# Random Attractors for Stochastic Three-Component Reversible Gray-Scott System on Infinite Lattices

Authors: Anhui Gu; Zhaojuan Wang; Shengfan Zhou
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We prove the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices.


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## Body

## 1. Introduction

Consider the following stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices:(1.1)duidt=d1(ui-1-2ui+ui+1)-(F+k)ui+ui2vi-Gui3+Nwi+ui∘dWdt,dvidt=d2(vi-1-2vi+vi+1)+F(1-vi)-ui2vi+Gui3+vi∘dWdt,dwidt=d3(wi-1-2wi+wi+1)+kui-(F+N)wi+wi∘dWdt,u(0)=u0=(ui0)i∈Z,v(0)=v0=(vi0)i∈Z,w(0)=w0=(wi0)i∈Z,
where i∈ℤ (the set of integers), (ui)i∈ℤ∈ℓ2,(vi)i∈ℤ∈ℓ2, and (wi)i∈ℤ∈ℓ2; all the parameters are positive constants; W is a Brownian motion on (Ω,ℱ,ℙ) and ∘ denotes the Stratonovich sense of the stochastic term.System (1.1) can be considered as a discrete model of stochastic three-component reversible Gray-Scott system in which the existence of a random attractor has been established [1]. When there is no stochastic term, system (1.1) can be considered as a discrete analogue of the following three-component reversible Gray-Scott system in R:(1.2)∂u∂t=d1Δu-(F+k)u+u2v-Gu3+Nw,∂v∂t=d2Δv+F(1-v)-u2v+Gu3,∂w∂t=d3Δw+ku-(F+N)w,
which was firstly introduced by Mahara et al. [2], then it was reduced to system (1.2) under some nondimensional transformations in You [3]. Also, the existence of a global attractor for the solution semiflow of (1.2) with Neumann boundary condition on a bounded domain of space dimension n≤3 was proved in [3].WhenG=0 and w=0, system (1.2) becomes the two-components Gray-Scott equations which was one of the models signified the seminal work of the Brussell school. The model originated from describing an isothermal, cubic autocatalytic, continuously fed and diffusive reactions of two chemicals (see [4–8]), but neglected the reversible factors. Indeed, the reversibility in the interactions of multispecies is an indispensable factor in many processes in natural and social sciences. If we take the reversibility into account, it yields system (1.2).Stochastic lattice differential equations have discrete spatial structures and take random influences into account. These random effects are not only introduced to compensate for the defects in some deterministic models, but are also rather intrinsic phenomena. Bates et al. [9] initiated the consideration of stochastic lattice dynamical systems with additive noises and Caraballo and Lu [10] was the first to consider the stochastic lattice dynamical systems with a multiplicative noise, and Han et al. [11] generalized the results of [9, 10] to a more general space. For more details and the quite recent results, we can refer to, for example, [12–15].Just like the models considered in biology, the discrete time models governed by difference equations are more appropriate than the continuous ones; we can also deal with the chemical and biochemical reactions in the same manner, see, for example, [16, 17] and the references therein. However, very few investigations are on this topic, especially for the stochastic three-component reversible Gray-Scott system on infinite lattices, is widely open, to the best of our knowledge.The paper is organized as follows. In Section2, we present some preliminaries and definitions. Section 3 is devoted to the existence of a random attractor.


## 2. Preliminaries

Let(Ω,ℱ,ℙ) be a probability space, where Ω is a subset of 𝒞0(R,R)={ω∈𝒞(R,R):ω(0)=0}, which endowed with the compact open topology (see [18]), ℱ is the Borel σ-algebra, and ℙ is the corresponding Wiener measure on Ω. Let θtω(·)=ω(·+t)-ω(t), t∈R, then (Ω,ℱ,ℙ,(θt)t∈R) is an ergodic metric dynamical system. Throughout the paper, we denote
(2.1)l2={x=(xi)i∈Z:xi∈R,∑i∈Z‍xi2<∞},
the Hilbert space equipped with the usual inner product and norm:
(2.2)〈x,y〉=∑i∈Z‍(xi,yi),‖x‖2=〈x,x〉,∀x=(xi)i∈Z,y=(yi)i∈Z∈l2.For the reader's convenience, we introduce some basic concepts related to random dynamical systems and random attractor, which are taken from [11, 18, 19]. Let (H,∥·∥H) be a separable Hilbert space and (Ω0,ℱ,ℙ) a probability space.Definition 2.1.
A stochastic process{φ(t,ω)}t≥0,ω∈Ω0 is a continuous random dynamical system over (Ω0,ℱ,ℙ,(θt)t∈R) if φ is (ℬ[0,∞)×ℱ×ℬ(H),ℬ(H))-measurable, and for all ω∈Ω0,(i)
the mappingφ(t,ω):H↦H, x↦φ(t,ω)x is continuous for every t≥0,(ii)
φ(0,ω) is the identity on H,(iii)
(cocycle property)φ(s+t,ω)=φ(t,θsω)φ(s,ω) for all s,t≥0.Definition 2.2.
(i) A set-valued mappingω↦B(ω)⊂H (we may write it as B(ω) for short) is said to be a random set if the mapping ω↦ distH(x,B(ω)) is measurable for any x∈H, where distH(x,D) is the distant in H between the element x and the set D⊂H.
(ii) A random setB(ω) is said to be bounded if there exist x0∈H and a random variable r(ω)>0 such that B(ω)⊂{x∈H:∥x-x0∥H≤r(ω),x0∈H} for all ω∈Ω0.
(iii) A random setB(ω) is called a compact random set if B(ω) is compact for all ω∈Ω0.
(iv) A random bounded setB(ω)⊂H is called tempered with respect to (θt)t∈R if for a.e. ω∈Ω0, limt→+∞e-γtsupx∈B(θ-tω)∥x∥H=0  for  all  γ>0. A random variable ω↦r(ω)∈R is said to be tempered with respect to (θt)t∈R if for a.e. ω∈Ω0, limt→+∞supt∈Re-γtr(θ-tω)=0  for  all  γ>0.
We consider a continuous random dynamical system (RDS){φ(t,ω)}t≥0,ω∈Ω0 over (Ω0,ℱ,ℙ,(θt)t∈R) and 𝒟(H) the set of all tempered random sets of 𝒟(H).Definition 2.3.
A random set𝒦 is called an absorbing set in 𝒟(H) if for all B∈𝒟(H) and a.e. ω∈Ω0 there exists tB(ω)>0 such that
(2.3)φ(t,θ-tω)B(θ-tω)⊂K(ω)∀t≥tB(ω).Definition 2.4.
A random set𝒜 is called a global random 𝒟(H) attractor (pullback 𝒟(H) attractor) for {φ(t,ω)}t≥0,ω∈Ω0 if the following hold:(i)
𝒜 is a random compact set, that is, ω↦d(x,𝒜(ω)) is measurable for every x∈H and 𝒜(ω)) is compact for a.e. ω∈Ω0;(ii)
𝒜 is strictly invariant, that is, for ω∈Ω0 and all t≥0, φ(t,ω)𝒜(ω)=𝒜(θtω);(iii)
𝒜 attracts all sets in 𝒟(H), that is, for all B∈𝒟(H) and a.e. ω∈Ω0, we have
(2.4)lim⁡t→+∞d(φ(t,θ-tω)B(θ-tω),A(ω))=0,where d(X,Y)=supx∈Xinfy∈Y∥x-y∥H is the Hausdorff semimetric (X⊆H,Y⊆H).Proposition 2.52.5 (see [11]).
Suppose that(a)
there exists a random bounded absorbing setK(ω)∈𝒟(ℓ2), ω∈Ω0, such that for any B(ω)∈𝒟(ℓ2) and all ω∈Ω0, there exists T(ω,B)>0 yielding φ(t,θ-tω,B(θ-tω))⊂K(ω) for all t≥T(ω,B);(b)
the RDS{φ(t,ω)}t≥0,ω∈Ω0 is random asymptotically null on K(ω), that is, for  any ϵ>0, there exist T(ϵ,ω,K)>0 and I0(ϵ,ω,K)∈ℕ such that
(2.5)sup⁡u∈K(ω)∑|i|>I0(ϵ,ω,K(ω))‍|φ(t,θ-tω,u(θ-tω))i|2≤ϵ2,∀t≥T(ϵ,ω,K(ω)).
Then the RDS{φ(t,ω,·)}t≥0,ω∈Ω0 possesses a unique global random 𝒟(H) attractor given by
(2.6)A~(ω)=⋂τ≥T(ω,K)‍⋃t≥τ‍φ(t,θ-tω,K(θ-tω))¯.


## 3. Existence of a Random Attractor

In this section, we will derive the random attractor of the stochastic three-component reversible Gray-Scott lattice system (1.1) with a multiplicative white noise.Forx=(xi)i∈ℤ, we define A,B,B* to be linear operators from ℓ2 to ℓ2 for i∈ℤ, as follows:
(3.1)(Ax)i=-xi-1+2xi-xi+1,(Bx)i=xi+1-xi,(B*x)i=xi-1-xi.
It is easy to show that A=BB*=B*B, (B*x,x′)=(x,Bx′) for all x,x′∈ℓ2, which implies that (Ax,x)≥0.In the sequel, we rewrite the system (1.1) with initial values (u0,v0,w0):=(u0i,v0i,w0i)i∈ℤ∈ℓ2×ℓ2×ℓ2=H as the following integral equations in H for t≥0,ω∈Ω:(3.2)u(t)=u0+∫0t[-d1Au(s)-(F+k)u(s)+u2(s)v(s)-Gu3(s)+Nw(s)]ds+∫0tu(s)∘dW,v(t)=v0+∫0t[-d2Av(s)+F(1-v(s))-u2(s)v(s)+Gu3(s)]ds+∫0tv(s)∘dW,w(t)=w0+∫0t[-d3Aw(s)+ku(s)-(F+N)w(s)]ds+∫0tw(s)∘dW,
where W is a two-sided Brownian motion on the same probability space (Ω,ℱ,ℙ). To prove that this system (3.2) generates a random dynamical system, we will transform it into a random differential equation system in H.Before performing this transformation, we need to recall some properties of the Ornstein-Uhlenbeck processes. Let(3.3)z(θtω)=-∫-∞0‍eτθtω(τ)dτ,t∈R,ω∈Ω.
We know that z(θtω) is an Ornstein-Uhlenbeck process on (Ω,ℱ,ℙ,(θt)t∈R) and solves the following one-dimensional stochastic differential equation (see [20] for details):
(3.4)z=-zdt+dw(t),z(-∞)=0,∀t≥0,ω∈Ω,
where w(t)(ω)=w(t,ω)=ω(t) for ω∈Ω,t∈R. In fact, we have the following.Lemma 3.13.1 (see [10, 18]).
There exists aθt-variant set Ω~∈ℱ of 𝒞0(R,R) of full ℙ measure such that, for ω∈Ω~, one has(i)
the random variable|z(ω)| is tempered;(ii)
the mapping(3.5)(t,ω)⟼z(θtω)=-∫-∞0‍esω(t+s)ds+ω(t)
is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;(iii)
(3.6)lim⁡t→±∞|z(θtω)|t=0,lim⁡t→±∞1t∫0t‍z(θsω)ds=0,(3.7)lim⁡t→±∞1t∫0t‍|z(θsω)|ds=E|z|<∞.Obviously,ez(θtω)1dH is clearly a homeomorphism in H, and the inverse operator e-z(θtω)1dH is well defined. It easily follows from (3.6) that ∥ez(θtω)1dH∥H and ∥e-z(θtω)1dH∥H have subexponential growth as t→±∞ for all ω∈Ω~, which implies that they are tempered. Since the mapping of θ on Ω~ has the same properties as the original one if we choose the trace σ-algebra with respect to Ω~ to be denoted also by ℱ, we can change our metric dynamical system with respect to Ω~, and still denoted by the symbols (Ω,ℱ,ℙ,(θt)t∈R).Let(3.8)g~(t,ω)=e-z(θtω)1dHg(t,ω),ω∈Ω,
where g~(t,ω)=(u~(t,ω),v~(t,ω),w~(t,ω)), and g(t,ω)=(u(t,ω),v(t,ω),w(t,ω)) is a solution of (3.2). Then system (1.1) can be written as the following random system with random coefficients but without white noise:
(3.9)du~dt=-d1Au~-(F+k-z(θtω))u~+e2z(θtω)u~2v~-Ge2z(θtω)u~3+Nw~,dv~dt=-d2Av~+Fe-z(θtω)-(F-z(θtω))v~-e2z(θtω)u~2v~+Ge2z(θtω)u~3,dw~dt=-d3Aw~+ku~-(F+N-z(θtω))w~
and an initial condition
(3.10)g~0(ω)=(u~(0),v~(0),w~(0))=(u~0,v~0,w~0)∈H.Now we establish the following result.Theorem 3.2.
LetT>0 and g~0(ω)∈H be fixed. Then the following properties hold:(i)
for everyω∈Ω, system (3.9) admits a unique solution g~(·,ω,g~0)∈𝒞([0,T],H),(ii)
the solutiong~(t,ω) of system (3.9) depends continuously on the initial data g~0, that is, for each ω∈Ω, the mapping g~0∈H↦g~(·,ω,g~0)∈𝒞([0,T],H) is continuous.Proof.
(1) Denote(3.11)L(g~)=(-(d1A+F+k)u~0Nw~0-(d2A+F)v~0ku~0-(d3A+F+N)w~),Ψ(g~,θtω)=(z(θtω)u~+e2z(θtω)u~2v~-Ge2z(θtω)u~3Fe-z(θtω)+z(θtω)v~-e2z(θtω)u~2v~+Ge2z(θtω)u~3z(θtω)w~).
Then system (3.9) can be written as
(3.12)dg~dt=L(g~)+Ψ(g~,θtω).
Since z(θtω) is continuous with respect to t, define
(3.13)Ψ(g~,ω)=(z(ω)u~+e2z(ω)u~2v~-Ge2z(ω)u~3Fe-z(ω)+z(ω)v~-e2z(ω)u~2v~+Ge2z(ω)u~3z(ω)w~),
then
(3.14)‖Ψ(g~,ω)‖H≤2F|e-z(ω)|+2(|z(ω)|+max⁡{1,G}e2z(ω))‖g~‖H+3(1+G)e2z(ω)‖g~‖H2.
For any g=(u,v,w)=(ui,vi,wi)i∈ℤ, g~=(u~,v~,w~)=(u~i,v~i,w~i)i∈ℤ∈H,
(3.15)‖Ψ(g,ω)-Ψ(g~,ω)‖H≤2(‖z(ω)|+3(1+G)e2z(ω)(‖g‖H2+‖g~‖H2))‖g-g~‖H.
For any bounded set D∈H with supg∈D∥g∥H≤r, define a random variable ϱD(ω) by
(3.16)ϱD(ω)=2(|z(ω)|+3(1+G)e2z(ω)r+max⁡{1,G}e2z(ω))r+2F|e-z(ω)|≥0.
Then
(3.17)∫ττ+1‍ϱD(θtω)dt=2∫ττ+1‍((|z(θtω)|+3(1+G)e2z(θtω)r+max⁡⁡{1,G}e2z(θtω))r+Fe-z(θtω))dt<∞,∀τ∈R,
and, for any g¯,g,g~∈D, we have
(3.18)‖Ψ(g¯,ω)‖H≤ϱD(ω),‖Ψ(g,ω)-Ψ(g~,ω)‖H≤ϱD(ω)‖g-g~‖H.
Then we obtain that L(g~)+Ψ(g~,θtω) is locally Lipschitz in g~ from H to H. By Proposition 2.1.1 in [19], problem (3.12) possesses a unique local solution g~(·,ω,g~0)∈𝒞1((0,Tmax),H), where (0,Tmax) is the maximal interval of existence of the solution of (3.12). Now, we will show that the local solution is a global one. Define
(3.19)r~(t)=Nkw~(t),μ=kN,
then system (3.9) can be written as
(3.20)du~dt=-d1Au~-(F+k-z(θtω))u~+e2z(θtω)u~2v~-Ge2z(θtω)u~3+kr~,dv~dt=-d2Av~+Fe-z(θtω)-(F-z(θtω))v~-e2z(θtω)u~2v~+Ge2z(θtω)u~3,μdr~dt=-μd3Ar~+ku~-(μF+k-z(θtω))r~.
Taking the inner products of (3.20) with Gu~(t), v~(t) and Gw~(t), respectively, and adding up the resulting equalities, we get
(3.21)12ddt(G‖u~‖2+‖v~‖2+μG‖r~‖2)+GF‖u~‖2+F2‖v~‖2+μGF‖r~‖2≤z(θtω)(G‖u~‖2+‖v~‖2+G‖r~‖2)+F2e-2z(θtω),
which implies that
(3.22)ddt(G‖u~‖2+‖v~‖2+Gμ‖w~‖2)+2GF‖u~‖2+F‖v~‖2+2GFμ‖w~‖2≤2z(θtω)(G‖u~‖2+‖v~‖2+Gμ2‖w~‖2)+Fe-2z(θtω).
Setting(3.23)λ=Fmin⁡{1,2G,2G/μ}max⁡{1,G,G/μ},C1=2max⁡{1,G,G/μ2}min⁡{1,G,G/μ},C2=Fmin⁡{1,G,G/μ},
then (3.22) yields
(3.24)ddt‖g~(t)‖H2+(λ-C1z(θtω))‖g~(t)‖H2≤C2e-2z(θtω).
Applying Gronwall's lemma to (3.24), we obtain that, for t≥0,
(3.25)‖g~(t,ω,g~0(ω))‖H2≤e-λt+C1∫0t‍z(θsω)ds‖g~0(ω)‖H2+C2e-λt+C1∫0t‍z(θsω)ds∫0t‍e-2z(θsω)+λs+C1∫s0‍z(θτω)dτds.
Denoting(3.26)α(ω)=C1∫0T‍|z(θsω)|ds,β(ω)=C2max⁡t∈[0,T](e-λt+C1∫0t‍z(θsω)ds∫0t‍e-2z(θsω)+λs+C1∫s0‍z(θτω)dτds),
we get
(3.27)‖g~(t,ω,g~0(ω))‖H2≤‖g~0(ω)‖H2eα(ω)+β(ω),
which implies that the solution g~ is defined in any interval [0,T].
(2) Letg0,g~0∈H and
(3.28)X(t):=g(t,ω,g0),Y(t):=g~(t,ω,g~0)
be the corresponding solutions of (3.12). Then, denoting Z(t)=(Z1,Z2,Z3)=X(t)−Y(t)=(u-u~,v-v~,w-w~), we have
(3.29)dZ1dt=-d1AZ1-(F+k-z(θtω))Z1+NZ3+e2z(θtω)(u2v-u~2v~)-Ge2z(θtω)(u3-u~3),dZ2dt=-d2AZ2-(F-z(θtω))Z2-e2z(θtω)(u2v-u~2v~)+Ge2z(θtω)(u3-u~3),dZ3dt=-d3AZ3+kZ1-(F+N-z(θtω))Z3.
Taking the inner product of (3.29) with Z1,Z2 and Z3, respectively, it yields
(3.30)ddt(‖Z1‖2+‖Z2‖2+‖Z3‖2)≤2z(θtω)(‖Z1‖2+‖Z2‖2+‖Z3‖2)+N2‖Z1‖2+k2‖Z3‖2+2e2z(θtω)(u2v-u~2v~,Z1-Z2)-2Ge2z(θtω)(u3-u~3,Z1-Z2).
Due to (3.27), we have
(3.31)e2z(θtω)(u2v-u~2v~,Z1-Z2)-Ge2z(θtω)(u3-u~3,Z1-Z2)≤2(1+G)e2z(θtω)(‖g‖H2+‖g~‖H2)(‖Z1‖2+‖Z2‖2)≤4(1+G)e2z(θtω)‖g~‖H2(‖Z1‖2+‖Z2‖2+‖Z3‖2).
Denoting δ=(1/2)max{N,k}, we obtain that
(3.32)ddt(‖Z1‖2+‖Z2‖2+‖Z3‖2)≤(δ+z(θtω)+8(1+G)e2z(θtω)‖g~‖H2)(‖Z1‖2+‖Z2‖2+‖Z3‖2).
By Gronwall's lemma, for t∈[0,T], we have
(3.33)‖Z1‖2+‖Z2‖2+‖Z3‖2≤(‖Z1(0)‖2+‖Z2(0)‖2+∥Z3(0))eδ′t+8(1+G)∫0te2z(θsω)‖g~(s,ω,g~0(ω))‖H2ds,
where δ′=δ+maxt∈[0,T]|z(θtω)|. According to (3.25), for t∈[0,T],
(3.34)∫0te2z(θsω)‖g~(s,ω,g~0(ω))‖H2ds≤‖g~0(ω)‖H2∫0te2z(θsω)+C1∫0s‍z(θτω)dτds+C2∫0te2z(θsω)+C1∫0s‍z(θτω)dτ∫0se-2z(θτω)-C1∫0tz(θςω)dςdτds∶=‖g~0(ω)‖H2∫0tξ(θsω)ds+C2∫0tξ(θsω)∫0sξ-1(θτω)dτds,
where ξ(θtω)=e2z(θtω)+C1∫0t‍z(θτω)dτ. Obviously, t↦lnξ(θtω) is continuous ℙ-a.s. Also, we have limt→+∞(lnξ(θtω)/t)=0, which implies that ξ(θtω) is a tempered random variable. Then by Proposition 4.3.3, [18], for given ɛ>0, there is an ɛ-slowly varying random variable R(ω) for which
(3.35)e-ɛ|t|R(ω)≤ξ(θtω)≤eɛ|t|R(ω),∀t∈R,ω∈Ω,
where R(ω),ω∈Ω satisfies
(3.36)e-ɛ|t|R(ω)≤R(θtω)≤eɛ|t|R(ω)∀t∈R,ω∈Ω.
Combining with (3.34) and (3.35), we easily conclude that for t∈[0,T],
(3.37)∫0t‍e2z(θsω)‖g~(s,ω,g~0(ω))‖H2ds≤‖g~0(ω)‖H2ɛR(ω)eɛT+C22ɛ2e2ɛT,
which implies
(3.38)sup⁡t∈[0,T]‖X(t)-Y(t)‖2≤(‖u0-u~0‖2+‖v0-v~0‖2+‖w0-w~0‖2)eδ′′T,
where δ′′=δ′+8(1+G)((∥g~0(ω)∥H2/ɛT)R(ω)eɛT+(C2/2ɛ2T)e2ɛT)<∞. If u0=u~0,v0=v~0 and w0=w~0, then the above inequality shows the uniqueness and continuous dependence on the initial data of the solution of (3.12). So the both results of the theorem hold.Theorem 3.3.
System (3.12) generates a continuous random dynamical system (φ(t))t≥0 over (Ω,ℱ,ℙ,(θt)t∈R), where φ(t,ω,g0)=g~(t,ω,g0) for g0∈H, t≥0 and for all ω∈Ω. Moreover, if one defines φ by
(3.39)ψ(t,ω,g0)=ez(θtω)φ(t,ω,e-z(ω)g0)
for g0∈H, t≥0 and for all ω∈Ω, then ψ is another random dynamical system for which the process (ω,t)↦ψ(t,ω,g0) solves (3.2) for any initial condition g0∈H.Proof.
The fact thatφ is continuous random dynamical system follows from Theorem 3.2. The measurability of ψ follows from the properties of the transformation (see [18, 19]). It follows directly the other statements.Note that the two random dynamical systems are equivalent. It is easy to check thatψ has a random attractor provided φ possesses a random attractor. Then, we only need to consider the random dynamical system φ.Now, we are in the position to study the existence of tempered random bounded absorbing set and global random attractor for the RDSφ in H.Lemma 3.4.
There exists a random bounded ballK(ω)∈𝒟(H) centered at 0 with random radius ρ(ω)>0 such that K(ω)∈𝒟(H) is a random absorbing set for φ in 𝒟(H); that is, for any B(ω)∈𝒟(H) and ω∈Ω, there exists TB(ω)>0 yielding φ(t,θ-tω,B(θ-tω))⊂K(ω), for all t≥TB(ω).Proof.
By substitutingω by θ-tω in (3.25), we have
(3.40)‖g~(t,θ-tω,g~0(θ-tω))‖H2≤e-λt+C1∫0t‍z(θs-tω)ds‖g~0(θ-tω)‖H2+C2e-λt+C1∫0t‍z(θs-tω)ds∫0t‍e-2z(θs-tω)+λs+C1∫s0‍z(θτ-tω)dτds≤e-λt+C1∫0t‍z(θs-tω)ds‖g~0(θ-tω)‖H2+C2∫-t0‍e-2z(θsω)+λ(s-t)+C1∫st‍z(θτω)dτds≤e-λt+C1∫-t0‍z(θsω)ds‖g~0(θ-tω)‖H2+C2∫-t0‍e-2z(θsω)+λs+C1∫s0‍z(θτω)dτds≤e-λt+C1∫-t0‍z(θsω)ds‖g~0(θ-tω)‖H2+C2∫-∞0‍e-2z(θsω)+λs+C1∫s0‍z(θτω)dτds.
By the properties of the Ornstein-Uhlenbeck process,
(3.41)∫-∞0‍e-2z(θsω)+λs+C1∫s0‍z(θτω)dτds<+∞.
Notice that {B(ω)}∈𝒟(H) is tempered, then for g~0(ω)∈B(θ-tω),
(3.42)lim⁡t→+∞e-λt+C1∫-t0‍z(θsω)ds‖g~0(θ-tω)‖H2=0.
We can choose
(3.43)ρ2(ω)=1+C2∫-∞0‍e-2z(θsω)+λs+C1∫s0‍z(θτω)dτds,
then K(ω) is a random absorbing set for φ in 𝒟(H), and K(ω)∈𝒟(H). Here, we remain only to check that
(3.44)lim⁡t→+∞e-γtρ2(θ-tω)=0.
Indeed, obviously we have
(3.45)e-γtρ2(θ-tω)=e-γt+C2e-γt∫-∞0‍e-2z(θs-tω)+λs+C1∫s0‍z(θτ-tω)dτds=e-γt︸→0ast→+∞+C2e-γt∫-∞-t‍e-2z(θsω)+λs+C1∫s-t‍z(θτω)dτds︸→0ast→+∞.Lemma 3.5.
The RDS{φ(t,ω,·)}t≥0,ω∈Ω generated by (3.9) is random asymptotically null on K(ω); that is, for any ϵ>0, there exist T(ϵ,ω,K)>0, ∀t≥T(ϵ,ω,K(ω)), and M(ϵ,ω,K(ω))∈ℕ such that
(3.46)sup⁡φ0∈K(ω)∑|i|>M(ϵ,ω,K(ω))‍|φ(t,θ-tω,φ0(θ-tω))i|H2≤ϵ2.Proof.
Choose a smooth cut-off function satisfying0≤ρ(s)≤1 for s∈ℝ+ and ρ(s)=0 for 0≤s≤1, ρ(s)=1 for s≥2. Suppose there exists a constant c such that |ρ′(s)|≤c for s∈ℝ+.
Setx=(ρ(|i|/M)u~i)i∈ℤ,y=(ρ(|i|/M)v~i)i∈ℤ, and z=(ρ(|i|/M)w~i)i∈ℤ. By taking the inner product of (3.20) with Gx, y, and Gz, respectively, we get
(3.47)G2ddt∑i∈Zρ(|i|M)|u~i|2=-Gd1(Au~,x)-G(F+k-z(θtω))∑i∈Zρ(|i|M)|u~i|2+kG(r~,x)+Ge2z(θtω)(u~2v~,x)-G2e2z(θtω)(u~3,x),12ddt∑i∈Zρ(|i|M)|v~i|2=-d2(Av~,y)-(F-z(θtω))∑i∈Zρ(|i|M)|v~i|2-Ge2z(θtω)(u~2v~,y)+G2e2z(θtω)(u~3,y)+(e-z(θtω),y),μG2ddt∑i∈Zρ(|i|M)|r~i|2=-μGd3(Ar~,z)-G(μF+k-z(θtω))∑i∈Zρ(|i|M)|r~i|2+kG(u~,z).
Due to [9, 10], we have
(3.48)-(Au~,x)≤2c‖u~‖2M,-(Av~,y)≤2c‖v~‖2M,-(Ar~,z)≤2c‖r~‖2M.
Combining with (3.47) to (3.48), we obtain
(3.49)12ddt∑i∈Zρ(|i|M)(G|u~i|2+|v~i|2+μG|r~i|2)+G(F-z(θtω))∑i∈Zρ(|i|M)|u~i|2+(F2-z(θtω))∑i∈Zρ(|i|M)|v~i|2+G(μF-z(θtω))∑i∈Zρ(|i|M)|r~i|2≤2cM(Gd1‖u~‖2+d2‖v~‖2+μGd3‖r~‖2)+F2e-2z(θtω),
that is,
(3.50)ddt∑i∈Zρ(|i|M)(G|u~i|2+|v~i|2+Gμ|w~i|2)+2GF∑i∈Zρ(|i|M)|u~i|2+F∑i∈Zρ(|i|M)|v~i|2+2GFμ∑i∈Zρ(|i|M)|w~i|2≤2z(θtω)∑i∈Zρ(|i|M)(G|u~i|2+|v~i|2+Gμ2|w~i|2)+4cM(Gd1‖u~‖2+d2‖v~‖2+Gμd3‖w~‖2)+Fe-2z(θtω).
Denote(3.51)C3=2cmax⁡{d1,d2,d3}C1,
then (3.50) yields
(3.52)ddt∑i∈Zρ(|i|M)|g~i|H2+(λ-C1z(θtω))∑i∈Zρ(|i|M)|g~i|H2≤C3M‖g~‖H2+C2e-2z(θtω).
By using Gronwall's lemma, for t≥TK=TK(ω), we have
(3.53)∑i∈Zρ(|i|M)|g~i(t,ω,g~0(ω))|H2≤e-λ(t-TK)+C1∫Tkt‍z(θsω)ds∑i∈Zρ(|i|M)|g~i(Tk,ω,g~0(ω))|H2+C3M∫Tkt‍e-λ(t-s)+C1∫st‍z(θτω)dτ‖g~(s,ω,g~0(ω))‖H2ds+C2∫Tkt‍e-λ(t-s)-2z(θsω)+C1∫st‍z(θτω)dτds.
Replace ω by θ-tω. We then estimate each term on the right-hand side of (3.53). From (3.25) with t replaced by TK and ω by θ-tω, respectively, it then follows that
(3.54)e-λ(t-TK)+C1∫Tkt‍z(θs-tω)ds∑i∈Zρ(|i|M)|g~i(Tk,θ-tω,g~0(θ-tω))|H2≤e-λt+C1∫0t‍z(θs-tω)ds‖g~0(ω)‖H2+C2∫0TK‍e-2z(θs-tω)+λ(s-t)+C1∫st‍z(θτ-tω)dτds≤e-λt+C1∫-t0‍z(θsω)ds‖g~0(ω)‖H2+C2∫-tTK-t‍e-2z(θsω)+λs+C1∫s0‍z(θτω)dτds.
Hence, by using (3.6), there is a T1(ϵ,ω,K(ω))>TK(ω), such that if t>T1(ϵ,ω,K(ω)),
(3.55)e-λ(t-TK)+C1∫Tkt‍z(θs-tω)ds∑i∈Zρ(|i|M)|g~i(Tk,θ-tω,g~0(θ-tω))|H2<ϵ23.
Next, we estimate
(3.56)C3M∫Tkt‍e-λ(t-s)+C1∫st‍z(θτ-tω)dτ‖g~(s,θ-tω,g~0(θ-tω))‖H2ds≤C3M(t-TK)e-λt+C1∫0t‍z(θr-tω)dr‖g~0(ω)‖H2+C2C3M∫TKt‍∫0s‍e-2z(θτ-tω)-λ(t-τ)+C1∫τt‍z(θr-tω)drdτds≤C3M(t-TK)e-λt+C1∫-t0‍z(θrω)dr‖g~0(ω)‖H2+C2C3M∫TKt‍∫-ts-t‍e-2z(θτω)+λτ+C1∫τ0‍z(θrω)drdτds≤C3M(t-TK)e-λt+C1∫-t0‍z(θrω)dr‖g~0(ω)‖H2+C2C3M(t-TK)∫-t0‍e-2z(θτω)+λτ+C1∫τ0‍z(θrω)drdτ.
By using (3.7), there exist T2(ϵ,ω,K(ω))>TK(ω) and M1(ϵ,ω,K(ω))>0, such that if t>T2(ϵ,ω,K(ω)) and M>M1(ϵ,ω,K(ω)), then
(3.57)C3M∫Tkt‍e-λ(t-s)+C1∫st‍z(θτ-tω)dτ‖g~(s,θ-tω,g~0(θ-tω))‖H2ds≤ϵ23.
By using (3.7) again, there exists T3(ϵ,ω,K(ω))>0, such that if T>T3(ϵ,ω,K(ω)), we have
(3.58)C2∫Tkt‍e-λ(t-s)-2z(θsω)+C1∫st‍z(θτω)dτds≤ϵ23.
Therefore, by letting
(3.59)T(ϵ,ω,K(ω))=max⁡{T1(ϵ,ω,K(ω)),T2(ϵ,ω,K(ω)),T3(ϵ,ω,K(ω))},M(ϵ,ω,K(ω))=M1(ϵ,ω,K(ω)),
we obtain, for t>T(ϵ,ω,K(ω)) and M>M(ϵ,ω,K(ω)),
(3.60)∑|i|≥2M|g~i(t,θ-tω,g~0(θ-tω))|H2≤∑i∈Zρ(|i|M)|g~i(t,θ-tω,g~0(θ-tω))|H2≤ϵ2,
which implies
(3.61)∑|i|>M(ϵ,ω,K(ω))|(φ(t,θ-tω,φ0(θ-tω))i|H2≤ϵ2,∀t≥T(ϵ,ω,K(ω)).
The proof is completed.Now, we have the main result.Theorem 3.6.
The random dynamical systemφ generated by system (3.9) has a unique global random attractor.Proof.
It is obvious from Lemmas3.4, and 3.5, and Proposition 2.5.



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*Source: 340789-2012-07-09.xml*