψ(t,ω,T(ω,ξ))=T(ϑtω,ϕ(t,ω,ξ)),foranyξ∈𝒞,
where, for every ω∈Ω, T(ω,ξ)=ξ+z0(ω) is a homeomorphism of 𝒞. The proof is complete.


## 4. Existence of Random Attractors

In this section, we prove the existence of a𝒟-random attractor for the random dynamical system ψ associated with the stochastic retarded reaction-diffusion equation (10)-(11) on ℝd. We first establish the existence of a 𝒟-random attractor for its conjugated random dynamical system ϕ, then the existence of a 𝒟-random attractor for ψ follows from the conjugation relation between ϕ and ψ. To this end, we will derive uniform estimates on the mild solutions of problem (27)-(28) when t→∞ with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness for ϕ. In particular, we will show that the tails of the solutions, that is, solutions evaluated at large value of |x|, are uniformly small when time is sufficiently large.From now on, we always assume that𝒟 is the collection of all tempered subsets of 𝒞 with respect to (Ω,ℱ,ℙ,(ϑt)t∈ℝ). The next lemma shows that ϕ has a random absorbing set in 𝒟.Lemma 13.
There existsK∈𝒟 such that K is a random absorbing set for ϕ in 𝒟; that is, for any B∈𝒟 and ℙ a.e. ω∈Ω, there exists TB(ω)>0 such that
(65)ϕ(t,ϑ-tω,B(ϑ-tω))⊆K(ω)∀t≥TB(ω).Proof.
Replacingω with ϑ-tω in (41) and (44), we get that, for all t≥0,
(66)∥v(t,ϑ-tω,v0(ϑ-tω))∥2≤[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]e-αt+c1∫0t‍eα(s-t)∥z(s,ϑ-tω)∥2ds+∥g∥2αβ≤[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]e-αt+c1∫-∞0‍eαs∥z(0,ϑsω)∥2ds+∥g∥2αβ,(67)∥vt(ϑ-tω,v0(ϑ-tω))∥𝒞2≤[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]eα(ν-t)+c1eαν∫0t‍eα(s-t)∥z(s,ϑ-tω)∥2ds+∥g∥2αβ≤[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]eα(ν-t)+c1eαν∫-∞0‍eαs∥z(0,ϑsω)∥2ds+∥g∥2αβ.
By assumption, B∈𝒟 is tempered. On the other hand, by Remark 6, ∥z0(ω)∥𝒞2 is also tempered. Therefore, if v0(ϑ-tω)∈B(ϑ-tω), then there exists TB(ω)>0 such that, for all t≥TB(ω),
(68)[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]e-αt≤1+r(ω),
where
(69)r(ω)=∫-∞0‍eαs∥z(0,ϑsω)∥2ds
is tempered by Remark 6. Then it follows from (66), (67), and (68) that for all t≥TB(ω),
(70)∥v(t,ϑ-tω,v0(ϑ-tω))∥2≤(c1+1)r(ω)+∥g∥2αβ+1,(71)∥vt(ϑ-tω,v0(ϑ-tω))∥𝒞2≤(c1+1)eανr(ω)+∥g∥2αβ+1.
Given ω∈Ω, we define
(72)K(ω)={ξ∈𝒞:∥ξ∥𝒞2≤r1(ω)},
where
(73)r1(ω)=(c1+1)eανr(ω)+∥g∥2αβ+1
is tempered. Then K∈𝒟. Further, (71) indicates that K is a random absorbing set for ϕ in 𝒟, which completes the proof.We next derive uniform estimates forv in H1(ℝd). From property (1) of Theorem 11 and the fact that [37, page 165]
(74)W1,2(ɛ,T;L2(ℝd))∩L2(ɛ,T;H2(ℝd))⊂C([ɛ,T];H1(ℝd)),
for any 0<ɛ<T<∞, we get that v∈C((0,∞);H1(ℝd)).Lemma 14.
LetB∈𝒟 and v0(ω)∈B(ω). Then for ℙ a.e. ω∈Ω, the solution v(t,ω,v0(ω)) of problem (27)-(28) satisfies, for all t≥TB(ω),
(75)∫tt+1‍∥vs(ϑ-t-1ω,v0(ϑ-t-1ω))∥𝒞2ds≤r2(ω),∫tt+1‍∥∇v(s,ϑ-t-1ω)∥2ds≤r3(ω),
where r2 and r3 are tempered and TB is the random function in Lemma 13.Proof.
Replacingω with ϑ-τ-1ω in (44), we get that, for all τ≥0 and t∈[τ,τ+1],
(76)∥vt(ϑ-τ-1ω,v0(ϑ-τ-1ω))∥𝒞2≤[(1+2νcf)∥v0(ϑ-τ-1ω)∥𝒞2+2νcf∥z0(ϑ-τ-1ω)∥𝒞2]eα(ν-t)+c1eαν∫0t‍eα(s-t)∥z(s,ϑ-τ-1ω)∥2ds+∥g∥2αβ.
Integrating (76) over the interval [τ,τ+1] leads to
(77)∫ττ+1‍∥vt(ϑ-τ-1ω,v0(ϑ-τ-1ω))∥𝒞2dt≤[(1+2νcf)∥v0(ϑ-τ-1ω)∥𝒞2+2νcf∥z0(ϑ-τ-1ω)∥𝒞2]×∫ττ+1‍eα(ν-t)dt+c1eαν∫ττ+1‍∫0t‍eα(s-t)∥z(0,ϑs-τ-1ω)∥2dsdt+∥g∥2αβ≤[(1+2νcf)∥v0(ϑ-τ-1ω)∥𝒞2+2νcf∥z0(ϑ-τ-1ω)∥𝒞2]eα(ν-τ)+c1eα(ν+1)r(ω)+∥g∥2αβ,
which, together with (68), implies that, for all t≥TB(ω),
(78)∫tt+1‍∥vs(ϑ-t-1ω,v0(ϑ-t-1ω))∥𝒞2ds≤[c1eα(ν+1)+eαν]r(ω)+∥g∥2αβ+eαν:=r2(ω).
Obviously, r2 is tempered. Integrating (37) over the interval [t,t+1] leads to
(79)eα(t+1)∥v(t+1)∥2-eαt∥v(t)∥2+2∫tt+1‍eαs∥∇v(s)∥2ds≤-(2λ-cf-α-β)∫tt+1‍eαs∥v(s)∥2ds+∥g∥2β∫tt+1‍eαsds+1cf∫tt+1‍eαs∥f(vs+zs)∥2ds.
Using the Young inequality and (A3), we obtain that
(80)1cf∫tt+1‍eαs∥f(vs+zs)∥2ds≤cf∫t-νt+1‍eαs∥v(s)+z(s)∥2ds≤cf∫tt+1‍eαs[(1+γ)∥v(s)∥2+(1+γ-1)∥z(s)∥2]ds+cf∫t-νt‍eαs∥v(s)+z(s)∥2ds.
It follows from (79) and (80) that
(81)eα(t+1)∥v(t+1)∥2-eαt∥v(t)∥2+2∫tt+1‍eαs∥∇v(s)∥2ds≤-(2λ-(2+γ)cf-α-β)∫tt+1‍eαs∥v(s)∥2ds+∥g∥2βeα(t+1)+∫t-νt‍eαs∥v(s)+z(s)∥2ds+c1∫tt+1‍eαs∥z(s)∥2ds.
Thus,
(82)2∫tt+1‍eαs∥∇v(s)∥2ds≤eαt∥v(t)∥2+∫t-νt‍eαs∥v(s)+z(s)∥2ds+c1∫tt+1‍eαs∥z(s)∥2ds+∥g∥2βeα(t+1).
Replacing ω with ϑ-τ-1ω in (82), we get from (71) that, for ℙ a.e. ω∈Ω and all t≥TB(ω),
(83)2∫tt+1‍∥∇v(s,ϑ-t-1ω)∥2ds≤∥v(t,ϑ-t-1ω)∥2+∫t-νt‍∥v(s,ϑ-t-1ω)+z(s,ϑ-t-1ω)∥2ds+c1∫tt+1‍eα(s-t)∥z(s,ϑ-t-1ω)∥2ds+∥g∥2βeα≤∥v(t,ϑ-t-1ω)∥2+2∫t-νt‍[∥v(s,ϑ-t-1ω)∥2+∥z(s,ϑ-t-1ω)∥2]ds+c1eα∫-10‍eαs∥z(0,ϑsω)∥2ds+∥g∥2βeα≤(1+2ν)∥vt(ϑ-t-1ω)∥𝒞2+2ν∥z0(ϑ-1ω)∥𝒞2+c1eαr(ω)+∥g∥2βeα≤(1+2ν)r1(ϑ-1ω)+2ν∥z0(ϑ-1ω)∥𝒞2+c1eαr(ω)+∥g∥2βeα.
Therefore, we have that, for all t≥TB(ω),
(84)∫tt+1‍∥∇v(s,ϑ-t-1ω)∥2ds≤1+2ν2r1(ϑ-1ω)+ν∥z0(ϑ-1ω)∥𝒞2+c12eαr(ω)+∥g∥22βeα:=r3(ω).
By Remark 6, r3 is tempered. Then the lemma follows from (78) and (84).Lemma 15.
LetB∈𝒟 and v0(ω)∈B(ω). Then for ℙ a.e. ω∈Ω, the solution v(t,ω,v0(ω)) of problem (27)-(28) satisfies, for all t≥TB(ω)+ν+1 and σ1,σ2∈[-ν,0],
(85)∥∇v(t,ϑ-tω,v0(ϑ-tω))∥2≤r4(ω),|∫t+σ1t+σ2‍∥Δv(s,ϑ-tω,v0(ϑ-tω))∥2ds|≤r5(ω),
where r4 and r5 are tempered and TB is the random function in Lemma 13.Proof.
Taking the inner product of (27) with Δv in L2(ℝd), we get that
(86)12ddt∥∇v∥2+λ∥∇v∥2+∥Δv∥2=-(f(vt+zt),Δv)-(g,Δv).
Using the Young inequality, we obtain that
(87)-(f(vt+zt),Δv)≤∥f(vt+zt)∥∥Δv∥≤14∥Δv∥2+∥f(vt+zt)∥2,-(g,Δv)≤∥g∥∥Δv∥≤14∥Δv∥2+∥g∥2.
It follows from (86) and (87) that
(88)12ddt∥∇v∥2+λ∥∇v∥2+12∥Δv∥2≤∥f(vt+zt)∥2+∥g∥2.
Thus,
(89)ddt∥∇v∥2+∥Δv∥2≤2∥f(vt+zt)∥2+2∥g∥2.
Let TB(ω) be the positive constant in Lemma 13, and take t≥TB(ω) and s∈(t,t+1). Integrating (89) over the interval [s,t+1] leads to
(90)∥∇v(t+1)∥2≤∥∇v(s)∥2+2∫st+1‍∥f(vτ+zτ)∥2dτ+2∫st+1‍∥g∥2dτ≤∥∇v(s)∥2+2∫tt+1‍∥f(vτ+zτ)∥2dτ+2∥g∥2.
Integrating the above with respect tos over the interval [t,t+1], we get that
(91)∥∇v(t+1)∥2≤∫tt+1‍∥∇v(s)∥2ds+2∫tt+1‍∥f(vs+zs)∥2ds+2∥g∥2.
Replacing ω with ϑ-t-1ω and by (A1)-(A2) and Lemmas 13 and 14, we find that, for all t≥TB(ω),
(92)∥∇v(t+1,ϑ-t-1ω,v0(ϑ-t-1ω))∥2≤∫tt+1‍∥∇v(s,ϑ-t-1ω,v0(ϑ-t-1ω))∥2ds+2∥g∥2+2∫tt+1‍∥f(vs(ϑ-t-1ω)+zs(ϑ-t-1ω))∥2ds≤∫tt+1‍∥∇v(s,ϑ-t-1ω,v0(ϑ-t-1ω))∥2ds+2∥g∥2+4lf2(p1(ω))∫tt+1‍[∥(vsϑ-t-1ω)∥𝒞2+∥zs(ϑ-t-1ω)∥𝒞2]ds≤r3(ω)+4lf2(p1(ω))×[r2(ω)+supσ∈[-1,0]∥z(0,ϑσω)∥2]+2∥g∥2,
where
(93)p1(ω)=supσ∈[-1,0]{r1(ϑσω)+∥z(0,ϑσω)∥}
is tempered by Remark 6. Then we have that, for all t≥TB(ω)+1,
(94)∥∇v(t,ϑ-tω,v0(ϑ-tω))∥2≤r3(ω)+4lf2(p1(ω))×[r2(ω)+supσ∈[-1,0]∥z(0,ϑσω)∥2]+2∥g∥2:=r4(ω).
By(A2), one can easily see that lf(p) is also tempered for any tempered random variable p. Hence, r4 is tempered.
Lett≥ν, -ν≤σ1≤σ2≤0. Integrating (89) over the interval [t+σ1,t+σ2] leads to
(95)∥∇v(t+σ2)∥2+∫t+σ1t+σ2‍∥Δv(s)∥2ds≤∥∇v(t+σ1)∥2+2∫t+σ1t+σ2‍∥f(vs+zs)∥2ds+2∥g∥2ds.
Replacing ω with ϑ-tω in (95) and by (A1)-(A2), Lemma 13, and (94), we find that, for all t≥TB(ω)+ν+1 and σ1,σ2∈[-ν,0],
(96)∫t+σ1t+σ2‍∥Δv(s,ϑ-tω)∥2ds≤∥∇v(t+σ1,ϑ-tω)∥2+2∫t+σ1t+σ2‍∥f(vs(ϑ-tω)+zs(ϑ-tω))∥2ds+2ν∥g∥2≤2ν∥g∥2+supσ∈[-ν,0]∥∇v(t+σ,ϑ-tω)∥2+4lf2(p2(ω))×∫t-νt‍[∥(vsϑ-tω)∥𝒞2+∥zs(ϑ-tω)∥𝒞2]ds≤4νlf2(p2(ω))supσ∈[-ν,0]{r1(ϑσω)+∥z(0,ϑσω)∥2}+supσ∈[-ν,0]r4(ϑσω)+2ν∥g∥2:=r5(ω),
where
(97)p2(ω)=supσ∈[-ν,0]{r1(ϑσω)+∥z(0,ϑσω)∥}
is tempered by Remark 6. Similar to r4, r5 is also tempered. Then the lemma follows from (94) and (96).Lemma 16.
LetB∈𝒟 and v0(ω)∈B(ω). Then for every ɛ>0 and ℙ a.e. ω∈Ω, there exist T*=T*(B,ω,ɛ)>0 and R*=R*(ω,ɛ)>0 such that the solution v(t,ω,v0(ω)) of problem (27)-(28) satisfies, for all t≥T*,
(98)sups∈[-ν,0]∫|x|≥R*‍|vt(s,ϑ-tω,v0(ϑ-tω))(x)|2dx≤ɛ.Proof.
Letρ be a smooth function defined on ℝ+ such that 0≤ρ(s)≤1 for all s≥0, and
(99)ρ(s)={0,0≤s≤1,1,s≥2.
Then there exists a positive deterministic constant c2 such that |ρ′(s)|≤c2 for all s≥0. Taking the inner product of (27) with ρ(|x|2/k2)v in L2(ℝd), we get that
(100)12ddt∫ℝd‍ρ(|x|2k2)|v|2dx+λ∫ℝd‍ρ(|x|2k2)|v|2dx-∫ℝd‍Δvρ(|x|2k2)vdx=∫ℝd‍f(vt+zt)ρ(|x|2k2)vdx+∫ℝd‍gρ(|x|2k2)vdx.
We now estimate the terms in (100). First, we have that
(101)-∫ℝd‍Δvρ(|x|2k2)vdx=∫ℝd‍|∇v|2ρ(|x|2k2)dx+∫ℝd‍vρ′(|x|2k2)2xk2·∇vdx=∫ℝd‍|∇v|2ρ(|x|2k2)dx+∫k≤|x|≤2k‍vρ′(|x|2k2)2xk2·∇vdx.
Note that the second term on the right-hand side of (101) is bounded by
(102)|∫k≤|x|≤2k‍vρ′(|x|2k2)2xk2·∇vdx|≤22k∫k≤|x|≤2k‍|v||ρ′(|x|2k2)||∇v|dx≤4c2k∫k≤|x|≤2k|v|‍|∇v|dx≤4c2k(∥v∥2+∥∇v∥2).
By (101) and (102), we find that
(103)-∫ℝd‍Δvρ(|x|2k2)vdx≥∫ℝd‍|∇v|2ρ(|x|2k2)dx-4c2k(∥v∥2+∥∇v∥2).
For the right-hand side of (100), applying the Young inequality, we obtain that
(104)∫ℝd‍f(vt+zt)ρ(|x|2k2)vdx≤cf2∫ℝd‍ρ(|x|2k2)|v|2dx+12cf∫ℝd‍ρ(|x|2k2)|f(vt+zt)|2dx,∫ℝd‍gρ(|x|2k2)vdx≤12β∫ℝd‍g2ρ(|x|2k2)dx+β2∫ℝd‍ρ(|x|2k2)|v|2dx.
Then it follows from (100), (103), and (104) that
(105)ddt∫ℝd‍ρ(|x|2k2)|v|2dx≤-(2λ-cf-β)∫ℝd‍ρ(|x|2k2)|v|2dx+1cf∫ℝd‍ρ(|x|2k2)|f(vt+zt)|2dx+1β∫ℝd‍g2ρ(|x|2k2)dx+8c2k(∥v∥2+∥∇v∥2).
Consequently,
(106)ddt(eαt∫ℝd‍ρ(|x|2k2)|v|2dx)≤-(2λ-cf-α-β)eαt∫ℝd‍ρ(|x|2k2)|v|2dx+1cfeαt∫ℝd‍ρ(|x|2k2)|f(vt+zt)|2dx+1βeαt∫ℝd‍g2ρ(|x|2k2)dx+8c2keαt(∥v∥2+∥∇v∥2).
Take T1=T1(B,ω)≥TB(ω)+ν+1. For all t>T1, integrating (106) over the interval [T1,t] leads to
(107)eαt∫ℝd‍ρ(|x|2k2)|v(t)|2dx-eαT1∫ℝd‍ρ(|x|2k2)|v(T1)|2dx≤-(2λ-cf-α-β)∫T1t‍eαs∫ℝd‍ρ(|x|2k2)|v(s)|2dxds+1cf∫T1t‍eαs∫ℝd‍ρ(|x|2k2)|f(vs+zs)|2dxds+1β∫T1t‍eαs∫ℝd‍g2ρ(|x|2k2)dxds+8c2k∫T1t‍eαs(∥v∥2+∥∇v∥2)ds.
Using the Young inequality and (A3), we obtain that
(108)1cf∫T1t‍eαs∫ℝd‍ρ(|x|2k2)|f(vt+zt)|2dxds≤cf∫T1-νt‍eαs∫ℝd‍ρ(|x|2k2)|v(s)+z(s)|2dxds≤cf∫T1t‍eαs∫ℝd‍ρ(|x|2k2)[(1+γ)|v(s)|2cf∫T1t‍eαs∫ℝd‍ρ(|x|2k2)+(1+γ-1)|z(s)|2]dxds+2cf∫T1-νT1‍eαs∫ℝd‍ρ(|x|2k2)(|v(s)|2+|z(s)|2)dxds.
By (107) and (108), we find that
(109)eαt∫ℝd‍ρ(|x|2k2)|v(t)|2dx-eαT1∫ℝd‍ρ(|x|2k2)|v(T1)|2dx≤-(2λ-(2+γ)cf-α-β)×∫T1t‍eαs∫ℝd‍ρ(|x|2k2)|v(s)|2dxds+2cf∫T1-νT1‍eαs∫ℝd‍ρ(|x|2k2)(|v(s)|2+|z(s)|2)dxds+c1∫T1t‍eαs∫ℝd‍ρ(|x|2k2)|z(s)|2dxds+eαtαβ∫ℝd‍ρ(|x|2k2)g2dx+8c2k∫T1t‍eαs(∥v∥2+∥∇v∥2)ds≤2cf∫T1-νT1‍eαs∫ℝd‍ρ(|x|2k2)(|v(s)|2+|z(s)|2)dxds+c1∫T1t‍eαs∫ℝd‍ρ(|x|2k2)|z(s)|2dxds+2eαtαβ∫ℝd‍ρ(|x|2k2)g2dx+8c2k∫T1t‍eαs(∥v∥2+∥∇v∥2)ds.
Then we have, for all t>T1,
(110)∫ℝd‍ρ(|x|2k2)|v(t)|2dx≤[(1+2νcf)∥vT1(ω)∥𝒞2+2νcf∥zT1(ω)∥𝒞2]eα(T1-t)+c1∫T1t‍eα(s-t)∫ℝd‍ρ(|x|2k2)|z(s)|2dxds+1αβ∫ℝd‍ρ(|x|2k2)g2dx+8c2k∫T1t‍eα(s-t)(∥v∥2+∥∇v∥2)ds.
If we take t≥T1+ν, then by (110) we find that, for all σ∈[-ν,0],
(111)∫ℝd‍ρ(|x|2k2)|v(t+σ)|2dx≤[(1+2νcf)∥vT1(ω)∥𝒞2+2νcf∥zT1(ω)∥𝒞2]eα(T1-t-σ)+c1∫T1t+σ‍eα(s-t-σ)∫ℝd‍ρ(|x|2k2)|z(s)|2dxds+1αβ∫ℝd‍ρ(|x|2k2)g2dx+8c2k∫T1t+σ‍eα(s-t-σ)(∥v∥2+∥∇v∥2)ds≤[(1+2νcf)∥vT1(ω)∥𝒞2+2νcf∥zT1(ω)∥𝒞2]eα(T1+ν-t)+c1eαν∫T1t‍eα(s-t)∫ℝd‍ρ(|x|2k2)|z(s)|2dxds+1αβ∫ℝd‍ρ(|x|2k2)g2dx+8c2keαν∫T1t‍eα(s-t)(∥v∥2+∥∇v∥2)ds.
Then we have, for all t≥T1+ν,
(112)supσ∈[-ν,0]∫ℝd‍ρ(|x|2k2)|vt(σ)|2dx≤[(1+2νcf)∥vT1(ω)∥𝒞2+2νcf∥zT1(ω)∥𝒞2]eα(T1+ν-t)+c1eαν∫T1t‍eα(s-t)∫ℝd‍ρ(|x|2k2)|z(s)|2dxds+1αβ∫ℝd‍ρ(|x|2k2)g2dx+8c2keαν∫T1t‍eα(s-t)(∥v∥2+∥∇v∥2)ds.
Replacing ω with ϑ-tω, we find that
(113)supσ∈[-ν,0]∫ℝd‍ρ(|x|2k2)|vt(σ,ϑ-tω,v0(ϑ-tω))|2dx≤[(1+2νcf)∥vT1(ϑ-tω)∥𝒞2+2νcf∥zT1(ϑ-tω)∥𝒞2]eα(T1+ν-t)+c1eαν∫T1t‍eα(s-t)∫ℝd‍ρ(|x|2k2)|z(0,ϑs-tω)|2dxds+8c2keαν∫T1t‍eα(s-t)(∥v(s,ϑ-tω)∥2+8c2keαν∫T1t‍eα(s-t)+∥∇v(s,ϑ-tω)∥2)ds+1αβ∫ℝd‍ρ(|x|2k2)g2dx.
We now estimate the terms in (113) as follows. First, replacing t with T1 and then replacing ω with ϑT1-tω in (67), we find that
(114)∥vT1(ϑ-tω,v0(ϑ-tω))∥𝒞2≤[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]eα(ν-T1)t+c1eανr(ϑT1-tω)+∥g∥2αβ.
Thus,
(115)[(1+2νcf)∥vT1(ϑ-tω)∥𝒞2+2νcf∥zT1(ϑ-tω)∥𝒞2]eα(T1+ν-t)≤(1+2νcf)[(1+2νcf)∥v0(ϑ-tω)∥𝒞2≤(1+2νcf)+2νcf∥z0(ϑ-tω)∥𝒞2]eα(2ν-t)+(1+2νcf)∥g∥2αβeα(T1+ν-t)+c1(1+2νcf)eανr(ϑT1-tω)eα(2ν+T1-t)+2νcf∥z0(ϑT1-tω)∥𝒞2eα(T1+ν-t).
Since r(ω) and ∥z0(ω)∥𝒞2 are tempered functions, B∈𝒟 is tempered set and v0(ϑ-tω)∈B(ϑ-tω); we find from (115) that, for every ɛ>0, there exists T2=T2(B,ω,ɛ)≥T1+ν such that, for all t≥T2,
(116)[(1+2νcf)∥vT1(ϑ-tω)∥𝒞2+2νcf∥zT1(ϑ-tω)∥𝒞2]eα(T1+ν-t)≤ɛ4.
Next, note that
(117)∫-∞0‍eαs∥z(0,ϑsω)∥2ds=∫-∞0‍eαs∫ℝd‍|z(0,ϑsω)|2dxds<∞.
By the Lebesgue theorem of dominated convergence, there exists R1=R1(ω,ɛ)>0 such that, for all k≥R1,
(118)∫-∞0‍eαs∫|x|≥k‍|z(0,ϑsω)|2dxds≤ɛ4c1eαν.
Then it follows from (118) that, for all t≥T2 and k≥R1,
(119)∫T1t‍eα(s-t)∫ℝd‍ρ(|x|2k2)|z(0,ϑs-tω)|2dxds≤∫-∞0‍eαs∫ℝd‍ρ(|x|2k2)|z(0,ϑsω)|2dxds≤∫-∞0‍eαs∫|x|≥k‍|z(0,ϑsω)|2dxds≤ɛ4c1eαν.For the third term on the right-side of (113), we get from (70) and (94) that, for all t≥TB(ω)+ν+1,
(120)∥v(t,ϑ-tω)∥2+∥∇v(t,ϑ-tω)∥2≤r5(ω)+(c1+1)r(ω)+∥g∥2αβ+1:=r6(ω).
Then by (120), we obtain that
(121)∫T1t‍eα(s-t)(∥v(s,ϑ-tω)∥2+∥∇v(s,ϑ-tω)∥2)ds≤∫T1t‍eα(s-t)r6(ϑs-tω)ds≤∫-∞0‍eαsr6(ϑsω)ds.
Since r6 is tempered, the last integral in (121) exists. Therefore, there exists R2=R2(ω,ɛ)>0 such that, for all k≥R2,
(122)8c2keαν∫T1t‍eα(s-t)(∥v(s,ϑ-tω)∥2+∥∇v(s,ϑ-tω)∥2)ds≤ɛ4.
Finally, since g∈L2(ℝd), there exists R3=R3(ω,ɛ)>0 such that, for all k≥R3(ω,ɛ),
(123)1αβ∫ℝd‍ρ(|x|2k2)g2dx≤1αβ∫|x|≥k‍g2dx≤ɛ4.
Taking R4=R4(ω,ɛ)=max{R1,R2,R3}, it follows from (116), (119), (122), and (123) that, for all t≥T2 and k≥R4,
(124)supσ∈[-ν,0]∫|x|≥2k‍|vt(σ,ϑ-tω,v0(ϑ-tω))(x)|2dx≤supσ∈[-ν,0]∫ℝd‍ρ(|x|2k2)|vt(σ,ϑ-tω,v0(ϑ-tω))|2dx≤ɛ,
which completes the lemma.Lemma 17.
LetB∈𝒟 and v0(ω)∈B(ω). Then the solution v(t,ω,v0(ω)) of problem (27)-(28) satisfies, for all t≥TB(ω)+ν+1 and σ,τ1,τ2∈[-ν,0],
(125)∥vt(σ,ϑ-tω,v0(ϑ-tω))∥H1(ℝd)2≤r7(ω),∥vt(τ1,ϑ-tω,v0(ϑ-tω))-vt(τ2,ϑ-tω,v0(ϑ-tω))∥≤r8(ω)|τ1-τ2|1/2,
where r7 and r8 are tempered random functions.Proof.
By (120), we find that, for all t≥TB(ω)+ν+1 and σ∈[-ν,0],
(126)∥vt(σ,ϑ-tω,v0(ϑ-tω))∥H1(ℝd)2=∥v(t+σ,ϑ-tω,v0(ϑ-tω))∥2+∥∇v(t+σ,ϑ-tω,v0(ϑ-tω))∥2≤supσ∈[-ν,0]r6(ϑσω):=r7(ω).
Next, by (A1)-(A2) and Lemma 14, we find that, for all t≥TB(ω)+ν+1 and τ1,τ2∈[-ν,0] (assuming τ1≤τ2 for simplicity),
(127)∫t+τ1t+τ2‍∥f(vs(ϑ-tω)+zs(ϑ-tω))∥ds≤lf(p2(ω))∫t+τ1t+τ2‍∥vs(ϑ-tω)+zs(ϑ-tω)∥𝒞ds≤lf(p2(ω))∫t+τ1t+τ2‍[∥vs(ϑ-tω)∥𝒞+∥zs(ϑ-tω)∥𝒞]ds≤lf(p2(ω))[supσ∈[-ν,0]r1(ϑσω)+supσ∈[-2ν,0]∥z(ϑσ)∥]×|τ1-τ2|,(128)∫t+τ1t+τ2‍∥v(s,ϑ-tω)∥ds≤supσ∈[-ν,0]r1(ϑσω)|τ1-τ2|.
By Lemma 15, we obtain that
(129)∫t+τ1t+τ2‍∥Δv(s,ϑ-tω)∥ds≤{|τ1-τ2|∫t+τ1t+τ2‍∥Δv(s,ϑ-tω)∥2ds}1/2≤r5(ω)|τ1-τ2|1/2.
Then it follows from (127), (128), and (129) that
(130)∥vt(τ2,ϑ-tω,v0(ϑ-tω))-vt(τ1,ϑ-tω,v0(ϑ-tω))∥=∥v(t+τ2,ϑ-tω,v0(ϑ-tω))-v(t+τ1,ϑ-tω,v0(ϑ-tω))∥≤∥∫t+τ1t+τ2‍v′(s,ϑ-tω)ds∥≤∫t+τ1t+τ2‍∥Δv(s,ϑ-tω)∥ds+λ∫t+τ1t+τ2‍∥v(s,ϑ-tω)∥ds+∫t+τ1t+τ2‍∥f(vs(ϑ-tω)+zs(ϑ-tω))∥ds+∫t+τ1t+τ2‍∥g∥ds≤r8(ω)|τ1-τ2|1/2,
where
(131)r8(ω)=r5(ω)+ν{(lf(p2(ω))+λ)supσ∈[-ν,0]r1(ϑσω)+lf(p2(ω))supσ∈[-2ν,0]|z(ϑσω)|+∥g∥}.
It is not difficult to see that r7 and r8 are tempered. The lemma follows from (127) and (131).Lemma 18.
The random dynamical systemϕ is 𝒟-pullback asymptotically compact in 𝒞; that is, for ℙ a.e. ω∈Ω, the sequence {ϕ(tn,ϑ-tnω,vn0(ϑ-tnω))}n=1∞ has a convergent subsequence in 𝒞 provided tn→∞, B∈𝒟, and vn0(ϑ-tnω)∈B(ϑ-tnω).Proof.
Denote byQk the set {x∈ℝd:|x|<k} for each k∈ℕ. Since tn→∞, there exists n0=n0(B,ω)∈ℕ such that tn>TB(ω)+ν+1 for all n≥n0. Then by Lemma 17, we find that, for all n≥n0 and σ∈[-ν,0],
(132)∥vtn(σ,ϑ-tnω,v0(ϑ-tnω))∥H1(Qk)2≤r7(ω).
By the compactness of embedding H1(Qk)↪L2(Qk), it follows from (132) that for each σ∈[-ν,0] the sequence {vtn(σ,ϑ-tnω,v0(ϑ-tnω))}n=1∞ is relatively compact in L2(Qk). On the other hand, by Lemma 17, we also find that, for all n≥n0 and τ1,τ2∈[-ν,0],
(133)∥vtn(τ1,ϑ-tnω,v0(ϑ-tnω))-vtn(τ2,ϑ-tnω,v0(ϑ-tnω))∥L2(Qk)≤r8(ω)|τ1-τ2|1/2.
Hence, the sequence {vtn(·,ϑ-tnω,v0(ϑ-tnω))}n=1∞ is equicontinuous. By the Ascoli-Arzelà theorem, for each k∈ℕ the sequence {vtn(·,ϑ-tnω,v0(ϑ-tnω))}n=1∞ is relatively compact in C([-ν,0];L2(Qk)). Then, by a diagonal procedure, we can extract a subsequence {tni} such that for each k∈ℕ, the sequence {vtni(·,ϑ-tniω,v0(ϑ-tniω))}i=1∞ converges to ξk(·,ω) in C([-ν,0];L2(Qk)).
Obviously, for fixedσ∈[-ν,0] and ω∈Ω, ξk+1(σ,ω)(·) coincides with ξk(σ,ω)(·) on Qk. Therefore, one can define unambiguously a measurable function ξ(σ,ω)(·) by stipulating that it is equal to ξk(σ,ω)(·) on Qk. By Lemma 13, we obtain that, for all σ∈[-ν,0], k∈ℕ, and i>n0,
(134)∫Qk‍|vtni(σ,ϑ-tniω,v0(ϑ-tniω))(x)|2dx≤supσ∈[-ν,0]∫ℝd‍|vtni(σ,ϑ-tniω,u0(ϑ-tniω))(x)|2dx≤r1(ω).
Thus,
(135)∫Qk‍|ξ(σ,ω)(x)|2dx=limi→∞∫Qk‍|vtni(σ,ϑ-tniω,v0(ϑ-tniω))(x)|2dx≤r1(ω),
which implies
(136)∥ξ(σ,ω)∥2≤supk∈ℕ∫Qk‍|ξ(σ,ω)(x)|2dx≤r1(ω).
Hence, ξ(σ,ω)∈L2(ℝd).
For everyB∈𝒟 and ɛ>0, by Lemma 16, there exist R*=R*(ω,ɛ)>0 and T*=T*(B,ω,ɛ)>0 such that, for all u0∈B and t≥T*,
(137)supσ∈[-ν,0]∫|x|≥R*‍|vt(σ,ϑ-tω,v0(ϑ-tω))|2dx≤ɛ28.
Since tni→∞, there exists q1=q1(ω,ɛ)∈ℕ such that tni≥T* for all i>q1. Then we get from (137) that, for all i>q1,
(138)supσ∈[-ν,0]∫|x|≥R*‍|vtni(σ,ϑ-tniω,v0(ϑ-tniω))(x)|2dx≤ɛ28.
Take a fixed integer k≥R*. Then we find from (138) that, for all integer l>k and σ∈[-ν,0],
(139)∫k≤|x|<l‍|ξ(σ,ω)(x)|2dx=limi→∞∫k≤|x|<l‍|vtni(σ,ϑ-tniω,v0(ϑ-tniω))(x)|2dx≤supσ∈[-ν,0],i>q1∫|x|≥k‍|vtni(σ,ϑ-tniω,v0(ϑ-tniω))(x)|2dx≤ɛ28,
which implies
(140)supσ∈[-ν,0]∫|x|≥k‍|ξ(σ,ω)(x)|2dx≤supσ∈[-ν,0],l>k∫k≤|x|<l‍|ξ(σ,ω)(x)|2dx≤ɛ28.
As the sequence {vtni(·,ϑ-tniω,v0(ϑ-tniω))}i=1∞ converges to ξ(·,ω) in C([-ν,0];L2(Qk)), there exists q2=q2(ω,ɛ)∈ℕ such that, for all i>q2,
(141)supσ∈[-ν,0]∫Qk‍|vtni(σ,ϑ-tniω,v0(ϑ-tniω))(x)-ξ(σ,ω)(x)|2dx≤ɛ22.
Let q=max{q1,q2}. Then it follows from (138), (140), and (141) that, for all i>q,
(142)supσ∈[-ν,0]∥vtni(σ,ϑ-tniω,v0(ϑ-tniω))-ξ(σ,ω)∥2≤supσ∈[-ν,0]∫Qk‍|utni(σ,ϑ-tniω,u0(ϑ-tniω))(x)≤supσ∈[-ν,0]∫Qk‍-ξ(σ,ω)(x)(σ,ϑ-tniω,u0(ϑ-tniω))|2dx+2supσ∈[-ν,0]∫|x|≥k‍|ξ(σ,ω)(x)|2dx+2supσ∈[-ν,0]∫|x|≥k‍|vtni(σ,ϑ-tniω,v0(ϑ-tniω))(x)|2dx≤ɛ2,
which shows that the sequence {vtni(·,ϑ-tniω,v0(ϑ-tniω))}i=1∞ converges to ξ(·,ω) in 𝒞. This completes the proof.We are now in a position to present our main result about the existence of a𝒟-random attractor for ψ in 𝒞.Theorem 19.
The random dynamical systemψ has a unique 𝒟-random attractor in 𝒞.Proof.
Notice thatϕ has a closed absorbing set K in 𝒟 by Lemma 13 and is 𝒟-pullback asymptotically compact in 𝒞 by Lemma 18. Hence, the existence of a unique 𝒟-random attractor {𝒜1(ω)}ω∈Ω for ϕ follows from Proposition 10 immediately.
Sinceψ and ϕ are conjugated by the random homeomorphism T(ω,ξ)=ξ+z0(ω) and z0(ω)∈𝒞 is tempered, then, by Proposition 1.8.3 in [33], ψ has a unique 𝒟-random attractor {𝒜2(ω)}ω∈Ω in 𝒞 which is given by
(143)𝒜2(ω)={ξ(ω)+z0(ω):ξ(ω)∈𝒜1(ω)}.
The proof is complete.



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*Source: 981576-2013-07-24.xml*