# Random Attractors of Stochastic Three-Component Reversible Gray-Scott System on Unbounded Domains

Authors: Anhui Gu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The existence of a pullback random attractor is established for a
stochastic three-component reversible Gray-Scott system on unbounded domains.
The Gray-Scott system is recast as a random dynamical system and
asymptotic compactness which is illustrated by using uniform, a priori estimates
for far-field values of solutions and a cutoff technique.


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## Body

## 1. Introduction

Consider the asymptotic behavior of solutions of the following stochastic three-component reversible Gray-Scott system with multiplicative noise defined in the entire spaceℝn×ℝn×ℝn:
(1.1)∂u~∂t=d1▵u~-(F+k)u~+u~2v~-Gu~3+Nw~+f1(x)+σu~∘dBtdt,∂v~∂t=d2▵v~-Fv~-u~2v~+Gu~3+f2(x)+σv~∘dBtdt,∂w~∂t=d3▵w~-(F+N)w~+ku~+f3(x)+σw~∘dBtdt,
with initial data
(1.2)u~(0,x)=u~0(x),v~(0,x)=v~0(x),w~(0,x)=w~0(x),x∈Rn,
where all the parameters are given positive constants; fi(i=1,2,3) are nonlinear functions satisfying certain conditions; Bt is a two-sided real-valued Wiener process on a probability space (Ω,ℱ,ℙ), Ω={ω∈C(ℝ,ℝ):ω(0)=0}, the Borel σ-algebra ℱ on Ω is generated by the compact open topology (see [1]), and ℙ is the corresponding Wiener measure on ℱ; ∘ denotes the Stratonovich sense of the stochastic term.Historically, whenw~=0,G=0,f1=f3=0,f2=F and there are no random terms (σ=0), system (1.1) reduces to the two-component Gray-Scott system which signified one of the Brussels schools led by the renowned physical chemist and Nobel Prize laureate (1977), Ilya Prigogine, which originated from describing an isothermal, cubic autocatalytic, continuously fed, and diffusive reactions of two chemicals (see [2–6]). The three-component reversible Gray-Scott model was firstly introduced by Mahara et al., which is based on the scheme of two reversible chemical or biochemical reactions [7]. Then in [8], You took some nondimensional transformations, the three-component reversible system was reduced to the system (1.1) without random forces. In [8], You considered the existence of global attractor for the system (1.1) with Neumann boundary condition on a bounded domain of space dimension n≤3 by the method of the rescaling and grouping estimation.Stochastic differential equations of this type arise from many chemical or biochemical systems when random spatiotemporal forcing is taken into consideration. These random perturbations are intrinsic effects in a variety of settings and spatial scales. They may be most obviously influential at the microscopic and smaller scales, but indirectly they play an important role in macroscopic phenomena. Recently, Gu [9] gave the existence of a compact random attractor for stochastic three-component reversible Gray-Scott system with multiplicative white noise in a bounded domain of ℝn(n≤3) when f1=f3=0,f2=F in system (1.1). As pointed in [10], the discussion of the same or similar coupled reaction-diffusion systems on a higher dimensional domain with the space dimension n>3 and on an unbounded domain is still open to the best of our knowledge. Here, we intend to investigate the dynamical behavior of the system (1.1) on unbounded domains and give a partly answer to the problems proposed in [10]. It is worth mentioning that Sobolev embedding is not compact on domains of infinite volume. This introduces a major obstacle for proving the existence of random attractors for partial differential equations on unbounded domains. For some deterministic equations, the difficulty caused by the unboundedness of domains can be overcome by the energy equation approach which developed by Ball in [11, 12] and used by many authors (see, e.g., [13–15]). In this paper, we will use the uniform estimates on the far-field values of solutions to circumvent the difficulty caused by the unboundedness of the domain. This idea was developed in [16] to prove the asymptotic compactness of solutions for autonomous parabolic equations on ℝn, and later extended to nonautonomous equations (see, e.g., [17–19]) and stochastic equations (see, e.g., [20–22]). Here, we will use the method of tail-estimates to investigate the asymptotic behavior of system (1.1) with initial data (1.2).The paper is organized as follows. In the next section, we recall the fundamental concepts and results for pullback random attractors for random dynamical systems. In Section3, we define a cocycle for the stochastic three-component reversible Gray-Scott system on ℝn×ℝn×ℝn. Section 4 is devoted to deriving the uniform estimates of solutions for large space and time variables. In the last section, we give the asymptotic compactness of the solution by using uniform estimates on the tails of solutions and then prove the existence of a pullback random attractor.The following notations will be used throughout the paper. We denote by∥·∥ and (,·,) the norm and inner product in L2(ℝn) or ℍ=[L2(ℝn)]3. Let 𝕌=[L6(ℝn)]3; 𝔼=[H1(ℝn)]3; ∥·∥L6 and ∥·∥𝕌 denote the norm in L6(ℝn) and 𝕌.


## 2. Preliminaries

In this section, we recall some basic concepts related to random attractors for random dynamical systems. We refer the reader to [1, 23, 24] for more details.Let(X,∥·∥X) be a separable Hilbert space with Borel σ-algebra ℬ(X), and let (Ω,ℱ,ℙ) be a probability space.Definition 2.1.
(
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=θtθs for all s,t∈ℝ and θtℙ=ℙ for all t∈ℝ.Definition 2.2.
A stochastic process{φ(t,ω)}t≥0,ω∈Ω is a continuous random dynamical system (RDS) over (Ω,ℱ,ℙ,(θt)t∈ℝ) if φ is (ℬ[0,∞)×ℱ×ℬ(X),ℬ(X))-measurable, and for all ω∈Ω,(i)
the mappingφ(t,ω):X↦X, x↦φ(t,ω)x is continuous for every t≥0;(ii)
φ(0,ω) is the identity on X;(iii)
(cocycle property)φ(s+t,ω)=φ(t,θsω)φ(s,ω) for all s,t≥0.Definition 2.3.
(i) A set-valued mappingω↦B(ω)⊂X (we may write it as B(ω) for short) is said to be a random set if the mapping ω↦distX(x,B(ω)) is measurable for any x∈X.
(ii) A random setB(ω) is said to be bounded if there exist x0∈X and a random variable r(ω)>0 such that B(ω)⊂{x∈X:∥x-x0∥X≤r(ω),x0∈X} for all ω∈Ω.
(iii) A random setB(ω) is called a compact random set if B(ω) is compact for all ω∈Ω.
(iv) A random bounded setB(ω)⊂X is called tempered with respect to (θt)t∈ℝ if for a.e. ω∈Ωlimt→+∞e-γtsupx∈B(θ-tω)∥x∥X=0forallγ>0. A random variable ω↦r(ω)∈ℝ is said to be tempered with respect to (θt)t∈ℝ if for a.e. ω∈Ω, limt→+∞e-γtsupt∈ℝ|r(θ-tω)|=0forallγ>0.We consider a continuous RDS{φ(t,ω)}t≥0,ω∈Ω over (Ω,ℱ,ℙ,(θt)t∈ℝ) and 𝒟 the set of all tempered random sets of X.Definition 2.4.
A random set𝒦(ω) is called an absorbing set in 𝒟 if for all B∈𝒟 and a.e. ω∈Ω there exist tB(ω)>0 such that
(2.1)φ(t,θ-tω)B(θ-tω)⊂K(ω)∀t≥tB(ω).Definition 2.5.
A random set𝒜 is called a global 𝒟-random attractor (or 𝒟-pullback attractor) for {φ(t,ω)}t≥0,ω∈Ω if the following hold:(i)
𝒜 is a random compact set, that is ω↦d(x,𝒜(ω)) is measurable for every x∈X and 𝒜(ω) is compact for a.e. ω∈Ω;(ii)
𝒜 is strictly invariant, that is, for ω∈Ω and all t≥0 one has φ(t,ω)𝒜(ω)=𝒜(θtω);(iii)
𝒜 attracts all sets in 𝒟, that is, for all B∈𝒟 and a.e. ω∈Ω we have
(2.2)lim⁡t→+∞d(φ(t,θ-tω)B(θ-tω),A(ω))=0,where d(Y,Z)=supy∈Yinfz∈Z∥y-z∥X is the Hausdorff semimetric (Y⊆X,Z⊆X).Proposition 2.6 (see [24]).
LetK(ω)∈𝒟 be a random absorbing set for the continuous RDS (φ(t))t≥0, which is closed and satisfies for a.e. ω∈Ω the following asymptotic compactness condition: each sequence xn∈φ(tn,θ-tnω,K(θ-tnω)) with tn→∞ has a convergent subsequence in X. Then, the cocycle φ has a unique global random attractor. 							(2.3)A(ω)=⋂τ≥tK(ω)‍⋃t≥τ‍φ(t,θt(ω),K(θt(ω)))¯.


## 3. RDS Generated by Stochastic Three-Component ReversibleGray-Scott System

In this section, we will give the basic setting of system (1.1) and show that it generates a random dynamical system. Let (Ω,ℱ,ℙ) be a probability space as in Section 1. Define (θt)t∈ℝ on Ω via θtω(·)=ω(·+t)-ω(t),t∈ℝ, then (Ω,ℱ,ℙ,(θt)t∈ℝ) is an ergodic metric dynamical system (see [1, 23]).Denoteg~=(u~,v~,w~)T, system (1.1) with initial data (1.2) can be rewritten as
(3.1)∂g~∂t=Ag~+Λ~(g~)+f(x)+σg~∘dBtdt,t>0,g~(0,x)=g~0(x),x∈Rn,
where
(3.2)A=(d1000d2000d3)▵,Λ~(g~)=(-(F+k)u~+u~2v~-Gu~3+Nw~-Fv~-u~2v~+Gu~3ku~-(F+N)w~),
and f(x)=(f1(x),f2(x),f3(x))T, here T denotes the transposition.For our purpose, it is convenient to transform the problem (3.1) into a deterministic system with a random parameter and then show that it generates a random dynamical system.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 [25] for details):
(3.4)dz=-zdt+dBt,z(-∞)=0,∀t≥0,ω∈Ω,
where Bt(ω)=B(t,ω)=ω(t) for ω∈Ω,t∈ℝ. In fact, from [1, 26], we know that the random variable z(ω) is tempered, and there is a θt-invariant set Ω~⊂Ω of full ℙ measure such that for ω∈Ω~, t↦z(θtω) is continuous in t; furthermore,
(3.5)lim⁡t→±∞|z(θtω)|t=lim⁡t→±∞1t∫0t‍z(θsω)ds=0.Let(3.6)u(t)=α(t)u~(t)v(t)=α(t)v~(t)withα(t)=e-σz(θtω)w(t)=α(t)w~(t),
then system (3.1) can be written as
(3.7)∂u∂t=d1▵u-(F+k-σz(θtω))u+α-2(t)u2v-Gα-2(t)u3+Nw+α(t)f1,∂v∂t=d2▵v-(F-σz(θtω))v-α-2(t)u2v+Gα-2(t)u3+α(t)f2,∂w∂t=d3▵w-(F+N-σz(θtω))w+ku+α(t)f3,
that is, g(t,·)=(u(t,·),v(t,·),w(t,·)), t>0  satisfies
(3.8)dgdt=Ag+Λ(g,ω)+α(t)f(x),t>0,g(0,x)=g0(x)=e-σz(ω)g~0(x),x∈Rn,
where
(3.9)Λ(g,ω)=(-(F+k-σz(θtω))u+α-2(t)u2v-Gα-2(t)u3+Nw-(F-σz(θtω))v-α-2(t)u2v+Gα-2(t)u3-(F+N-σz(θtω))w+ku).
We will consider (3.8) for ω∈Ω~ and still use Ω instead of Ω~ from now on.As in the case of a bounded domain with Dirichlet boundary conditions which are studied in [27], for Λ:𝔼∩𝕌↦ℍ is locally Lipschitz continuous and f∈ℍ∩𝕌, by a Galerkin method, one can show that for ℙ-a.e. ω∈Ω and for all g0∈ℍ, (3.8) has a unique solution g(·,ω,g0)∈C([0,∞),ℍ)∩L2((0,T);𝔼) with g(0,ω,g0)=g0 for every T>0. Similarly to [28], because of the continuous nonlinearity Λ, one may take the domain to be a sequence of balls with radius approaching ∞ to deduce the existence of a weak solution to (3.8) on ℝn×ℝn×ℝn. Furthermore, one may get that g(t,ω,g0) is unique and continuous with respect to g0 in ℍ for all t≥0. Then, (3.8) generates a continuous random dynamical system (φ(t))t≥0 over (Ω,ℱ,ℙ,(θt)t∈ℝ) according to the conditions (i)–(iii) in Definition 2.2, where
(3.10)φ(t,ω,g~0)=g(t,ω,g0),∀(t,ω,g~0)∈R+×Ω×H.
We now define a mapping ϕ:ℝ+×Ω×ℍ→ℍ by
(3.11)ϕ(t,ω,g~0)=g~(t,ω,g~0)=g(t,ω,e-σz(ω)g~0)eσz(θtω),
for all (t,ω,g~0)∈ℝ+×Ω×ℍ.Thenϕ is a continuous random dynamical system associated with problem (3.1) on ℝn×ℝn×ℝn.We remark that the two random dynamical systems are conjugated to each other; thus, the inverse transformation ofφ is a solution of the original system. For more details on the conjugate theory of stochastic and random differential equations, we can refer to [29]. Thus, in the following sections, we only need to consider the existence of a random attractor of φ.


## 4. Uniform Estimates of Solutions

In this section, we establish the uniform estimates on the solution of the stochastic Gray-Scott system onℝn×ℝn×ℝn when t→∞ in order to derive the existence of a bounded random absorbing set and the asymptotic compactness of the random dynamical system associated with the problem. Particularly when time is sufficiently large, we will show that the tails of the solutions for large space variables are uniformly small.We always assume that𝒟 is the collection of all tempered subsets of ℍ with respect to (Ω,ℱ,ℙ,(θt)t∈ℝ). The next lemma implied that φ has a random absorbing set in 𝒟.Lemma 4.1.
Assume thatf∈ℍ. Then, there exists a random ball {𝒜(ω)}∈𝒟 centered at 0 with radius R-(ω)>0 such that {𝒜(ω)} is a random absorbing set for φ in 𝒟, that is, for any {B(ω)}∈𝒟 and ℙ-a.e. ω∈Ω, there is TB(ω)>0 such that
(4.1)φ(t,θ-tω,B(θ-tω))⊆A(ω)∀t>TB(ω).Proof.
Define(4.2)W(t,x)=Nkw(t,x),μ=kN,
then the system (3.8) becomes
(4.3)∂u∂t=d1▵u-(F+k-σz(θtω))u+α-2(t)u2v-Gα-2(t)u3+kW+α(t)f1,(4.4)∂v∂t=d2▵v-(F-σz(θtω))v-α-2(t)u2v+Gα-2(t)u3+α(t)f2,(4.5)μ∂W∂t=μd3▵W-(μF+k-μσz(θtω))W+ku+α(t)f3.
Take the inner products ((4.3),Gu(t)), ((4.4),v(t)), and ((4.5),GW(t)). Then summing up the resulting equalities, we get
(4.6)ddt(G‖u‖2+‖v‖2+μG‖W‖2)+2d1G‖∇u‖2+2d2‖∇v‖2+2μGd3‖∇W‖2+2(G(F+k)‖u‖2+F‖v‖2+G(μF+k)‖W‖2)=4kG∫Rn‍uWdx+2∫Rn‍α(t)uf1dx+2∫Rn‍α(t)vf2dx+2∫Rn‍α(t)Wf3dx+2σz(θtω)(G‖u‖2+‖v‖2+μG‖W‖2)-2α-2(t)∫Rn‍(Gu2-uv)2dx,
that is,
(4.7)ddt(G‖u‖2+‖v‖2+μG‖W‖2)+2d1G‖∇u‖2+2d2‖∇v‖2+2μGd3‖∇W‖2≤(2σz(θtω)-F)(G‖u‖2+‖v‖2+μG‖W‖2)+α2(t)GF(‖f1‖2+G‖f2‖2+1μ‖f3‖2).
Let c1=max{1,G,1/μ}/GFmin{1,G,1/μ} and apply Gronwall lemma to (4.7) for t≥0, we have
(4.8)‖g(t,ω,g0(ω))‖2≤e2σ∫0t‍z(θsω)ds-Ft‖g0(ω)‖2+c1‖f‖2e2σ∫0t‍z(θsω)ds-Ft∫0t‍e-2σz(θsω)-2σ∫0s‍z(θτω)dτ+Fsds.
By substituting ω with θ-tω in (4.8), it yields that
(4.9)‖g(t,θ-tω,g0(θ-tω))‖2≤e2σ∫0t‍z(θs-tω)ds-Ft‖g0(θ-tω)‖2+c1‖f‖2e2σ∫0t‍z(θs-tω)ds-Ft∫0t‍e-2σz(θs-tω)-2σ∫0s‍z(θτ-tω)dτ+Fsds≤e2σ∫-t0‍z(θsω)ds-Ft‖g0(θ-tω)‖2+c1‖f‖2∫-t0‍e-2σz(θsω)-2σ∫s0‍z(θτω)dτ+Fsds≤e2σ∫-t0‍z(θsω)ds-Ft‖g0(θ-tω)‖2+c1‖f‖2∫-∞0‍e-2σz(θsω)-2σ∫s0‍z(θτω)dτ+Fsds.
By (3.5), we obtain
(4.10)∫-∞0‍e-2σz(θsω)-2σ∫s0‍z(θτω)dτ+Fsds<+∞.
Notice that B(ω)∈𝒟 is tempered, then for any g0(θ-tω)∈B(θ-tω),
(4.11)lim⁡t→+∞e2σ∫-t0‍z(θsω)ds-Ft‖g0(θ-tω)‖2=0.
Denote
(4.12)R-2(ω)=1+c1‖f‖2∫-∞0‍e-2σz(θsω)-2σ∫s0‍z(θτω)dτ+Fsds,
then {𝒜(ω)}∈𝒟 is a random absorbing set for φ in 𝒟, which completes the proof.Lemma 4.2.
Assume thatf∈ℍ. Then there exists a tempered random variable R~(ω)>0 such that for any B(ω)∈𝒟 and g0(ω)∈B(ω), there exists a TB(ω)>0 such that the solution φ of (3.8) satisfies for ℙ-a.e. ω∈Ω, for all t≥TB(ω),
(4.13)∫tt+1‍‖∇φ(s,θ-t-1ω,g0(θ-t-1ω))‖2ds≤R~(ω).Proof.
By replacingt and ω with T~ and θ-tω in (4.8), for T~>0, we get
(4.14)‖g(T~,θ-tω,g0(θ-tω))‖2≤e2σ∫0T~‍z(θs-tω)ds-FT~‖g0(θ-tω)‖2+c1‖f‖2e2σ∫0T~‍z(θs-tω)ds-Ft∫0T~‍e-2σz(θs-tω)-2σ∫0s‍z(θτ-tω)dτ+Fsds.
Multiplying both sides of (4.14) with e2σ∫T~t‍z(θs-tω)ds-F(t-T~), we have for t≥T~(4.15)e2σ∫T~t‍z(θs-tω)ds-F(t-T~)‖g(T~,θ-tω,g0(θ-tω))‖2≤e2σ∫0t‍z(θs-tω)ds-Ft‖g0(θ-tω)‖2+c1‖f‖2∫0T~‍e-2σz(θs-tω)+2σ∫st‍z(θτ-tω)dτ+F(s-t)ds.
Apply Gronwall lemma to (4.7) again and denote d0=min{d1,d2,d3},c2=1/d0, then for all t≥T~,
(4.16)‖g(t,ω,g0(ω))‖2≤e2σ∫T~t‍z(θsω)ds-F(t-T~)‖g(T~,ω,g0(ω))‖2+c1‖f‖2∫T~t‍e-2σz(θsω)+2σ∫st‍z(θτω)dτ+F(s-t)ds-2d0∫T~t‍e2σ∫st‍z(θτω)dτ+F(s-t)‖∇g(s,ω,g0(ω))‖2ds,
which implies that
(4.17)∫T~t‍e2σ∫st‍z(θτω)dτ+F(s-t)‖∇g(s,ω,g0(ω))‖2ds≤c22e2σ∫T~t‍z(θsω)ds-F(t-T~)‖g(T~,ω,g0(ω))‖2+c1c22‖f‖2∫T~t‍e-2σz(θsω)+2σ∫st‍z(θτω)dτ+F(s-t)ds.
Replacing ω in (4.17) with θ-tω and by (4.15), we obtain
(4.18)∫T~t‍e2σ∫st‍z(θτ-tω)dτ+F(s-t)‖∇g(s,θ-tω,g0(θ-tω))‖2ds≤c22e2σ∫T~t‍z(θs-tω)ds-F(t-T~)‖g(T~,θ-tω,g0(θ-tω))‖2+c1c22‖f‖2∫T~t‍e-2σz(θs-tω)+2σ∫st‍z(θτ-tω)dτ+F(s-t)ds≤c22e2σ∫0t‍z(θs-tω)ds-Ft‖g0(θ-tω)‖2+c1c22‖f‖2∫0t‍e-2σz(θs-tω)+2σ∫st‍z(θτ-tω)dτ+F(s-t)ds=c22e2σ∫-t0‍z(θsω)ds-Ft‖g0(θ-tω)‖2+c1c22‖f‖2∫-t0‍e-2σz(θsω)+2σ∫s0‍z(θτω)dτ+Fsds.
By substituting t and T~ for t+1 and t in (4.18), we find that
(4.19)∫tt+1‍e2σ∫st+1‍z(θτ-t-1ω)dτ+F(s-t-1)‖∇g(s,θ-t-1ω,g0(θ-t-1ω))‖2ds≤c22e2σ∫-t-10‍z(θsω)ds-F(t+1)‖g0(θ-t-1ω)‖2+c1c22‖f‖2∫-t-10‍e-2σz(θsω)+2σ∫s0‍z(θτω)dτ+Fsds.
We know for s∈[t,t+1],
(4.20)∫tt+1‍e2σ∫st+1‍z(θτ-t-1ω)dτ+F(s-t-1)‖∇g(s,θ-t-1ω,g0(θ-t-1ω))‖2ds≥∫tt+1‍e-2σmax⁡0≤τ≤1⁡|z(θτω)|-F‖∇g(s,θ-t-1ω,g0(θ-t-1ω))‖2ds.
Due to (3.5) and the temperedness of g0(ω), there exists a TB(ω)>0 such that for t≥TB(ω), and by (4.19) and (4.20), we get
(4.21)∫tt+1‍‖∇g(s,θ-t-1ω,g0(θ-t-1ω))‖2ds≤c22e2σ∫-t-10‍z(θsω)ds+2σmax⁡0≤τ≤1|z(θτω)|-Ft‖g0(θ-t-1ω)‖2+c1c22‖f‖2∫-t-10‍e-2σz(θsω)+2σmax⁡0≤τ≤1|z(θτω)|+2σ∫s0‍z(θτω)dτ+F(s+1)ds≤1+c1c22‖f‖2∫-∞0‍e-2σz(θsω)+2σmax⁡0≤τ≤1|z(θτω)|+2σ∫s0‍z(θτω)dτ+F(s+1)ds:=R~(ω).
By (3.5) again, we have R~(ω) is tempered and the proof is completed.We need the proposition to prove the next result.Proposition 4.3.
Assume thatf∈ℍ∩𝕌. Then, there exists a tempered random variable R^(ω)>0 such that for any B(ω)∈𝒟 and g0(ω)∈B(ω), there exists a TB(ω)>0 such that the solution φ of (3.8) satisfies for ℙ-a.e. ω∈Ω for all t≥TB(ω)(4.22)∫tt+1‍‖φ(s,θ-t-1ω,g0(θ-t-1ω))‖U6ds≤R^(ω).Proof.
LetV(t,x)=v(t,x)/G, then (4.3)–(4.5) can be written as
(4.23)∂u∂t=d1▵u-(F+k-σz(θtω))u+α-2(t)u2v-Gα-2(t)u3+kW+α(t)f1,∂V∂t=d2▵V-(F-σz(θtω))V-α-2(t)u2V+α-2(t)u3+1Gα(t)f2,μ∂W∂t=μd3▵W-(μF+k-μσz(θtω))W+ku+α(t)f3.
Taking the inner products of (4.23) with u5(t), GV5(t), and W5(t), and summing up the resulting equalities, we get
(4.24)16ddt(‖u‖L66+G‖V‖L66+μ‖W‖L66)+5(d1‖u2∇u‖2+d2G‖V2∇V‖2+μd3G‖W2∇W‖2)=-(F+k-σz(θtω))‖u‖L66-G(F-σz(θtω))‖V‖L66-(μF+k-μσz(θtω))‖W‖L66+k∫Rn‍u5Wdx+k∫Rn‍uW5dx-Gα-2(t)∫Rn‍(u8-u7V-u3V5+u2V6)dx+∫Rn‍α(t)f1u5dx+∫Rn‍α(t)f2V5dx+∫Rn‍α(t)f3W5dx,
that is,
(4.25)ddt(‖u‖L66+G‖V‖L66+μ‖W‖L66)≤-6(F-σz(θtω))‖u‖L66-6G(F-σz(θtω))‖V‖L66-6(μF-μσz(θtω))‖W‖L66+6∫Rn‍α(t)f1u5dx+6∫Rn‍α(t)f2V5dx+6∫Rn‍α(t)f3W5dx≤(6σz(θtω)-F)‖u‖L66+G(6σz(θtω)-F)‖V‖L66+μ(6σz(θtω)-F)‖W‖L66+1F5α6(t)(‖f1‖L66+1G5‖f2‖L66+1μ5‖f3‖L66).
Furthermore, let
(4.26)c3=max⁡{1,1/G5,1/μ5}F5min⁡{1,1/G5,1/μ5},
and apply Gronwall lemma, for ν≥0, we can deduce from (4.25) that
(4.27)‖g(ν,ω,g0(ω))‖U6≤e6σ∫0ν‍z(θsω)ds-Fν‖g0(ω)‖U6+c3‖f‖U6e6σ∫0ν‍z(θsω)ds-Fν∫0ν‍e-6σz(θsω)-6σ∫0s‍z(θτω)dτ+Fsds.
Letting TB(ω) be a positive variable in Lemma 4.2, t≥TB(ω), integrating (4.27) for ν∈(t,t+1), we have
(4.28)∫tt+1‍‖g(ν,ω,g0(ω))‖U6dT≤e6σ∫0t+1‍z(θsω)ds-Ft‖g0(ω)‖U6+c3‖f‖U6e6σ∫0t+1‍z(θsω)ds-Ft∫0t+1‍e-6σz(θsω)-6σ∫0s‍z(θτω)dτ+Fsds.
Replacing ω by θ-t-1ω in (4.28), we obtain
(4.29)∫tt+1‍‖g(ν,θ-t-1ω,g0(θ-t-1ω))‖U6dν≤e6σ∫0t+1‍z(θs-t-1ω)ds-Ft‖g0(θ-t-1ω)‖U6+c3‖f‖U6e6σ∫0t+1‍z(θs-t-1ω)ds-Ft∫0t+1‍e-6σz(θs-t-1ω)-6σ∫0s‍z(θτ-t-1ω)dτ+Fsds≤e6σ∫-t-10‍z(θsω)ds-Ft‖g0(θ-t-1ω)‖U6+c3‖f‖U6e6σ∫-t-10‍z(θsω)ds-Ft∫-t-10‍e-6σz(θsω)-6σ∫s0‍z(θτω)dτ+Fsds≤1+c3‖f‖U6∫-∞0‍e-6σz(θsω)-6σ∫s0‍z(θτω)dτ+Fsds:=R^(ω).
The last but one line in (4.29) due to B(ω)∈𝒟 is tempered, then for any g0(θ-t-1ω)∈B(θ-t-1ω)⊂B(θ-tω),
(4.30)lim⁡t→+∞e6σ∫-t-10‍z(θsω)ds-Ft‖g0(θ-t-1ω)‖U6=0.
The proof is completed.Lemma 4.4.
Assume thatf∈ℍ∩𝕌. Then, there exists a tempered random variable R~(ω)>0 such that for any B(ω)∈𝒟 and g0(ω)∈B(ω), there exists a TB(ω)>0 such that the solution φ of (3.8) satisfies for ℙ-a.e. ω∈Ω, for all t≥TB(ω),
(4.31)‖∇φ(t,θ-tω,g0(θ-tω))‖2≤R(ω).Proof.
Taking the inner products of (4.3)–(4.5) with -Δu,-Δv,-ΔW, respectively, and summing up the three resulting equalities, we have
(4.32)12ddt(‖∇u‖2+‖∇v‖2+μ‖∇W‖2)+d1‖▵u‖2+d2‖▵v‖2+μd3‖▵W‖2+(F+k-σz(θtω))‖∇u‖2+(F-σz(θtω))‖∇v‖2+(μF+k-μσz(θtω))‖∇W‖2=-α-2(t)∫Rn‍u2v▵udx+Gα-2(t)∫Rn‍u3▵udx+α-2(t)∫Rn‍u2v▵vdx-Gα-2(t)∫Rn‍u3▵vdx-2k∫Rn‍u▵Wdx-∫Rn‍α(t)Δuf1dx-∫Rn‍α(t)Δvf2dx-∫Rn‍α(t)ΔWf3dx.
By Hölder inequality, it yields
(4.33)ddt(‖∇u‖2+‖∇v‖2+μ‖∇W‖2)+(F-2σz(θtω))(‖∇u‖2+‖∇v‖2+μ‖∇W‖2)≤(2d1+2d2)α-4(t)∫Rn‍u4v2dx+(2d1+2d2)G2α-4(t)∫Rn‍u6dx+1d1α2(t)‖f1‖2+1d2α2(t)‖f2‖2+1μd3α2(t)‖f3‖2≤1d0(83+4G2)α-4(t)(‖u‖L66+‖v‖L66)+1d0α2(t)‖f‖2,
which implies that
(4.34)ddt‖∇g‖2≤(2σz(θtω)-F)‖∇g‖2+c4α-4(t)‖g‖U6+c5α2(t)‖f‖2,
where c4=c2(8/3+4G2)/min{1,1/μ}andc5=c2/min{1,1/μ}.
LetTB(ω) be a positive variable in Lemma 4.2, t≥TB(ω) and s∈(t,t+1). Then integrate (4.34) over (s,t+1), we get
(4.35)‖∇g(t+1,ω,g0(ω))‖2≤‖∇g(s,ω,g0(ω))‖2+∫st+1‍(2σz(θτω)-F)‖∇g(τ,ω,g0(ω))‖2dτ+c4∫st+1‍α-4(τ)‖g(τ,ω,g0(ω))‖U6dτ+c5∫st+1‍α2(τ)‖f‖2dτ.
Now integrating (4.35) with respect to s over (t,t+1), we find that
(4.36)‖∇g(t+1,ω,g0(ω))‖2≤(1-F)∫tt+1‍‖∇g(τ,ω,g0(ω))‖2dτ+2σ∫tt+1‍z(θτω)‖∇g(τ,ω,g0(ω))‖2dτ+c4∫tt+1‍α-4(τ)‖g(τ,ω,g0(ω))‖U6dτ+c5∫tt+1‍α2(τ)‖f‖2dτ.
Since z(ω) is tempered and z(θtω) is continuous in t, there exists a tempered variable r(ω) such that
(4.37)|z(θtω)|≤e|t|r(ω),∀t∈Rn.
Then, replacingω with θ-t-1ω, by Lemma 4.2 and Proposition 4.3, it follows in (4.36) that
(4.38)‖∇g(t+1,θ-t-1ω,g0(θ-t-1ω))‖2≤(1-F)∫tt+1‍‖∇g(τ,θ-t-1ω,g0(θ-t-1ω))‖2dτ+2σ∫tt+1‍z(θτ-t-1ω)‖∇g(τ,θ-t-1ω,g0(θ-t-1ω))‖2dτ+c4∫tt+1‍e4σz(θτ-t-1ω)‖g(τ,θ-t-1ω,g0(θ-t-1ω))‖U6dτ+c5∫tt+1‍e-2σz(θτ-t-1ω)‖f‖2dτ≤(1-F+2eσr(ω))∫tt+1‍‖∇g(τ,θ-t-1ω,g0(θ-t-1ω))‖2dτ+c4e4σmax⁡-1≤τ≤0⁡|z(θτω)|∫tt+1‍‖g(τ,θ-t-1ω,g0(θ-t-1ω))‖U6dτ+c5∫-10‍e-2σz(θτω)‖f‖2dτ≤(1-F+2eσr(ω))R~(ω)+c4e4σmax⁡-1≤τ≤0⁡|z(θτω)|R^(ω)+c5∫-10‍e-2σz(θτω)‖f‖2dτ:=R(ω).
Note that f∈ℍ∩𝕌. It is easy to see that R(ω) is tempered. This completes the proof.Lemma 4.5.
Assume thatf∈ℍ. Let {B(ω)}∈𝒟 and g0(ω)∈B(ω). Then, for every ϵ>0, there exist T*=T(ϵ,ω,B)>0 and K*=K(ϵ,ω)>0 such that the solution φ of problem (3.8) satisfies for ℙ-a.e. ω∈Ω, forallt≥T,
(4.39)∫|x|≥K*‍|φ(t,θ-tω,g0(θ-tω))|2dx≤ϵ.Proof.
Choose a smooth cutoff 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+.
Taking the inner product of (4.3), (4.4), and (4.5) with Gρ(|x|2/K2)u, ρ(|x|2/K2)v and Gρ(|x|2/K2)W in L2(ℝn), respectively, we get
(4.40)G2ddt∫Rn‍ρ(|x|2K2)|u|2dx-d1G∫Rn‍ρ(|x|2K2)u▵udx+G(F+k-σz(θtω))∫Rn‍ρ(|x|2K2)|u|2dx=G∫Rn‍ρ(|x|2K2)α-2(t)u3vdx-G2∫Rn‍ρ(|x|2K2)α-2(t)u4dx+kG∫Rn‍ρ(|x|2K2)uWdx+G∫Rn‍ρ(|x|2K2)α(t)f1udx,12ddt∫Rn‍ρ(|x|2K2)|v|2dx-d2∫Rn‍ρ(|x|2K2)v▵vdx+(F-σz(θtω))∫Rn‍ρ(|x|2K2)|v|2dx=-∫Rn‍ρ(|x|2K2)α-2(t)u2v2dx+G∫Rn‍ρ(|x|2K2)α-2(t)vu3dx+∫Rn‍ρ(|x|2K2)α(t)f2vdx,μG2ddt∫Rn‍ρ(|x|2K2)|W|2dx-μGd3∫Rn‍ρ(|x|2K2)W▵Wdx+G(μF+k-σz(θtω))∫Rn‍ρ(|x|2K2)|W|2dx=kG∫Rn‍ρ(|x|2K2)Wudx+G∫Rn‍ρ(|x|2K2)α(t)f3Wdx.
Adding up the three equalities, we have(4.41)ddt(G∫Rn‍ρ(|x|2K2)|u|2dx+∫Rn‍ρ(|x|2K2)|v|2dx+μG∫Rn‍ρ(|x|2K2)|W|2dx)-2(d1G∫Rn‍ρ(|x|2K2)u▵udx+d2∫Rn‍ρ(|x|2K2)v▵vdx+μGd3∫Rn‍ρ(|x|2K2)W▵Wdx)+G(F-2σz(θtω))∫Rn‍ρ(|x|2K2)|u|2dx+(F-2σz(θtω))∫Rn‍ρ(|x|2K2)|v|2dx+G(μF-2σz(θtω))∫Rn‍ρ(|x|2K2)|W|2dx≤G2Fα2(t)∫Rn‍ρ(|x|2K2)|f1(x,t)|2dx+G2μFα2(t)∫Rn‍ρ(|x|2K2)|f3(x,t)|2dx+12F∫Rn‍α2(t)ρ(|x|2K2)|f2(x,t)|2dx.
We know that(4.42)∫Rn‍ρ(|x|2K2)u▵udx≤-∫Rn‍ρ(|x|2K2)|∇u|2dx+c6K(‖u‖2+‖∇u‖2),
where c6 is a positive constant which depends on c. Then from (4.41) and (4.42), we get
(4.43)ddt(G∫Rn‍ρ(|x|2K2)|u|2dx+∫Rn‍ρ(|x|2K2)|v|2dx+μG∫Rn‍ρ(|x|2K2)|W|2dx)+G(F-2σz(θtω))∫Rn‍ρ(|x|2K2)|u|2dx+(F-2σz(θtω))∫Rn‍ρ(|x|2K2)|v|2dx+G(μF-2σz(θtω))∫Rn‍ρ(|x|2K2)|W|2dx≤G2Fα2(t)∫Rn‍ρ(|x|2K2)|f1(x,t)|2dx+G2μFα2(t)∫Rn‍ρ(|x|2K2)|f3(x,t)|2dx+12F∫Rn‍α2(t)ρ(|x|2K2)|f2(x,t)|2dx+2d0c6K(G‖u‖2+‖v‖2+μG‖W‖2)+2d0c6K(G‖∇u‖2+‖∇v‖2+μG‖∇W‖2).
Defining
(4.44)c7=max⁡{1,G,G/μ}min⁡{1,G,G/μ},c8=2d0c6c7,d0=max⁡{d1,d2,d3},
and by Gronwall lemma, for any t≥T~, we obtain
(4.45)∫Rn‍ρ(|x|2K2)|g(t,ω,g0(ω))|2dx≤e2σ∫T~t‍z(θτω)dτ-F(t-T~)∫Rn‍ρ(|x|2K2)|g(T~,ω,g0(ω))|2dx+c7∫T~t‍e2σ∫st‍z(θτω)dτ-F(t-s)-2σz(θsω)∫Rn‍ρ(|x|2K2)f2dxds+c8K∫T~t‍e2σ∫st‍z(θτω)dτ-F(t-s)‖g(s,ω,g0(ω))‖2ds+c8K∫T~t‍e2σ∫st‍z(θτω)dτ-F(t-s)‖∇g(s,ω,g0(ω))‖2ds.
By replacing ω by θ-tω, it then follows from (4.45) that
(4.46)∫Rn‍ρ(|x|2K2)|g(t,θ-tω,g0(θ-tω))|2dx≤e2σ∫T~t‍z(θτ-tω)dτ-F(t-T~)∫Rn‍ρ(|x|2K2)|g(T~,θ-tω,g0(θ-tω))|2dx+c7∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)-2σz(θs-tω)∫Rn‍ρ(|x|2K2)f2dxds+c8K∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)‖g(s,θ-tω,g0(θ-tω))‖2ds+c8K∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)‖∇g(s,θ-tω,g0(θ-tω))‖2ds.
We now estimate each term in (4.46) on the right-hand side one by one. By substituting t by T~ and ω by θ-tω in (4.8) and combine with the first term of (4.46), we get
(4.47)e2σ∫T~t‍z(θτ-tω)dτ-F(t-T~)∫Rn‍ρ(|x|2K2)|g(T~,θ-tω,g0(θ-tω))|2dx≤e2σ∫T~t‍z(θτ-tω)dτ-F(t-T~)({∫0s‍}∫0T~‍e2σ∫0T~‍z(θs-tω)ds-FT~‖g0(θ-tω)‖2+c1‖f‖2e2σ∫0T~‍z(θs-tω)ds-FT~∫0T~‍e-2σz(θs-tω)-2σ∫0s‍z(θτ-tω)dτ+Fsds)=e2σ∫0t‍z(θτ-tω)dτ-Ft‖g0(θ-tω)‖2+c1‖f‖2∫0T~‍e-2σz(θs-tω)+2σ∫st‍z(θτ-tω)dτ-F(t-s)ds.
Obviously, there existsT1=T1(ϵ,ω,B)>T~ such that for t>T1,
(4.48)e2σ∫T~t‍z(θτ-tω)dτ-F(t-T~)∫Rn‍ρ(|x|2K2)|g(T~,θ-tω,g0(θ-tω))|2dx≤ϵ.
For the second term on the right-hand side of (4.46), since f∈ℍ, there exist T2=T2(ϵ,ω,B)>T~ and K1=K1(ϵ,ω)>0 such that for all t>T2 and K>K1, then
(4.49)c7∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)-2σz(θs-tω)∫Rn‍ρ(|x|2K2)f2dxds≤c7∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)-2σz(θs-tω)∫|x|≥K‍ρ(|x|2K2)f2dxds≤ϵ.
For the third term, by replacing t by s and ω by θ-tω in (4.8)
(4.50)c8K∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)‖g(s,θ-tω,g0(θ-tω))‖2ds≤c8K∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)({∫0s‍}e2σ∫0s‍z(θs-tω)ds-Fs‖g0(θ-tω)‖2+c1‖f‖2e2σ∫0s‍z(θs-tω)ds-Fs∫0s‍e-2σz(θν-tω)-2σ∫0ν‍z(θτ-tω)dτ+Fνdν)ds≤c8K(t-T~)e2σ∫0t‍z(θs-tω)ds-Ft‖g0(θ-tω)‖2+c8c1‖f‖2K∫T~t‍∫0s‍e-2σz(θν-tω)+2σ∫νt‍z(θτ-tω)dτ-F(t-ν)dνds.
Since f∈ℍ, there exist T3=T3(ϵ,ω,B)>T~ and K2=K2(ϵ,ω)>0 such that for all t>T3 and K>K2, we have
(4.51)c8K∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)‖g(s,θ-tω,g0(θ-tω))‖2ds≤ϵ.
Finally, we estimate the last term on the right-hand side of (4.46). Since f∈ℍ, by using (4.18), there exist T4=T4(ϵ,ω,B)>T~ and K3=K3(ϵ,ω)>0 such that for all t>T4 and K>K3, we obtain
(4.52)c8K∫T~t‍e2σ∫st‍z(θτ-tω)dτ-F(t-s)‖∇g(s,θ-tω,g0(θ-tω))‖2ds≤ϵ.
Now, denoting
(4.53)T*=max⁡{T1,T2,T3,T4},K*=max⁡{K1,K2,K3},
and combining with (4.48), (4.49), (4.51), and (4.52), we get
(4.54)∫|x|≥K*‍ρ(|x|2K2)|g(t,θ-tω,g0(θ-tω))|2dx≤4ϵ,
which completes the proof.


## 5. Random Attractor

In this section, we will prove the existence of a𝒟-random attractor for the RDS φ associated with the stochastic Gray-Scott system (3.8) on ℍ. According to Lemma 4.1 that φ has a closed random absorbing set in 𝒟, which along with the 𝒟-pullback asymptotic compactness will obtain the existence of a unique 𝒟-random attractor. Next, we will establish the 𝒟-pullback asymptotic compactness of φ by using the uniform estimate on the tails of solutions.Lemma 5.1.
Assume thatf∈ℍ∩𝕌. The RDS φ is 𝒟-pullback asymptotically compact in ℍ, that is, for ℙ-a.e. ω∈Ω, the sequence {φ(tn,θ-tnω,g0,n(θ-tnω))} has a convergent subsequence in ℍ if tn→∞, where {B(ω)}∈𝒟 and g0,n(θ-tnω)∈B(θ-tnω).Proof.
Combining with Lemmas4.1 and 4.4, one can find that the proof is slightly a modification of the proof of Lemma 5.2 in [22] (or see Lemma 5.1 in [20]). Thus, we omitted it here.We are now in the position to state the main resultTheorem 5.2.
Assume thatf∈ℍ∩𝕌. The random dynamical system φ has a unique 𝒟-random attractor in ℍ.Proof.
Notice thatφ has a closed random absorbing set in 𝒟 by Lemma 4.1 and the is 𝒟-pullback asymptotically compact in ℍ by Lemma 5.1. Hence, the existence of a unique 𝒟-random attractor in ℍ follows from Proposition 2.6 consequently.Remark 5.3.
In the original three-component reversible Gray-Scott system (see [8]), the first constant F appearing in the second variable v-section does not depend on the space variable x. Here we have to affiliate the constant F to f2(x), or else, it will give a obstacle to establish the uniform estimates of solutions when t→∞. For example, when estimating the term 2∫ℝ‍α(t)vf2dx, we have 2∫ℝ‍α(t)vf2dx≤F∥v∥2+(f22α2(t)/F)|Dom|, where |Dom| denotes the volume of the domain. Actually, we know that it is maybe of infinite dimension on unbounded domain, which is different from the bounded case discussed in [9].



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*Source: 419472-2012-08-30.xml*