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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 810198, 15 pages
http://dx.doi.org/10.1155/2012/810198
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.Β Makinde

Copyright Β© 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.

Abstract

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 π’ͺ be an open bounded set of 𝐑𝑛(𝑛≀3) with a locally Lipschitz continuous boundary πœ•π’ͺ. We consider the stochastic three-component reversible Gray-Scott system with multiplicative noiseπœ•Μƒπ‘’πœ•π‘‘=𝑑1β–΅Μƒπ‘’βˆ’(𝐹+π‘˜)̃𝑒+̃𝑒2Μƒπ‘£βˆ’πΊΜƒπ‘’3+𝑁𝑀+πœŽΜƒπ‘’βˆ˜π‘‘π΅π‘‘,πœ•Μƒπ‘£π‘‘π‘‘πœ•π‘‘=𝑑2□̃̃𝑣+𝐹(1βˆ’π‘£)βˆ’Μƒπ‘’2̃𝑣+𝐺̃𝑒3Μƒ+πœŽπ‘£βˆ˜π‘‘π΅π‘‘,πœ•ξ‚π‘€π‘‘π‘‘πœ•π‘‘=𝑑3□𝑀+π‘˜Μƒπ‘’βˆ’(𝐹+𝑁)𝑀+πœŽπ‘€βˆ˜π‘‘π΅π‘‘,𝑑𝑑(1.1) where ̃𝑒=̃𝑒(π‘₯,𝑑), ̃̃𝑣=𝑣(π‘₯,𝑑), and 𝑀=𝑀(π‘₯,𝑑) are real-valued functions on π’ͺΓ—[𝜏,+∞), πœβˆˆπ‘; all the parameters are arbitrarily given positive constants; 𝐡𝑑 is a Brownian motion and ∘ denotes the Stratonovich sense of the stochastic term. In this work, we consider the homogenous Neumann boundary conditionπœ•Μƒπ‘’πœ•Μƒπ‘£πœ•π‚(π‘₯,𝑑)=πœ•ξ‚π‘€πœ•π‚(π‘₯,𝑑)=πœ•π‚(π‘₯,𝑑)=0,π‘₯βˆˆπœ•π’ͺ,(1.2) where πœ•/πœ•π‚ is the outward normal derivative, and with an initial conditioñ𝑒(π‘₯,0)=̃𝑒0̃̃𝑣(π‘₯),𝑣(π‘₯,0)=0𝑀(π‘₯),𝑀(π‘₯,0)=0(π‘₯),π‘₯∈π’ͺ.(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 𝑛≀3 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 [3–5]. 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 [7–9], 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, β€œnew” 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 (Ξ©,β„±,𝑃) be a probability space, and {πœƒπ‘‘βˆΆΞ©β†’Ξ©,π‘‘βˆˆπ‘} is a family of measure preserving transformations such that (𝑑,πœ”)β†¦πœƒπ‘‘πœ” is measurable, πœƒ0=𝑖𝑑, and πœƒπ‘‘+𝑠=πœƒπ‘‘πœƒπ‘  for all 𝑠,π‘‘βˆˆπ‘. The flow πœƒπ‘‘ together with the probability space (Ξ©,β„±,𝑃,(πœƒπ‘‘)π‘‘βˆˆπ‘) is called as a measurable dynamical system.

A random dynamical system (RDS) on a Polish space (𝑋,𝑑) with Borel 𝜎-algebra ℬ over πœƒ on (Ξ©,β„±,𝑃) is a measurable mapπœ‘βˆΆπ‘+Γ—Ξ©Γ—π‘‹βŸΆπ‘‹,(𝑑,πœ”,π‘₯)βŸΌπœ‘(𝑑,πœ”)π‘₯,(2.1) such that 𝑃-almost surely (𝑃-a.s.) we have(i)πœ‘(0,πœ”)=𝑖𝑑 on 𝑋;(ii)(cocycle property) πœ‘(𝑑+𝑠,πœ”)=πœ‘(𝑑,πœƒπ‘ πœ”)πœ‘(𝑠,πœ”) for all 𝑠,𝑑β‰₯0.

An RDS is continuous or differentiable if πœ‘(𝑑,πœ”)βˆΆπ‘‹β†’π‘‹ is continuous or differentiable.

A map π΅βˆΆΞ©β†’2𝑋 is said to be a closed (compact) random set if 𝐡(πœ”) is closed (compact) for 𝑃-a.s. πœ”βˆˆΞ© and if πœ”β†¦π‘‘(π‘₯,𝐡(πœ”)) is 𝑃-a.s. measurable for all π‘₯βˆˆπ‘‹.

Consider the product Hilbert spaces 𝐿2(π’ͺ), 𝐻=[𝐿2(π’ͺ)]3, and 𝐻1(π’ͺ), 𝐸=[𝐻1(π’ͺ)]3 with the usual inner products and norms(ξ€œπ‘’,𝑣)=Ω𝑒𝑣𝑑π‘₯,|𝑒|=(𝑒,𝑒)1/2βˆ€π‘’,π‘£βˆˆπΏ2(π’ͺ),((𝑒,𝑣))=𝑛𝑖=1𝐷𝑖𝑒,𝐷𝑖𝑣+𝐹(𝑒,𝑣),‖𝑒‖=((𝑒,𝑒))1/2βˆ€π‘’,π‘£βˆˆπ»1(π’ͺ).(2.2) Obviously, for fixed 𝐹 the scalar product and norm defined above are equivalent to the usual scalar product and norm in 𝐻1(π’ͺ). And the norm of 𝐿𝑝(π’ͺ) will be denoted by |β‹…|𝐿𝑝 if 𝑝≠2,𝑋=(𝐿6(π’ͺ))3.

Define the unbounded positive linear operator𝐴(𝐷(𝐴)⟢𝐻)=Ξ”βˆ’πΉπΌ,(2.3) whereξ‚βŽ›βŽœβŽœβŽœβŽœβŽπ‘‘Ξ”=1β–΅000𝑑2β–΅000𝑑3β–΅βŽžβŽŸβŽŸβŽŸβŽŸβŽ ,𝐻𝐷(𝐴)=(πœ™,πœ‘,πœ“)∈2ξ€»(π’ͺ)3βˆΆπœ•πœ™=πœ•π‚πœ•πœ‘=πœ•π‚πœ•πœ“ξ‚Ό.πœ•π‚=0onπœ•π’ͺ(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 𝐢0-semigroup {𝑒𝐴𝑑,𝑑β‰₯0} on the Hilbert space 𝐻. Its spectral set consists of only nonnegative eigenvalues, denoted by πœ‡π‘–=πœ†π‘–+𝐹, 𝑖β‰₯0, where πœ†π‘– are the corresponding eigenvalues of Δ satisfying0=πœ†0<πœ†1β‰€πœ†2β‰€β‹―β‰€πœ†π‘–ξ€·πœ†β‰€β‹―π‘–ξ€ΈβŸΆβˆžasπ‘–βŸΆβˆž.(2.5) By the fact that 𝐻1(π’ͺ)β†ͺ𝐿6(π’ͺ) is a continuous embedding for 𝑛≀3 and by the generalized Ḧolder inequality, one has||𝑒2𝑣||≀|𝑒|2𝐿6|𝑣|𝐿6,for𝑒,π‘£βˆˆπΏ6(π’ͺ).(2.6) Therefore, the nonlinear mapping defined on 𝐸,𝑓̃𝑀=βŽ›βŽœβŽœβŽœβŽœβŽΜƒπ‘’,𝑣,βˆ’π‘˜Μƒπ‘’+̃𝑒2Μƒπ‘£βˆ’πΊΜƒπ‘’3𝑀+π‘πΉβˆ’Μƒπ‘’2̃𝑣+𝐺̃𝑒3ξ‚π‘€βŽžβŽŸβŽŸβŽŸβŽŸβŽ π‘˜Μƒπ‘’βˆ’π‘βˆΆπΈβŸΆπ»,(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 noise𝑑̃𝑔𝑑𝑑=𝐴̃𝑔+𝑓(̃𝑔)+πœŽΜƒπ‘”βˆ˜π‘‘π΅π‘‘π‘‘π‘‘,(2.8) and an initial conditioñ𝑔(0)=̃𝑔0=̃𝑒0,̃𝑣0,𝑀0ξ€Έβˆˆπ»,(2.9) where ̃̃𝑔(𝑑)=(̃𝑒(𝑑,β‹…),𝑣(𝑑,β‹…),𝑀(𝑑,β‹…)). 𝐡𝑑 is a one-dimensional two-sided Wiener process on a probability space (Ξ©,β„±,𝑃), whereΞ©={πœ”βˆˆπΆ(𝐑,π‘π‘š)βˆΆπœ”(0)=0},(2.10) the Borel 𝜎-algebra β„± on Ξ© is generated by the compact open topology, and 𝑃 is the corresponding Wiener measure on β„±. We can define a family of measure-preserving and ergodic transformations (a flow) {πœƒπ‘‘}π‘‘βˆˆπ‘ byπœƒπ‘‘πœ”(β‹…)=πœ”(β‹…+𝑑)βˆ’πœ”(𝑑).(2.11) By means of the change of variables̃𝑒(𝑑)=𝛼(𝑑)̃𝑒(𝑑),𝑣(𝑑)=𝛼(𝑑)𝑣(𝑑)𝑀(𝑑)=𝛼(𝑑)𝑀(𝑑),with𝛼(𝑑)=π‘’βˆ’πœŽπ΅π‘‘,(2.12) system (1.1) can be written as πœ•π‘’πœ•π‘‘=𝑑1β–΅π‘’βˆ’(𝐹+π‘˜)𝑒+π›Όβˆ’2(𝑑)𝑒2π‘£βˆ’πΊπ›Όβˆ’2(𝑑)𝑒3+𝑁𝑀,(2.13)πœ•π‘£πœ•π‘‘=𝑑2□𝑣+𝐹(𝛼(𝑑)βˆ’π‘£)βˆ’π›Όβˆ’2(𝑑)𝑒2𝑣+πΊπ›Όβˆ’2(𝑑)𝑒3,(2.14)πœ•π‘€πœ•π‘‘=𝑑3□𝑀+π‘˜π‘’βˆ’(𝐹+𝑁)𝑀.(2.15) That is 𝑔(𝑑,β‹…)=(𝑒(𝑑,β‹…),𝑣(𝑑,β‹…),𝑀(𝑑,β‹…)) satisfies𝑑𝑔𝑑𝑑=𝐴𝑔+𝑓(𝑔,πœ”),(2.16) with initial condition𝑔(0)=̃𝑔(0)=𝑔0=𝑒0,𝑣0,𝑀0ξ€Έβˆˆπ»,(2.17) whereξ‚βŽ›βŽœβŽœβŽœβŽœβŽπ‘“(𝑔,πœ”)=βˆ’π‘˜π‘’+π›Όβˆ’2(𝑑)𝑒2π‘£βˆ’πΊπ›Όβˆ’2(𝑑)𝑒3+π‘π‘€πΉβˆ’π›Όβˆ’2(𝑑)𝑒2𝑣+πΊπ›Όβˆ’2(𝑑)𝑒3βŽžβŽŸβŽŸβŽŸβŽŸβŽ π‘˜π‘’βˆ’π‘π‘€.(2.18) Due to the fact that 𝐻1(π’ͺ)β†ͺ𝐿6(π’ͺ) and (2.6), we know that 𝑓(𝑔,πœ”) is locally Lipschitz continuous with respect to 𝑔 and bounded for every πœ”βˆˆΞ©. By the same method in [17, Chapters II and XV], we can prove for 𝑃-a.s. every πœ”βˆˆΞ© the local existence and uniqueness of the weak solution 𝑔(𝑑), π‘‘βˆˆ[𝜏,𝑇] for some 𝜏<𝑇, of (2.16) with 𝑔(𝜏)=𝑔0, which is continuously depending on the initial data 𝑔0∈𝐻 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 πœ”βˆˆΞ©, the following statements hold for all 𝜏<𝑇.(i) If ̃𝑔(0,πœ”)∈𝐻, then ̃𝑔(𝑑,πœ”) lies in [𝐢(𝜏,𝑇);𝐻)∩𝐢1((𝜏,𝑇);𝐻)∩𝐿2([𝜏,𝑇);𝐸).(2.19)(ii)̃𝑔(𝑑,̃𝑔(0,πœ”)) is jointly continuous in 𝑑 and ̃𝑔(0,πœ”) 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 πœ”βˆˆΞ©. Hence the solution mapping𝑆(𝑑,πœ”)βˆΆΜƒπ‘”(𝜏,πœ”)βŸΌΜƒπ‘”(𝑑,πœ”)(2.20) generates an RDS. So the transformation𝑆(𝑑,πœ”)βˆΆπ›Όβˆ’1(𝑑)̃𝑔(𝜏,πœ”)βŸΌπ›Όβˆ’1(𝑑)̃𝑔(𝑑,πœ”)(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(𝑑,πœ”).

3. Existence of a Random Attractor

A random set 𝒦(πœ”) is said to absorb the set π΅βŠ‚π‘‹ for an RDS πœ‘ if 𝑃-a.s. there exists 𝑑𝐡(πœ”) such thatπœ‘ξ€·π‘‘,πœƒβˆ’π‘‘πœ”ξ€Έπ΅βŠ‚π’¦(πœ”)βˆ€π‘‘β‰₯𝑑𝐡(πœ”).(3.1) A random set π’œ(πœ”) is said to be a random attractor associated to the RDS πœ‘ if 𝑃-a.s.:(i)π’œ(πœ”) is a random compact set, that is, 𝑃-a.s. πœ”βˆˆΞ©, π’œ(πœ”) is compact, and for all π‘₯βˆˆπ‘‹ and 𝑃-a.s. the map π‘₯↦dist(π‘₯,π’œ(πœ”)) is measurable.(ii)πœ‘(𝑑,πœ”)π’œ(πœ”)=𝐴(πœƒπ‘‘πœ”) for all 𝑑β‰₯0 (invariance).(iii)For all bounded π΅βŠ‚π‘‹, limπ‘‘β†’βˆžξ€·πœ‘ξ€·dist𝑑,πœƒβˆ’π‘‘πœ”ξ€Έξ€Έπ΅,π’œ(πœ”)=0,(3.2) where dist(β‹…,β‹…) denotes the Hausdorff semidistance: dist(π‘Œ,𝑍)=supinfπ‘₯βˆˆπ‘Œπ‘¦βˆˆπ‘π‘‘(π‘₯,𝑦),π‘Œ,π‘βˆˆπ‘‹.(3.3)

Proposition 3.1 (see [14, 15]). Let πœ™ be an RDS on a Polish space (𝑋,𝑑) with Borel 𝜎-algebra ℬ over the flow {πœƒπ‘‘}π‘‘βˆˆπ‘ on a probability space (Ξ©,β„±,𝑃). Suppose there exists a random compact set 𝒦(πœ”) such that for any bounded nonrandom set π΅βŠ‚π‘‹β€‰β€‰π‘ƒ-a.s ξ€·πœ‘ξ€·dist𝑑,πœƒβˆ’π‘‘πœ”ξ€Έξ€Έπ΅,𝒦(πœ”)⟢0asπ‘‘βŸΆ+∞.(3.4) Then the set π’œ(πœ”)=ξšπ΅βŠ‚π‘‹Ξ›π΅(πœ”)(3.5) is a unique random attractor for πœ™, where the union is taken over all bounded π΅βŠ‚π‘‹ and Λ𝐡(πœ”) is the omega-limit set of 𝐡 given by Λ𝐡(πœ”)=𝑠β‰₯0ξšπ‘‘β‰₯π‘ πœ™ξ€·π‘‘,πœƒβˆ’π‘‘πœ”ξ€Έπ΅.(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 π‘Ÿ1(πœ”)>0, depending on 𝐹, 𝐺, 𝜎, and πœ‡, such that for all 𝜌>0 there exists 𝑑(πœ”)β‰€βˆ’1 such that the following holds 𝑃-a.s. For all 𝑑0≀𝑑(πœ”), and for all ̃𝑔0∈𝐻 with |̃𝑔0|β‰€πœŒ, the solution 𝑔(𝑑,πœ”;𝑑0,𝛼(𝑑0)̃𝑔0) of system (2.16) over [𝑑0,∞), with 𝑔(𝑑0)=𝛼(𝑑0,πœ”)̃𝑔0, satisfies the inequality ||π‘”ξ€·βˆ’1,πœ”;𝑑0𝑑,𝛼0ξ€Έ,πœ”Μƒπ‘”0ξ€Έ||2β‰€π‘Ÿ21(πœ”).(3.7)

Proof. Define π‘π‘Š(𝑑,π‘₯)=π‘˜π‘˜π‘€(𝑑,π‘₯),πœ‡=𝑁.(3.8) Then (2.13)–(2.15) can be written as πœ•π‘’πœ•π‘‘=𝑑1β–΅π‘’βˆ’(𝐹+π‘˜)𝑒+π›Όβˆ’2(𝑑)𝑒2π‘£βˆ’πΊπ›Όβˆ’2(𝑑)𝑒3+π‘˜π‘Š,(3.9)πœ•π‘£πœ•π‘‘=𝑑2□𝑣+𝐹(𝛼(𝑑)βˆ’π‘£)βˆ’π›Όβˆ’2(𝑑)𝑒2𝑣+πΊπ›Όβˆ’2(𝑑)𝑒3πœ‡,(3.10)πœ•π‘Šπœ•π‘‘=πœ‡π‘‘3β–΅π‘Š+π‘˜π‘’βˆ’(πœ‡πΉ+π‘˜)π‘Š.(3.11) Taking the inner products of (3.9)–(3.11) with 𝐺𝑒(𝑑), 𝑣(𝑑), and πΊπ‘Š(𝑑), respectively. Then sum up the resulting equalities. By the Neumann boundary condition (1.2), we get 12𝑑𝑑𝑑𝐺|𝑒|2+|𝑣|2||π‘Š||+πœ‡πΊ2+𝐺(𝐹+π‘˜)|𝑒|2+𝐹|𝑣|2||π‘Š||+𝐺(πœ‡πΉ+π‘˜)2+𝑑1𝐺‖𝑒‖2+𝑑2‖𝑣‖2+πœ‡πΊπ‘‘3β€–π‘Šβ€–2β‰€ξ€œΞ©ξ€œπΉπ›Ό(𝑑)𝑣𝑑π‘₯+2π‘˜πΊΞ©π‘’π‘Šπ‘‘π‘₯βˆ’π›Όβˆ’2ξ€œ(𝑑)Ω𝐺𝑒2ξ€Έβˆ’π‘’π‘£2𝑑π‘₯β‰€π‘˜πΊ|𝑒|2||π‘Š||+π‘˜πΊ2+𝐹2|𝑣|2+𝐹𝛼2(𝑑)2||π’ͺ||,(3.12) where |π’ͺ| denotes the volume of π’ͺ. Set 𝑑𝑑=min1,𝑑2,𝑑3ξ€Ύ,𝐢1=𝐹||π’ͺ||min{1,𝐺,𝐺/πœ‡}.(3.13) Then (3.12) yields 𝑑||||𝑑𝑑𝑔(𝑑)2+𝑑‖𝑔(𝑑)β€–2||||+𝐹𝑔(𝑑)2≀𝐢1𝛼2(𝑑).(3.14) Applying Gronwall's inequality to (3.14) and then integrating in [𝑑0,βˆ’1], with 𝑑0β‰€βˆ’1 we have ||||𝑔(βˆ’1)2≀||𝛼𝑑0𝑔0||2π‘’βˆ’πΉ(βˆ’1βˆ’π‘‘0)+𝐢1ξ€œπ‘‘βˆ’10π‘’βˆ’πΉ(βˆ’1βˆ’π‘ )𝛼2(𝑠)𝑑𝑠≀𝑒𝐹𝑒𝐹𝑑0||𝛼𝑑0𝑔0||2+𝐢1ξ€œπ‘‘βˆ’10𝑒𝐹𝑠𝛼2ξ‚Ά.(𝑠)𝑑𝑠(3.15) Consequently, give 𝐡(0,𝜌)βŠ‚π», 𝑃-a.s. there exists 𝑑(πœ”,𝜌)β‰€βˆ’1 such that for 𝑑0≀𝑑(πœ”,𝜌) and all 𝑔0∈𝐡(0,𝜌), ||π‘”ξ€·βˆ’1,πœ”;𝑑0𝑑,𝛼0̃𝑔0ξ€Έ||2β‰€π‘Ÿ21(πœ”),(3.16) with π‘Ÿ21(πœ”)=𝑒𝐹1+𝐢1ξ€œβˆ’1βˆ’βˆžπ‘’πΉπ‘ π›Ό2ξ‚Ά(𝑠)𝑑𝑠.(3.17) Indeed, it is enough to choose 𝑑(πœ”,𝜌) such that 𝑒𝐹𝑑0𝛼2𝑑0ξ€ΈπœŒ2≀1,(3.18) and take into account (3.15) and the fact that 𝑃-a.s. 𝑒𝐹𝑠𝛼2(𝑠)=π‘’πΉπ‘ π‘’βˆ’2πœŽπ‘Šπ‘ β†’0 as π‘ β†’βˆ’βˆž.

If we now return to (3.14) and integrate for π‘‘βˆˆ[βˆ’1,0], we have||||𝑔(𝑑)2≀||||𝑔(βˆ’1)2π‘’βˆ’πΉ(𝑑+1)+𝐢1ξ€œπ‘‘βˆ’1π‘’βˆ’πΉ(π‘‘βˆ’π‘ )𝛼2(π‘‘ξ€œπ‘ )𝑑𝑠,0βˆ’1‖𝑔(𝑠)β€–2||||𝑑𝑠≀𝑔(βˆ’1)2+𝐢1ξ€œ0βˆ’1𝛼2(𝑠)𝑑𝑠.(3.19) Thus, we can conclude that given 𝐡(0,𝜌)βŠ‚π» and 𝑃-a.s. there exists 𝑑(πœ”,𝜌)β‰€βˆ’1 such that for 𝑑0≀𝑑(πœ”,𝜌) and for all 𝑔0∈𝐡(0,𝜌),||𝑔𝑑,πœ”;𝑑0,𝑔0ξ€Έ||2β‰€π‘’βˆ’πΉ(𝑑+1)π‘Ÿ21(πœ”)+𝐢1ξ€œπ‘‘βˆ’1π‘’βˆ’πΉ(π‘‘βˆ’π‘ )𝛼2(π‘‘ξ€œπ‘ )𝑑𝑠,0βˆ’1‖‖𝑔(𝑠,πœ”;𝑑0,𝑔0)β€–β€–2π‘‘π‘ β‰€π‘Ÿ21(πœ”)+𝐢1ξ€œ0βˆ’1𝛼2(𝑠)𝑑𝑠.(3.20)

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

Proposition 3.3. There exists a random variable π‘Ÿ3(πœ”)>0, depending on πœ†1, 𝜎, and 𝑑, such that for all πœŒξ…ž>0 there exists 𝑑(πœ”)β‰€βˆ’1 such that the following holds 𝑃-a.s. For all 𝑑0≀𝑑(πœ”) and for all ̃𝑔0∈𝐻 with |̃𝑔0|β‰€πœŒξ…ž, the solution 𝑔(𝑑,πœ”;𝑑0,𝛼(𝑑0)𝑔0) of system (2.16) over [𝑑0,∞), with 𝑔(𝑑0)=𝛼(𝑑0,πœ”)̃𝑔0, satisfies the inequality ||π‘”ξ€·βˆ’1,πœ”;𝑑0𝑑,𝛼0ξ€Έ,πœ”Μƒπ‘”0ξ€Έ||6π‘‹β‰€π‘Ÿ23(πœ”).(3.21) Also, for π‘‘βˆˆ[βˆ’1,0] one has ||||𝑔(𝑑)6π‘‹β‰€π‘’βˆ’πΉ(𝑑+1)π‘Ÿ23(πœ”)+𝐢4ξ€œπ‘‘βˆ’1π‘’βˆ’πΉ(π‘‘βˆ’πœ)𝛼6(𝜏)π‘‘πœ.(3.22)

Proof. Letting 𝑉(𝑑,π‘₯)=𝑣(𝑑,π‘₯)/𝐺, (3.9)–(3.11) can be written as πœ•π‘’πœ•π‘‘=𝑑1β–΅π‘’βˆ’(𝐹+π‘˜)𝑒+πΊπ›Όβˆ’2(𝑑)𝑒2π‘‰βˆ’πΊπ›Όβˆ’2(𝑑)𝑒3+π‘˜π‘Š,(3.23)πœ•π‘‰πœ•π‘‘=𝑑2𝐹□𝑣+𝐺𝛼(𝑑)βˆ’πΉπ‘‰βˆ’π›Όβˆ’2(𝑑)𝑒2𝑉+π›Όβˆ’2(𝑑)𝑒3πœ‡,(3.24)πœ•π‘Šπœ•π‘‘=πœ‡π‘‘3β–΅π‘Š+π‘˜π‘’βˆ’(πœ‡πΉ+π‘˜)π‘Š.(3.25) Take the inner products ((3.23), 𝑒5(𝑑)), ((3.24), 𝐺𝑉5(𝑑)) and ((3.25), π‘Š5(𝑑)) and sum up the resulting equalities. By the Neumann boundary condition, we get 16𝑑𝑑𝑑|𝑒|6𝐿6||𝑉||+𝐺6𝐿6||π‘Š||+πœ‡6𝐿6𝑑+51‖‖𝑒2β€–β€–βˆ‡π‘’2+𝑑2𝐺‖‖𝑉2β€–β€–βˆ‡π‘‰2+πœ‡π‘‘3πΊβ€–β€–π‘Š2β€–β€–βˆ‡π‘Š2ξ‚ξ€œ=βˆ’(𝐹+π‘˜)Ω𝑒6ξ€œπ‘‘π‘ +𝐹Ω𝛼(𝑑)𝑉5ξ€œπ‘‘π‘₯βˆ’πΊπΉΞ©π‘‰6ξ€œπ‘‘π‘₯βˆ’(πœ‡πΉ+π‘˜)Ξ©π‘Š6ξ€œπ‘‘π‘₯+π‘˜Ξ©π‘’5ξ€œπ‘Šπ‘‘π‘₯+π‘˜Ξ©π‘’π‘Š5𝑑π‘₯βˆ’πΊπ›Όβˆ’2ξ€œ(𝑑)Ω𝑒8βˆ’π‘’7π‘‰βˆ’π‘’3𝑉5+𝑒2𝑉6𝑑π‘₯.(3.26) By using Young's inequality, we obtain βˆ’πΊπ›Όβˆ’2(ξ€œπ‘‘)Ω𝑒8βˆ’π‘’7π‘‰βˆ’π‘’3𝑉5+𝑒2𝑉6ξ€Έπ‘˜ξ€œπ‘‘π‘₯≀0,Ω𝑒5ξ€œπ‘Šπ‘‘π‘₯+π‘˜Ξ©π‘’π‘Š5𝑑π‘₯β‰€π‘˜|𝑒|6𝐿6||π‘Š||+π‘˜6𝐿6.(3.27) From (3.27), (3.26) yields 𝑑𝑑𝑑|𝑒|6𝐿6||𝑉||+𝐺6𝐿6||π‘Š||+πœ‡6𝐿6+𝐹|𝑒|6𝐿6||𝑉||+𝐺6𝐿6||π‘Š||+πœ‡6𝐿6≀𝐹||π’ͺ||𝐺5𝛼6(𝑑),(3.28) that is, 𝑑𝑑𝑑|𝑒|6𝐿6+πΊβˆ’5|𝑣|6𝐿6+πœ‡βˆ’5|𝑀|6𝐿6+𝐹|𝑒|6𝐿6+πΊβˆ’5|𝑣|6𝐿6+πœ‡βˆ’5|𝑀|6𝐿6≀𝐹||π’ͺ||𝐺5𝛼6(𝑑).(3.29) By denoting 𝐢4=𝐹||π’ͺ||𝐺5ξ€½min1,πΊβˆ’5,πœ‡βˆ’5ξ€Ύ,(3.30) then (3.29) implies that 𝑑||||𝑑𝑑𝑔(𝑑)6𝑋||||+𝐹𝑔(𝑑)6𝑋≀𝐢4𝛼6(𝑑).(3.31) Integrating in [𝑑0,βˆ’1], with 𝑑0β‰€βˆ’1 we have ||||𝑔(βˆ’1)6𝑋≀||𝛼𝑑0𝑔0||6π‘‹π‘’βˆ’πΉ(βˆ’1βˆ’π‘‘0)+𝐢4ξ€œπ‘‘βˆ’10π‘’βˆ’πΉ(βˆ’1βˆ’π‘ )𝛼6(𝑠)𝑑𝑠.(3.32) Consequently, given 𝐡(0,πœŒξ…ž)βŠ‚π», 𝑃-a.s. there exists 𝑑(πœ”,πœŒξ…ž)β‰€βˆ’1 such that for all 𝑑0≀𝑑(πœ”,πœŒξ…ž) and for all 𝑔0∈𝐡(0,πœŒξ…ž), ||π‘”ξ€·βˆ’1,πœ”,𝑑0𝑑,𝛼0𝑔0ξ€Έ||6π‘‹β‰€π‘Ÿ23(πœ”),(3.33) with π‘Ÿ23(πœ”)=𝑒𝐹1+𝐢4ξ€œβˆ’1βˆ’βˆžπ‘’πΉπ‘ π›Ό6ξ‚Ά(𝑠)𝑑𝑠.(3.34) In fact, it is enough to choose 𝑑(πœ”,πœŒξ…ž) to satisfy 𝑒𝐹𝑑0𝛼6𝑑0ξ€ΈπœŒξ…ž6≀1,(3.35) and take into (3.32) and the fact that 𝑃-a.s. 𝑒𝐹𝑠𝛼6(𝑠)=π‘’πΉπ‘ π‘’βˆ’6πœŽπ‘Šπ‘ β†’0 as π‘ β†’βˆ’βˆž. Also, from (3.32) and for π‘‘βˆˆ[βˆ’1,0] we get ||||𝑔(𝑑)6π‘‹β‰€π‘’βˆ’πΉ(𝑑+1)π‘Ÿ23(πœ”)+𝐢4ξ€œπ‘‘βˆ’1π‘’βˆ’πΉ(π‘‘βˆ’πœ)𝛼6(𝜏)π‘‘πœ.(3.36)

Lemma 3.4. There exists a random variable π‘Ÿ2(πœ”)>0, depending on 𝐹, 𝐺, 𝑁, 𝑑, π‘˜, and 𝜎, such that for all 𝜌>0 there exists 𝑑(πœ”)β‰€βˆ’1 such that the following holds 𝑃-a.s. For all 𝑑0≀𝑑(πœ”) and for all ̃𝑔0∈𝐻 with |̃𝑔0|β‰€πœŒ, there exists a unique solution 𝑔(𝑑,πœ”;𝑑0,𝛼(𝑑0)𝑔0) of system (2.16) over [𝑑0,∞), with 𝑔(𝑑0)=𝛼(𝑑0,πœ”)̃𝑔0, and put ̃𝑔(𝑑,πœ”;𝑑0,̃𝑔0)∢=π›Όβˆ’1(𝑑,πœ”)𝑔(𝑑,πœ”;𝑑0,̃𝑔0). Then ‖‖̃𝑔0,πœ”;𝑑0,̃𝑔0ξ€Έβ€–β€–2β‰€π‘Ÿ22(πœ”).(3.37)

Proof. To get a bound in 𝐸, we multiply (2.13)–(2.15) by βˆ’Ξ”π‘’, βˆ’Ξ”π‘£, and βˆ’Ξ”π‘€, respectively. Add up the three equalities, and due to the Neumann boundary condition, we have 12𝑑𝑑𝑑‖𝑒‖2+‖𝑣‖2+‖𝑀‖2ξ€Έ+𝑑1|□𝑒|2+𝑑2|□𝑣|2+𝑑3|□𝑀|2+(𝐹+π‘˜)‖𝑒‖2+𝐹‖𝑣‖2+(𝐹+𝑁)‖𝑀‖2=βˆ’π›Όβˆ’2ξ€œ(𝑑)Ω𝑒2𝑣□𝑒𝑑π‘₯+πΊπ›Όβˆ’2ξ€œ(𝑑)Ω𝑒3ξ€œβ–΅π‘’π‘‘π‘₯βˆ’π‘Ξ©ξ€œπ‘€β–΅π‘’π‘‘π‘₯βˆ’πΉπ›Ό(𝑑)Ω□𝑣𝑑π‘₯+π›Όβˆ’2ξ€œ(𝑑)Ω𝑒2𝑣□𝑣𝑑π‘₯βˆ’πΊπ›Όβˆ’2ξ€œ(𝑑)Ω𝑒3ξ€œβ–΅π‘£π‘‘π‘₯βˆ’π‘˜Ξ©β‰€π‘‘π‘’β–΅π‘€π‘‘π‘₯12|□𝑒|2+𝑑22|□𝑣|2+𝑑32|□𝑀|2+π›Όβˆ’4ξ‚΅1(𝑑)𝑑1+1𝑑2ξ‚Άξ€œΞ©π‘’4𝑣2+𝑁𝑑π‘₯2𝑑1ξ€œΞ©π‘€2π‘˜π‘‘π‘₯+22𝑑3ξ€œΞ©π‘’2𝐺𝑑π‘₯+2π›Όβˆ’4(𝑑)𝑑2ξ€œΞ©π‘’6𝑑π‘₯,(3.38) that is, 𝑑𝑑𝑑‖𝑒‖2+‖𝑣‖2+‖𝑀‖2ξ€Έξ€·+𝐹‖𝑒‖2+‖𝑣‖2+‖𝑀‖2ξ€Έβ‰€π›Όβˆ’4ξ‚΅4(𝑑)3𝑑1+43𝑑2+2𝐺2𝑑2ξ‚Άξ€œΞ©ξ€·π‘’6+𝑣6𝑑π‘₯+2𝑁2𝑑1|𝑀|2+π‘˜2𝑑3|𝑒|2≀𝐢2π›Όβˆ’4(||𝑔||𝑑)6𝑋+𝐢3||𝑔||2.(3.39) Here 𝑑=min{𝑑1,𝑑2,𝑑3}, 𝐢2=(1/𝑑)(8/3+2𝐺2), 𝐢3=(1/𝑑)max{2𝑁2,π‘˜2}. Then from (3.39) we have 𝑑𝑑𝑑‖𝑔‖2+𝐹‖𝑔‖2≀𝐢2π›Όβˆ’4||𝑔||(𝑑)6𝑋+𝐢3||𝑔||2.(3.40) Integrating (3.40) in [𝑠,0], π‘ βˆˆ[βˆ’1,0], we obtain ‖𝑔(0)β€–2≀‖𝑔(𝑠)β€–2+𝐢2ξ€œ0π‘ π›Όβˆ’4||||(𝜏)𝑔(𝜏)6π‘‹π‘‘πœ+𝐢3ξ€œ0𝑠||||𝑔(𝜏)2π‘‘πœ.(3.41) Integrating (3.41) in [βˆ’1,0] and by (3.20), ‖‖𝑔(0)2β‰€ξ€œ0βˆ’1‖𝑔(𝜏)β€–2π‘‘πœ+𝐢2ξ€œ0βˆ’1π›Όβˆ’4||||(𝜏)𝑔(𝜏)6π‘‹π‘‘πœ+𝐢3ξ€œ0βˆ’1||||𝑔(𝜏)2≀1π‘‘πœπ‘‘+𝐢3πΉπ‘Ÿξ‚Άξ‚΅21(πœ”)+𝐢1ξ€œ0βˆ’1𝛼2ξ‚Ά(𝜏)π‘‘πœ+𝐢2supβˆ’1≀𝑑≀0π›Όβˆ’4ξ€œ(𝑑)0βˆ’1||||𝑔(𝜏)6𝑋.(3.42) It is now straightforward from (3.31) and (3.36) that ‖𝑔(0)β€–2=‖̃𝑔(0)β€–2≀1𝑑+𝐢3πΉπ‘Ÿξ‚Άξ‚΅21(πœ”)+𝐢1ξ€œ0βˆ’1𝛼2ξ‚Ά+𝐢(𝜏)π‘‘πœ2𝐹supβˆ’1≀𝑑≀0π›Όβˆ’4ξ‚΅π‘Ÿ(𝑑)23(πœ”)+𝐢4ξ€œ0βˆ’1𝛼6ξ‚Ά.(𝜏)π‘‘πœ(3.43) Consequently, 𝑃-a.s. there exists π‘Ÿ2(πœ”) such that given 𝜌>0, there exists ̃𝑑(πœ”)β‰€βˆ’1 such that for all 𝑑0≀̃𝑑(πœ”) and ̃𝑔0∈𝐻 with |̃𝑔0|β‰€πœŒ, ‖‖̃𝑔0,πœ”;𝑑0,̃𝑔0ξ€Έβ€–β€–2β‰€π‘Ÿ22(πœ”),(3.44) where π‘Ÿ22𝐢(πœ”)=5+𝐢6𝑒𝐹+𝐢1𝐢5ξ€œβˆ’1βˆ’βˆžπ‘’πΉ(𝑠+1)𝛼2(𝑠)𝑑𝑠+𝐢1𝐢5ξ€œ0βˆ’1𝛼2(𝑠)𝑑𝑠+𝐢4𝐢6ξ€œβˆ’1βˆ’βˆžπ‘’πΉ(𝑠+1)𝛼6(𝑠)𝑑𝑠+𝐢4𝐢6ξ€œ0βˆ’1𝛼6𝐢(𝑠)𝑑𝑠,5=ξ‚΅1𝑑+𝐢3𝐹,𝐢6=𝐢2𝐹supβˆ’1≀𝑑≀0π›Όβˆ’4(𝑑).(3.45)

Thus, we can have the main result.

Theorem 3.5. The RDS 𝑆(𝑑,πœ”) has a nonempty compact random attractor π’œ(πœ”).

Proof. This follows from Lemma 3.2 and Lemma 3.4 combined with the embedding of 𝐸β†ͺ𝐻 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 πœ•ξ‚π‘ˆπœ•π‘‘=𝑑1β–΅ξ‚ξ‚π‘ˆβˆ’(𝐹+π‘˜)π‘ˆ+2𝑒𝑣𝑒2πœŽπ‘Šπ‘‘ξ‚π‘ˆ+𝑒2𝑉𝑒2πœŽπ‘Šπ‘‘βˆ’3𝐺𝑒2πœŽπ‘Šπ‘‘π‘’2ξ‚ξ‚‹πœ•ξ‚π‘‰π‘ˆ+π‘π‘Š,πœ•π‘‘=𝑑1β–΅ξ‚ξ‚π‘‰βˆ’πΉπ‘‰βˆ’2𝑒𝑣𝑒2πœŽπ‘Šπ‘‘ξ‚π‘ˆβˆ’π‘’2𝑉𝑒2πœŽπ‘Šπ‘‘+3𝐺𝑒2𝑒2πœŽπ‘Šπ‘‘ξ‚πœ•ξ‚‹π‘Šπ‘ˆ,πœ•π‘‘=𝑑1β–΅ξ‚‹ξ‚ξ‚‹π‘Š+π‘˜π‘ˆβˆ’(𝐹+𝑁)π‘Š.(3.46) Letting Φ(𝑑)=(π‘ˆ,𝑉,π‘Š),π‘ž(𝑑)=̃𝑔(𝑑)βˆ’π‘”(𝑑)βˆ’Ξ¦(𝑑), then ξ€·π‘žπ‘ž(𝑑)=1(𝑑),π‘ž2(𝑑),π‘ž3ξ€Έ=ξ‚€(𝑑)Μƒπ‘’βˆ’ξ‚Μƒπ‘’βˆ’π‘ˆ,π‘£βˆ’ξ‚ξ‚π‘£βˆ’π‘‰,π‘€βˆ’ξ‚‹π‘Šξ‚π‘€βˆ’(3.47) satisfies πœ•π‘ž1πœ•π‘‘=𝑑1β–΅π‘ž1βˆ’(𝐹+π‘˜)π‘ž1ξ‚€+𝛼(𝑑)̃𝑒2Μƒπ‘£βˆ’π‘’2π‘£ξ‚Μƒβˆ’2̃𝑒𝑣𝑒2πœŽπ‘Šπ‘‘ξ‚(𝑑)π‘ˆβˆ’Μƒπ‘’2𝑒2πœŽπ‘Šπ‘‘ξ‚π‘‰+3𝐺̃𝑒2𝑒2πœŽπ‘Šπ‘‘ξ‚ξ‚€π‘ˆβˆ’πΊπ›Ό(𝑑)̃𝑒3βˆ’π‘’3+π‘π‘ž3,πœ•π‘ž2πœ•π‘‘=𝑑1β–΅π‘ž2βˆ’πΉπ‘ž2ξ‚€βˆ’π›Ό(𝑑)̃𝑒2Μƒπ‘£βˆ’π‘’2𝑣̃+2̃𝑒𝑣𝑒2πœŽπ‘Šπ‘‘ξ‚π‘ˆ+̃𝑒2𝑒2πœŽπ‘Šπ‘‘ξ‚π‘‰βˆ’3𝐺̃𝑒2𝑒2πœŽπ‘Šπ‘‘ξ‚ξ‚€π‘ˆ+𝐺𝛼(𝑑)̃𝑒3βˆ’π‘’3,πœ•π‘ž3πœ•π‘‘=𝑑1β–΅π‘ž3+π‘˜π‘ž1βˆ’(𝐹+𝑁)π‘ž3.(3.48) Here, let ̃𝑔(𝑑)=̃𝑔𝑑,πœ”;0,̃𝑔0ξ€Έ=̃𝑒𝑑,πœ”;0,̃𝑒0ξ€Έ,̃𝑣̃𝑣𝑑,πœ”;0,0ξ€Έ,𝑀𝑀𝑑,πœ”;0,0,𝑔(𝑑)=𝑔𝑑,πœ”;0,𝑔0ξ€Έ=𝑒𝑑,πœ”;0,𝑒0ξ€Έ,𝑣𝑑,πœ”;0,𝑣0ξ€Έ,𝑀𝑑,πœ”;0,𝑀0ξ€Έξ€Έ(3.49) be two solutions of system (1.1). From (3.48), it seems hard to get the conclusion.

Acknowledgments

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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