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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 810198, 15 pages
Research Article

Random Attractors for Stochastic Three-Component Reversible Gray-Scott System with Multiplicative White Noise

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China

Received 25 December 2011; Revised 26 February 2012; Accepted 26 February 2012

Academic Editor: Oluwole D. Makinde

Copyright © 2012 Anhui Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper is devoted to proving the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with multiplicative white noise.

1. Introduction

Let 𝒪 be an open bounded set of 𝐑𝑛(𝑛3) with a locally Lipschitz continuous boundary 𝜕𝒪. We consider the stochastic three-component reversible Gray-Scott system with multiplicative noise𝜕̃𝑢𝜕𝑡=𝑑1̃𝑢(𝐹+𝑘)̃𝑢+̃𝑢2̃𝑣𝐺̃𝑢3+𝑁𝑤+𝜎̃𝑢𝑑𝐵𝑡,𝜕̃𝑣𝑑𝑡𝜕𝑡=𝑑2̃̃𝑣+𝐹(1𝑣)̃𝑢2̃𝑣+𝐺̃𝑢3̃+𝜎𝑣𝑑𝐵𝑡,𝜕𝑤𝑑𝑡𝜕𝑡=𝑑3𝑤+𝑘̃𝑢(𝐹+𝑁)𝑤+𝜎𝑤𝑑𝐵𝑡,𝑑𝑡(1.1) where ̃𝑢=̃𝑢(𝑥,𝑡), ̃̃𝑣=𝑣(𝑥,𝑡), and 𝑤=𝑤(𝑥,𝑡) are real-valued functions on 𝒪×[𝜏,+), 𝜏𝐑; all the parameters are arbitrarily given positive constants; 𝐵𝑡 is a Brownian motion and denotes the Stratonovich sense of the stochastic term. In this work, we consider the homogenous Neumann boundary condition𝜕̃𝑢𝜕̃𝑣𝜕𝝂(𝑥,𝑡)=𝜕𝑤𝜕𝝂(𝑥,𝑡)=𝜕𝝂(𝑥,𝑡)=0,𝑥𝜕𝒪,(1.2) where 𝜕/𝜕𝝂 is the outward normal derivative, and with an initial conditioñ𝑢(𝑥,0)=̃𝑢0̃̃𝑣(𝑥),𝑣(𝑥,0)=0𝑤(𝑥),𝑤(𝑥,0)=0(𝑥),𝑥𝒪.(1.3)

The three-component reversible Gray-Scott model was firstly introduced by Mahara et al. [1]. Recently in [2], You gave the existence of global attractor for system (1.1) when 𝜎=0 with Neumann boundary condition (1.2) on a bounded domain of space dimension 𝑛3 by the method of the rescaling and grouping estimate. However, the reactions and diffusions are often affected by stochastic factors then it is important and meaningful to take the asymptotic behavior of solutions to consideration. Particularly, the dynamics of certain systems frequently follows some self-organization process where the development of new, complex structures takes place primarily in and through the system itself. This self-organization is normally triggered by internal variation processes, which are usually called fluctuations or noise, that have a positive influence on the system. For instance, recent theoretical studies and experiments with cultured glial cells and the Belousov-Zhabotinsky reaction have shown that noise may play a constructive role on the dynamical behavior of spatially extended systems [35]. Therefore, one cannot ignore the role of noise in chemical and biological self-organization and its relationship with the environmental selection of emergent patterns [6]. In [79], the influence of additive noise on Gray-Scott systems was discussed. As pointed in [10, 11], the effects of additive and multiplicative noises are fundamentally different in nonlinear systems. While the effect of additive noise does not depend on the state of the system, the effect of multiplicative noise is state dependent. Natural systems in which the effect of noise on the system's dynamics does depend on the recent state are autocatalytic chemical reactions or growth processes in developmental biology. More generally speaking, in each system whose dynamics shows some degree of self-referentiality, the effect of exogenous noise will depend on the recent system's state. If noise is multiplicative, “new” phenomena can occur; that is, the noisy system can exhibit behavior, which is qualitatively different from that of the deterministic system, a phenomenon that has been coined noise-induced transitions.

A fundamental problem in the study of dynamics of a stochastic partial differential equation is to show that it generates a random dynamical system (or stochastic flow). One of the most interesting concepts of the theory of random dynamical systems is the random attractor, which was introduced in the 90s of the last century (see [12]). An attractor for an autonomous dynamical system is a compact set in the phase space, attracting the image of particular sets of initial states under the evolution of the dynamical system. However, the random case is more complicated, because random attractors depend on the random parameter and have their own temporal dynamics induced by the noise (cf. the definition in Section 3). Moreover, the existence of a random attractor to the stochastic reversible Gray-Scott system, especially of three components, is widely open to the best of our knowledge. According to methodology of [2] of nondissipative coupling of three variables and the coefficients barrier, we consider system (1.1)–(1.3), which gives partly an answer to the problems of random perturbations proposed in [13]. In this paper, we use the notions and frameworks in [12, 14, 15] to study the stochastic three-component reversible Gray-Scott system with multiplicative white noise.

The paper is organized as follows. In Section 2, we give the existence and uniqueness of solution. Section 3 is devoted to the existence of a random attractor.

2. Existence and Uniqueness of Solutions

Let (Ω,,𝑃) be a probability space, and {𝜃𝑡ΩΩ,𝑡𝐑} is a family of measure preserving transformations such that (𝑡,𝜔)𝜃𝑡𝜔 is measurable, 𝜃0=𝑖𝑑, and 𝜃𝑡+𝑠=𝜃𝑡𝜃𝑠 for all 𝑠,𝑡𝐑. The flow 𝜃𝑡 together with the probability space (Ω,,𝑃,(𝜃𝑡)𝑡𝐑) is called as a measurable dynamical system.

A random dynamical system (RDS) on a Polish space (𝑋,𝑑) with Borel 𝜎-algebra over 𝜃 on (Ω,,𝑃) is a measurable map𝜑𝐑+×Ω×𝑋𝑋,(𝑡,𝜔,𝑥)𝜑(𝑡,𝜔)𝑥,(2.1) such that 𝑃-almost surely (𝑃-a.s.) we have(i)𝜑(0,𝜔)=𝑖𝑑 on 𝑋;(ii)(cocycle property) 𝜑(𝑡+𝑠,𝜔)=𝜑(𝑡,𝜃𝑠𝜔)𝜑(𝑠,𝜔) for all 𝑠,𝑡0.

An RDS is continuous or differentiable if 𝜑(𝑡,𝜔)𝑋𝑋 is continuous or differentiable.

A map 𝐵Ω2𝑋 is said to be a closed (compact) random set if 𝐵(𝜔) is closed (compact) for 𝑃-a.s. 𝜔Ω and if 𝜔𝑑(𝑥,𝐵(𝜔)) is 𝑃-a.s. measurable for all 𝑥𝑋.

Consider the product Hilbert spaces 𝐿2(𝒪), 𝐻=[𝐿2(𝒪)]3, and 𝐻1(𝒪), 𝐸=[𝐻1(𝒪)]3 with the usual inner products and norms(𝑢,𝑣)=Ω𝑢𝑣𝑑𝑥,|𝑢|=(𝑢,𝑢)1/2𝑢,𝑣𝐿2(𝒪),((𝑢,𝑣))=𝑛𝑖=1𝐷𝑖𝑢,𝐷𝑖𝑣+𝐹(𝑢,𝑣),𝑢=((𝑢,𝑢))1/2𝑢,𝑣𝐻1(𝒪).(2.2) Obviously, for fixed 𝐹 the scalar product and norm defined above are equivalent to the usual scalar product and norm in 𝐻1(𝒪). And the norm of 𝐿𝑝(𝒪) will be denoted by ||𝐿𝑝 if 𝑝2,𝑋=(𝐿6(𝒪))3.

Define the unbounded positive linear operator𝐴(𝐷(𝐴)𝐻)=Δ𝐹𝐼,(2.3) where𝑑Δ=1000𝑑2000𝑑3,𝐻𝐷(𝐴)=(𝜙,𝜑,𝜓)2(𝒪)3𝜕𝜙=𝜕𝝂𝜕𝜑=𝜕𝝂𝜕𝜓.𝜕𝝂=0on𝜕𝒪(2.4) By the Lumer-Phillips theorem and the generation theorem for analytic semigroup [16], the operator 𝐴 in (2.3) is linear, sectorial, closed, and defined and is the generator of an analytic 𝐶0-semigroup {𝑒𝐴𝑡,𝑡0} on the Hilbert space 𝐻. Its spectral set consists of only nonnegative eigenvalues, denoted by 𝜇𝑖=𝜆𝑖+𝐹, 𝑖0, where 𝜆𝑖 are the corresponding eigenvalues of Δ satisfying0=𝜆0<𝜆1𝜆2𝜆𝑖𝜆𝑖as𝑖.(2.5) By the fact that 𝐻1(𝒪)𝐿6(𝒪) is a continuous embedding for 𝑛3 and by the generalized Ḧolder inequality, one has||𝑢2𝑣|||𝑢|2𝐿6|𝑣|𝐿6,for𝑢,𝑣𝐿6(𝒪).(2.6) Therefore, the nonlinear mapping defined on 𝐸,𝑓̃𝑤=̃𝑢,𝑣,𝑘̃𝑢+̃𝑢2̃𝑣𝐺̃𝑢3𝑤+𝑁𝐹̃𝑢2̃𝑣+𝐺̃𝑢3𝑤𝑘̃𝑢𝑁𝐸𝐻,(2.7) is locally Lipschitz continuous. Thus, the initial boundary problem (1.1)–(1.3) is formulated as an initial value problem of the stochastic three-component reversible Gray-Scott system with multiplicative noise𝑑̃𝑔𝑑𝑡=𝐴̃𝑔+𝑓(̃𝑔)+𝜎̃𝑔𝑑𝐵𝑡𝑑𝑡,(2.8) and an initial conditioñ𝑔(0)=̃𝑔0=̃𝑢0,̃𝑣0,𝑤0𝐻,(2.9) where ̃̃𝑔(𝑡)=(̃𝑢(𝑡,),𝑣(𝑡,),𝑤(𝑡,)). 𝐵𝑡 is a one-dimensional two-sided Wiener process on a probability space (Ω,,𝑃), whereΩ={𝜔𝐶(𝐑,𝐑𝑚)𝜔(0)=0},(2.10) the Borel 𝜎-algebra on Ω is generated by the compact open topology, and 𝑃 is the corresponding Wiener measure on . We can define a family of measure-preserving and ergodic transformations (a flow) {𝜃𝑡}𝑡𝐑 by𝜃𝑡𝜔()=𝜔(+𝑡)𝜔(𝑡).(2.11) By means of the change of variables̃𝑢(𝑡)=𝛼(𝑡)̃𝑢(𝑡),𝑣(𝑡)=𝛼(𝑡)𝑣(𝑡)𝑤(𝑡)=𝛼(𝑡)𝑤(𝑡),with𝛼(𝑡)=𝑒𝜎𝐵𝑡,(2.12) system (1.1) can be written as 𝜕𝑢𝜕𝑡=𝑑1𝑢(𝐹+𝑘)𝑢+𝛼2(𝑡)𝑢2𝑣𝐺𝛼2(𝑡)𝑢3+𝑁𝑤,(2.13)𝜕𝑣𝜕𝑡=𝑑2𝑣+𝐹(𝛼(𝑡)𝑣)𝛼2(𝑡)𝑢2𝑣+𝐺𝛼2(𝑡)𝑢3,(2.14)𝜕𝑤𝜕𝑡=𝑑3𝑤+𝑘𝑢(𝐹+𝑁)𝑤.(2.15) That is 𝑔(𝑡,)=(𝑢(𝑡,),𝑣(𝑡,),𝑤(𝑡,)) satisfies𝑑𝑔𝑑𝑡=𝐴𝑔+𝑓(𝑔,𝜔),(2.16) with initial condition𝑔(0)=̃𝑔(0)=𝑔0=𝑢0,𝑣0,𝑤0𝐻,(2.17) where𝑓(𝑔,𝜔)=𝑘𝑢+𝛼2(𝑡)𝑢2𝑣𝐺𝛼2(𝑡)𝑢3+𝑁𝑤𝐹𝛼2(𝑡)𝑢2𝑣+𝐺𝛼2(𝑡)𝑢3𝑘𝑢𝑁𝑤.(2.18) Due to the fact that 𝐻1(𝒪)𝐿6(𝒪) and (2.6), we know that 𝑓(𝑔,𝜔) is locally Lipschitz continuous with respect to 𝑔 and bounded for every 𝜔Ω. By the same method in [17, Chapters II and XV], we can prove for 𝑃-a.s. every 𝜔Ω the local existence and uniqueness of the weak solution 𝑔(𝑡), 𝑡[𝜏,𝑇] for some 𝜏<𝑇, of (2.16) with 𝑔(𝜏)=𝑔0, which is continuously depending on the initial data 𝑔0𝐻 and turns out to be a strong solution on (𝜏,𝑇] by [16, Theorem 48.5]. One can show that for 𝑃-a.s. every 𝜔Ω, the following statements hold for all 𝜏<𝑇.(i) If ̃𝑔(0,𝜔)𝐻, then ̃𝑔(𝑡,𝜔) lies in [𝐶(𝜏,𝑇);𝐻)𝐶1((𝜏,𝑇);𝐻)𝐿2([𝜏,𝑇);𝐸).(2.19)(ii)̃𝑔(𝑡,̃𝑔(0,𝜔)) is jointly continuous in 𝑡 and ̃𝑔(0,𝜔) in [𝜏,𝑇)×𝐻.(iii)The solution mapping of (2.16) satisfies the property of an RDS.

This system has a unique solution for every 𝜔Ω. Hence the solution mapping𝑆(𝑡,𝜔)̃𝑔(𝜏,𝜔)̃𝑔(𝑡,𝜔)(2.20) generates an RDS. So the transformation𝑆(𝑡,𝜔)𝛼1(𝑡)̃𝑔(𝜏,𝜔)𝛼1(𝑡)̃𝑔(𝑡,𝜔)(2.21) also determines an RDS corresponding to system (1.1).

We will prove the existence of a nonempty compact random attractor for the RDS 𝑆(𝑡,𝜔).

3. Existence of a Random Attractor

A random set 𝒦(𝜔) is said to absorb the set 𝐵𝑋 for an RDS 𝜑 if 𝑃-a.s. there exists 𝑡𝐵(𝜔) such that𝜑𝑡,𝜃𝑡𝜔𝐵𝒦(𝜔)𝑡𝑡𝐵(𝜔).(3.1) A random set 𝒜(𝜔) is said to be a random attractor associated to the RDS 𝜑 if 𝑃-a.s.:(i)𝒜(𝜔) is a random compact set, that is, 𝑃-a.s. 𝜔Ω, 𝒜(𝜔) is compact, and for all 𝑥𝑋 and 𝑃-a.s. the map 𝑥dist(𝑥,𝒜(𝜔)) is measurable.(ii)𝜑(𝑡,𝜔)𝒜(𝜔)=𝐴(𝜃𝑡𝜔) for all 𝑡0 (invariance).(iii)For all bounded 𝐵𝑋, lim𝑡𝜑dist𝑡,𝜃𝑡𝜔𝐵,𝒜(𝜔)=0,(3.2) where dist(,) denotes the Hausdorff semidistance: dist(𝑌,𝑍)=supinf𝑥𝑌𝑦𝑍𝑑(𝑥,𝑦),𝑌,𝑍𝑋.(3.3)

Proposition 3.1 (see [14, 15]). Let 𝜙 be an RDS on a Polish space (𝑋,𝑑) with Borel 𝜎-algebra over the flow {𝜃𝑡}𝑡𝐑 on a probability space (Ω,,𝑃). Suppose there exists a random compact set 𝒦(𝜔) such that for any bounded nonrandom set 𝐵𝑋  𝑃-a.s 𝜑dist𝑡,𝜃𝑡𝜔𝐵,𝒦(𝜔)0as𝑡+.(3.4) Then the set 𝒜(𝜔)=𝐵𝑋Λ𝐵(𝜔)(3.5) is a unique random attractor for 𝜙, where the union is taken over all bounded 𝐵𝑋 and Λ𝐵(𝜔) is the omega-limit set of 𝐵 given by Λ𝐵(𝜔)=𝑠0𝑡𝑠𝜙𝑡,𝜃𝑡𝜔𝐵.(3.6)

Now, we will show the existence of a random attractor for the RDS (2.16).

Lemma 3.2. There exists a random variable 𝑟1(𝜔)>0, depending on 𝐹, 𝐺, 𝜎, and 𝜇, such that for all 𝜌>0 there exists 𝑡(𝜔)1 such that the following holds 𝑃-a.s. For all 𝑡0𝑡(𝜔), and for all ̃𝑔0𝐻 with |̃𝑔0|𝜌, the solution 𝑔(𝑡,𝜔;𝑡0,𝛼(𝑡0)̃𝑔0) of system (2.16) over [𝑡0,), with 𝑔(𝑡0)=𝛼(𝑡0,𝜔)̃𝑔0, satisfies the inequality ||𝑔1,𝜔;𝑡0𝑡,𝛼0,𝜔̃𝑔0||2𝑟21(𝜔).(3.7)

Proof. Define 𝑁𝑊(𝑡,𝑥)=𝑘𝑘𝑤(𝑡,𝑥),𝜇=𝑁.(3.8) Then (2.13)–(2.15) can be written as 𝜕𝑢𝜕𝑡=𝑑1𝑢(𝐹+𝑘)𝑢+𝛼2(𝑡)𝑢2𝑣𝐺𝛼2(𝑡)𝑢3+𝑘𝑊,(3.9)𝜕𝑣𝜕𝑡=𝑑2𝑣+𝐹(𝛼(𝑡)𝑣)𝛼2(𝑡)𝑢2𝑣+𝐺𝛼2(𝑡)𝑢3𝜇,(3.10)𝜕𝑊𝜕𝑡=𝜇𝑑3𝑊+𝑘𝑢(𝜇𝐹+𝑘)𝑊.(3.11) Taking the inner products of (3.9)–(3.11) with 𝐺𝑢(𝑡), 𝑣(𝑡), and 𝐺𝑊(𝑡), respectively. Then sum up the resulting equalities. By the Neumann boundary condition (1.2), we get 12𝑑𝑑𝑡𝐺|𝑢|2+|𝑣|2||𝑊||+𝜇𝐺2+𝐺(𝐹+𝑘)|𝑢|2+𝐹|𝑣|2||𝑊||+𝐺(𝜇𝐹+𝑘)2+𝑑1𝐺𝑢2+𝑑2𝑣2+𝜇𝐺𝑑3𝑊2Ω𝐹𝛼(𝑡)𝑣𝑑𝑥+2𝑘𝐺Ω𝑢𝑊𝑑𝑥𝛼2(𝑡)Ω𝐺𝑢2𝑢𝑣2𝑑𝑥𝑘𝐺|𝑢|2||𝑊||+𝑘𝐺2+𝐹2|𝑣|2+𝐹𝛼2(𝑡)2||𝒪||,(3.12) where |𝒪| denotes the volume of 𝒪. Set 𝑑𝑑=min1,𝑑2,𝑑3,𝐶1=𝐹||𝒪||min{1,𝐺,𝐺/𝜇}.(3.13) Then (3.12) yields 𝑑||||𝑑𝑡𝑔(𝑡)2+𝑑𝑔(𝑡)2||||+𝐹𝑔(𝑡)2𝐶1𝛼2(𝑡).(3.14) Applying Gronwall's inequality to (3.14) and then integrating in [𝑡0,1], with 𝑡01 we have ||||𝑔(1)2||𝛼𝑡0𝑔0||2𝑒𝐹(1𝑡0)+𝐶1𝑡10𝑒𝐹(1𝑠)𝛼2(𝑠)𝑑𝑠𝑒𝐹𝑒𝐹𝑡0||𝛼𝑡0𝑔0||2+𝐶1𝑡10𝑒𝐹𝑠𝛼2.(𝑠)𝑑𝑠(3.15) Consequently, give 𝐵(0,𝜌)𝐻, 𝑃-a.s. there exists 𝑡(𝜔,𝜌)1 such that for 𝑡0𝑡(𝜔,𝜌) and all 𝑔0𝐵(0,𝜌), ||𝑔1,𝜔;𝑡0𝑡,𝛼0̃𝑔0||2𝑟21(𝜔),(3.16) with 𝑟21(𝜔)=𝑒𝐹1+𝐶11𝑒𝐹𝑠𝛼2(𝑠)𝑑𝑠.(3.17) Indeed, it is enough to choose 𝑡(𝜔,𝜌) such that 𝑒𝐹𝑡0𝛼2𝑡0𝜌21,(3.18) and take into account (3.15) and the fact that 𝑃-a.s. 𝑒𝐹𝑠𝛼2(𝑠)=𝑒𝐹𝑠𝑒2𝜎𝑊𝑠0 as 𝑠.

If we now return to (3.14) and integrate for 𝑡[1,0], we have||||𝑔(𝑡)2||||𝑔(1)2𝑒𝐹(𝑡+1)+𝐶1𝑡1𝑒𝐹(𝑡𝑠)𝛼2(𝑑𝑠)𝑑𝑠,01𝑔(𝑠)2||||𝑑𝑠𝑔(1)2+𝐶101𝛼2(𝑠)𝑑𝑠.(3.19) Thus, we can conclude that given 𝐵(0,𝜌)𝐻 and 𝑃-a.s. there exists 𝑡(𝜔,𝜌)1 such that for 𝑡0𝑡(𝜔,𝜌) and for all 𝑔0𝐵(0,𝜌),||𝑔𝑡,𝜔;𝑡0,𝑔0||2𝑒𝐹(𝑡+1)𝑟21(𝜔)+𝐶1𝑡1𝑒𝐹(𝑡𝑠)𝛼2(𝑑𝑠)𝑑𝑠,01𝑔(𝑠,𝜔;𝑡0,𝑔0)2𝑑𝑠𝑟21(𝜔)+𝐶101𝛼2(𝑠)𝑑𝑠.(3.20)

To prove the absorption at time 𝑡=0, we need the following proposition.

Proposition 3.3. There exists a random variable 𝑟3(𝜔)>0, depending on 𝜆1, 𝜎, and 𝑑, such that for all 𝜌>0 there exists 𝑡(𝜔)1 such that the following holds 𝑃-a.s. For all 𝑡0𝑡(𝜔) and for all ̃𝑔0𝐻 with |̃𝑔0|𝜌, the solution 𝑔(𝑡,𝜔;𝑡0,𝛼(𝑡0)𝑔0) of system (2.16) over [𝑡0,), with 𝑔(𝑡0)=𝛼(𝑡0,𝜔)̃𝑔0, satisfies the inequality ||𝑔1,𝜔;𝑡0𝑡,𝛼0,𝜔̃𝑔0||6𝑋𝑟23(𝜔).(3.21) Also, for 𝑡[1,0] one has ||||𝑔(𝑡)6𝑋𝑒𝐹(𝑡+1)𝑟23(𝜔)+𝐶4𝑡1𝑒𝐹(𝑡𝜏)𝛼6(𝜏)𝑑𝜏.(3.22)

Proof. Letting 𝑉(𝑡,𝑥)=𝑣(𝑡,𝑥)/𝐺, (3.9)–(3.11) can be written as 𝜕𝑢𝜕𝑡=𝑑1𝑢(𝐹+𝑘)𝑢+𝐺𝛼2(𝑡)𝑢2𝑉𝐺𝛼2(𝑡)𝑢3+𝑘𝑊,(3.23)𝜕𝑉𝜕𝑡=𝑑2𝐹𝑣+𝐺𝛼(𝑡)𝐹𝑉𝛼2(𝑡)𝑢2𝑉+𝛼2(𝑡)𝑢3𝜇,(3.24)𝜕𝑊𝜕𝑡=𝜇𝑑3𝑊+𝑘𝑢(𝜇𝐹+𝑘)𝑊.(3.25) Take the inner products ((3.23), 𝑢5(𝑡)), ((3.24), 𝐺𝑉5(𝑡)) and ((3.25), 𝑊5(𝑡)) and sum up the resulting equalities. By the Neumann boundary condition, we get 16𝑑𝑑𝑡|𝑢|6𝐿6||𝑉||+𝐺6𝐿6||𝑊||+𝜇6𝐿6𝑑+51𝑢2𝑢2+𝑑2𝐺𝑉2𝑉2+𝜇𝑑3𝐺𝑊2𝑊2=(𝐹+𝑘)Ω𝑢6𝑑𝑠+𝐹Ω𝛼(𝑡)𝑉5𝑑𝑥𝐺𝐹Ω𝑉6𝑑𝑥(𝜇𝐹+𝑘)Ω𝑊6𝑑𝑥+𝑘Ω𝑢5𝑊𝑑𝑥+𝑘Ω𝑢𝑊5𝑑𝑥𝐺𝛼2(𝑡)Ω𝑢8𝑢7𝑉𝑢3𝑉5+𝑢2𝑉6𝑑𝑥.(3.26) By using Young's inequality, we obtain 𝐺𝛼2(𝑡)Ω𝑢8𝑢7𝑉𝑢3𝑉5+𝑢2𝑉6𝑘𝑑𝑥0,Ω𝑢5𝑊𝑑𝑥+𝑘Ω𝑢𝑊5𝑑𝑥𝑘|𝑢|6𝐿6||𝑊||+𝑘6𝐿6.(3.27) From (3.27), (3.26) yields 𝑑𝑑𝑡|𝑢|6𝐿6||𝑉||+𝐺6𝐿6||𝑊||+𝜇6𝐿6+𝐹|𝑢|6𝐿6||𝑉||+𝐺6𝐿6||𝑊||+𝜇6𝐿6𝐹||𝒪||𝐺5𝛼6(𝑡),(3.28) that is, 𝑑𝑑𝑡|𝑢|6𝐿6+𝐺5|𝑣|6𝐿6+𝜇5|𝑤|6𝐿6+𝐹|𝑢|6𝐿6+𝐺5|𝑣|6𝐿6+𝜇5|𝑤|6𝐿6𝐹||𝒪||𝐺5𝛼6(𝑡).(3.29) By denoting 𝐶4=𝐹||𝒪||𝐺5min1,𝐺5,𝜇5,(3.30) then (3.29) implies that 𝑑||||𝑑𝑡𝑔(𝑡)6𝑋||||+𝐹𝑔(𝑡)6𝑋𝐶4𝛼6(𝑡).(3.31) Integrating in [𝑡0,1], with 𝑡01 we have ||||𝑔(1)6𝑋||𝛼𝑡0𝑔0||6𝑋𝑒𝐹(1𝑡0)+𝐶4𝑡10𝑒𝐹(1𝑠)𝛼6(𝑠)𝑑𝑠.(3.32) Consequently, given 𝐵(0,𝜌)𝐻, 𝑃-a.s. there exists 𝑡(𝜔,𝜌)1 such that for all 𝑡0𝑡(𝜔,𝜌) and for all 𝑔0𝐵(0,𝜌), ||𝑔1,𝜔,𝑡0𝑡,𝛼0𝑔0||6𝑋𝑟23(𝜔),(3.33) with 𝑟23(𝜔)=𝑒𝐹1+𝐶41𝑒𝐹𝑠𝛼6(𝑠)𝑑𝑠.(3.34) In fact, it is enough to choose 𝑡(𝜔,𝜌) to satisfy 𝑒𝐹𝑡0𝛼6𝑡0𝜌61,(3.35) and take into (3.32) and the fact that 𝑃-a.s. 𝑒𝐹𝑠𝛼6(𝑠)=𝑒𝐹𝑠𝑒6𝜎𝑊𝑠0 as 𝑠. Also, from (3.32) and for 𝑡[1,0] we get ||||𝑔(𝑡)6𝑋𝑒𝐹(𝑡+1)𝑟23(𝜔)+𝐶4𝑡1𝑒𝐹(𝑡𝜏)𝛼6(𝜏)𝑑𝜏.(3.36)

Lemma 3.4. There exists a random variable 𝑟2(𝜔)>0, depending on 𝐹, 𝐺, 𝑁, 𝑑, 𝑘, and 𝜎, such that for all 𝜌>0 there exists 𝑡(𝜔)1 such that the following holds 𝑃-a.s. For all 𝑡0𝑡(𝜔) and for all ̃𝑔0𝐻 with |̃𝑔0|𝜌, there exists a unique solution 𝑔(𝑡,𝜔;𝑡0,𝛼(𝑡0)𝑔0) of system (2.16) over [𝑡0,), with 𝑔(𝑡0)=𝛼(𝑡0,𝜔)̃𝑔0, and put ̃𝑔(𝑡,𝜔;𝑡0,̃𝑔0)=𝛼1(𝑡,𝜔)𝑔(𝑡,𝜔;𝑡0,̃𝑔0). Then ̃𝑔0,𝜔;𝑡0,̃𝑔02𝑟22(𝜔).(3.37)

Proof. To get a bound in 𝐸, we multiply (2.13)–(2.15) by Δ𝑢, Δ𝑣, and Δ𝑤, respectively. Add up the three equalities, and due to the Neumann boundary condition, we have 12𝑑𝑑𝑡𝑢2+𝑣2+𝑤2+𝑑1|𝑢|2+𝑑2|𝑣|2+𝑑3|𝑤|2+(𝐹+𝑘)𝑢2+𝐹𝑣2+(𝐹+𝑁)𝑤2=𝛼2(𝑡)Ω𝑢2𝑣𝑢𝑑𝑥+𝐺𝛼2(𝑡)Ω𝑢3𝑢𝑑𝑥𝑁Ω𝑤𝑢𝑑𝑥𝐹𝛼(𝑡)Ω𝑣𝑑𝑥+𝛼2(𝑡)Ω𝑢2𝑣𝑣𝑑𝑥𝐺𝛼2(𝑡)Ω𝑢3𝑣𝑑𝑥𝑘Ω𝑑𝑢𝑤𝑑𝑥12|𝑢|2+𝑑22|𝑣|2+𝑑32|𝑤|2+𝛼41(𝑡)𝑑1+1𝑑2Ω𝑢4𝑣2+𝑁𝑑𝑥2𝑑1Ω𝑤2𝑘𝑑𝑥+22𝑑3Ω𝑢2𝐺𝑑𝑥+2𝛼4(𝑡)𝑑2Ω𝑢6𝑑𝑥,(3.38) that is, 𝑑𝑑𝑡𝑢2+𝑣2+𝑤2+𝐹𝑢2+𝑣2+𝑤2𝛼44(𝑡)3𝑑1+43𝑑2+2𝐺2𝑑2Ω𝑢6+𝑣6𝑑𝑥+2𝑁2𝑑1|𝑤|2+𝑘2𝑑3|𝑢|2𝐶2𝛼4(||𝑔||𝑡)6𝑋+𝐶3||𝑔||2.(3.39) Here 𝑑=min{𝑑1,𝑑2,𝑑3}, 𝐶2=(1/𝑑)(8/3+2𝐺2), 𝐶3=(1/𝑑)max{2𝑁2,𝑘2}. Then from (3.39) we have 𝑑𝑑𝑡𝑔2+𝐹𝑔2𝐶2𝛼4||𝑔||(𝑡)6𝑋+𝐶3||𝑔||2.(3.40) Integrating (3.40) in [𝑠,0], 𝑠[1,0], we obtain 𝑔(0)2𝑔(𝑠)2+𝐶20𝑠𝛼4||||(𝜏)𝑔(𝜏)6𝑋𝑑𝜏+𝐶30𝑠||||𝑔(𝜏)2𝑑𝜏.(3.41) Integrating (3.41) in [1,0] and by (3.20), 𝑔(0)201𝑔(𝜏)2𝑑𝜏+𝐶201𝛼4||||(𝜏)𝑔(𝜏)6𝑋𝑑𝜏+𝐶301||||𝑔(𝜏)21𝑑𝜏𝑑+𝐶3𝐹𝑟21(𝜔)+𝐶101𝛼2(𝜏)𝑑𝜏+𝐶2sup1𝑡0𝛼4(𝑡)01||||𝑔(𝜏)6𝑋.(3.42) It is now straightforward from (3.31) and (3.36) that 𝑔(0)2=̃𝑔(0)21𝑑+𝐶3𝐹𝑟21(𝜔)+𝐶101𝛼2+𝐶(𝜏)𝑑𝜏2𝐹sup1𝑡0𝛼4𝑟(𝑡)23(𝜔)+𝐶401𝛼6.(𝜏)𝑑𝜏(3.43) Consequently, 𝑃-a.s. there exists 𝑟2(𝜔) such that given 𝜌>0, there exists ̃𝑡(𝜔)1 such that for all 𝑡0̃𝑡(𝜔) and ̃𝑔0𝐻 with |̃𝑔0|𝜌, ̃𝑔0,𝜔;𝑡0,̃𝑔02𝑟22(𝜔),(3.44) where 𝑟22𝐶(𝜔)=5+𝐶6𝑒𝐹+𝐶1𝐶51𝑒𝐹(𝑠+1)𝛼2(𝑠)𝑑𝑠+𝐶1𝐶501𝛼2(𝑠)𝑑𝑠+𝐶4𝐶61𝑒𝐹(𝑠+1)𝛼6(𝑠)𝑑𝑠+𝐶4𝐶601𝛼6𝐶(𝑠)𝑑𝑠,5=1𝑑+𝐶3𝐹,𝐶6=𝐶2𝐹sup1𝑡0𝛼4(𝑡).(3.45)

Thus, we can have the main result.

Theorem 3.5. The RDS 𝑆(𝑡,𝜔) has a nonempty compact random attractor 𝒜(𝜔).

Proof. This follows from Lemma 3.2 and Lemma 3.4 combined with the embedding of 𝐸𝐻 and Proposition 3.1.

Remark 3.6. It is necessary and interesting for us to consider the Hausdorff dimension of the random attractor which is generated by the stochastic three-component reversible Gray-Scott system with multiplicative white noise, but it seems impossibile to apply the results in [18, 19] directly because of the higher-order terms. In order to verify the differentiability properties for the cocycle generated by the random system, we need to check condition (2.7)–(2.13) in [19]. Considering the linearized equation (2.13)–(2.15), we have 𝜕𝑈𝜕𝑡=𝑑1𝑈(𝐹+𝑘)𝑈+2𝑢𝑣𝑒2𝜎𝑊𝑡𝑈+𝑢2𝑉𝑒2𝜎𝑊𝑡3𝐺𝑒2𝜎𝑊𝑡𝑢2𝜕𝑉𝑈+𝑁𝑊,𝜕𝑡=𝑑1𝑉𝐹𝑉2𝑢𝑣𝑒2𝜎𝑊𝑡𝑈𝑢2𝑉𝑒2𝜎𝑊𝑡+3𝐺𝑢2𝑒2𝜎𝑊𝑡𝜕𝑊𝑈,𝜕𝑡=𝑑1𝑊+𝑘𝑈(𝐹+𝑁)𝑊.(3.46) Letting Φ(𝑡)=(𝑈,𝑉,𝑊),𝑞(𝑡)=̃𝑔(𝑡)𝑔(𝑡)Φ(𝑡), then 𝑞𝑞(𝑡)=1(𝑡),𝑞2(𝑡),𝑞3=(𝑡)̃𝑢̃𝑢𝑈,𝑣𝑣𝑉,𝑤𝑊𝑤(3.47) satisfies 𝜕𝑞1𝜕𝑡=𝑑1𝑞1(𝐹+𝑘)𝑞1+𝛼(𝑡)̃𝑢2̃𝑣𝑢2𝑣̃2̃𝑢𝑣𝑒2𝜎𝑊𝑡(𝑡)𝑈̃𝑢2𝑒2𝜎𝑊𝑡𝑉+3𝐺̃𝑢2𝑒2𝜎𝑊𝑡𝑈𝐺𝛼(𝑡)̃𝑢3𝑢3+𝑁𝑞3,𝜕𝑞2𝜕𝑡=𝑑1𝑞2𝐹𝑞2𝛼(𝑡)̃𝑢2̃𝑣𝑢2𝑣̃+2̃𝑢𝑣𝑒2𝜎𝑊𝑡𝑈+̃𝑢2𝑒2𝜎𝑊𝑡𝑉3𝐺̃𝑢2𝑒2𝜎𝑊𝑡𝑈+𝐺𝛼(𝑡)̃𝑢3𝑢3,𝜕𝑞3𝜕𝑡=𝑑1𝑞3+𝑘𝑞1(𝐹+𝑁)𝑞3.(3.48) Here, let ̃𝑔(𝑡)=̃𝑔𝑡,𝜔;0,̃𝑔0=̃𝑢𝑡,𝜔;0,̃𝑢0,̃𝑣̃𝑣𝑡,𝜔;0,0,𝑤𝑤𝑡,𝜔;0,0,𝑔(𝑡)=𝑔𝑡,𝜔;0,𝑔0=𝑢𝑡,𝜔;0,𝑢0,𝑣𝑡,𝜔;0,𝑣0,𝑤𝑡,𝜔;0,𝑤0(3.49) be two solutions of system (1.1). From (3.48), it seems hard to get the conclusion.


The authors would like to thank the referees for many helpful suggestions and comments. Also more thanks to Professor Shengfan Zhou and Professor Yuncheng You for their helpful discussions, advice, and assistance. This work was supported by the National Natural Science Foundation of China under Grant 11071165 and Guangxi Provincial Department of Research Project under Grant 201010LX166.


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