Journal of Applied Mathematics

Volume 2012 (2012), Article ID 810198, 15 pages

http://dx.doi.org/10.1155/2012/810198

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

^{1}Department of Mathematics, Shanghai Normal University, Shanghai 200234, China^{2}College 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.

#### 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 be an open bounded set of with a locally Lipschitz continuous boundary . We consider the stochastic three-component reversible Gray-Scott system with multiplicative noise where , , and are real-valued functions on , ; all the parameters are arbitrarily given positive constants; is a Brownian motion and denotes the Stratonovich sense of the stochastic term. In this work, we consider the homogenous Neumann boundary condition where is the outward normal derivative, and with an initial condition

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 with Neumann boundary condition (1.2) on a bounded domain of space dimension 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, “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 be a probability space, and is a family of measure preserving transformations such that is measurable, , and for all . The flow together with the probability space is called as a measurable dynamical system.

A random dynamical system (RDS) on a Polish space with Borel -algebra over on is a measurable map such that -almost surely (-a.s.) we have(i) on ;(ii)(cocycle property) for all .

An RDS is continuous or differentiable if is continuous or differentiable.

A map is said to be a closed (compact) random set if is closed (compact) for -a.s. and if is -a.s. measurable for all .

Consider the product Hilbert spaces , , and , with the usual inner products and norms Obviously, for fixed the scalar product and norm defined above are equivalent to the usual scalar product and norm in . And the norm of will be denoted by if .

Define the unbounded positive linear operator where By the Lumer-Phillips theorem and the generation theorem for analytic semigroup [16], the operator in (2.3) is linear, sectorial, closed, and defined and is the generator of an analytic -semigroup on the Hilbert space . Its spectral set consists of only nonnegative eigenvalues, denoted by , , where are the corresponding eigenvalues of satisfying By the fact that is a continuous embedding for and by the generalized Hlder inequality, one has Therefore, the nonlinear mapping defined on , 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 noise and an initial condition where . is a one-dimensional two-sided Wiener process on a probability space , where the Borel -algebra on is generated by the compact open topology, and is the corresponding Wiener measure on . We can define a family of measure-preserving and ergodic transformations (a flow) by By means of the change of variables system (1.1) can be written as That is satisfies with initial condition where Due to the fact that and (2.6), we know that is locally Lipschitz continuous with respect to and bounded for every . By the same method in [17, Chapters II and XV], we can prove for -a.s. every the local existence and uniqueness of the weak solution , for some , of (2.16) with , which is continuously depending on the initial data and turns out to be a strong solution on by [16, Theorem 48.5]. One can show that for -a.s. every , the following statements hold for all . If , then lies in is jointly continuous in and in .The solution mapping of (2.16) satisfies the property of an RDS.

This system has a unique solution for every . Hence the solution mapping generates an RDS. So the transformation also determines an RDS corresponding to system (1.1).

We will prove the existence of a nonempty compact random attractor for the RDS .

#### 3. Existence of a Random Attractor

A random set is said to absorb the set for an RDS if -a.s. there exists such that A random set is said to be a random attractor associated to the RDS if -a.s.: is a random compact set, that is, -a.s. , is compact, and for all and -a.s. the map is measurable. for all (invariance).For all bounded , where denotes the Hausdorff semidistance:

Proposition 3.1 (see [14, 15]). *Let be an RDS on a Polish space with Borel -algebra over the flow on a probability space . Suppose there exists a random compact set such that for any bounded nonrandom set -a.s
**
Then the set
**
is a unique random attractor for , where the union is taken over all bounded and is the omega-limit set of given by
*

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

Lemma 3.2. *There exists a random variable , depending on , , , and , such that for all there exists such that the following holds -a.s. For all , and for all with , the solution of system (2.16) over , with , satisfies the inequality
*

*Proof. *Define
Then (2.13)–(2.15) can be written as
Taking the inner products of (3.9)–(3.11) with , , and , respectively. Then sum up the resulting equalities. By the Neumann boundary condition (1.2), we get
where denotes the volume of . Set
Then (3.12) yields
Applying Gronwall's inequality to (3.14) and then integrating in , with we have
Consequently, give , -a.s. there exists such that for and all ,
with
Indeed, it is enough to choose such that
and take into account (3.15) and the fact that -a.s. as .

If we now return to (3.14) and integrate for , we have Thus, we can conclude that given and -a.s. there exists such that for and for all ,

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

Proposition 3.3. *There exists a random variable , depending on , , and , such that for all there exists such that the following holds -a.s. For all and for all with , the solution of system (2.16) over , with , satisfies the inequality
**
Also, for one has
*

*Proof. *Letting , (3.9)–(3.11) can be written as
Take the inner products ((3.23), ), ((3.24), ) and ((3.25), ) and sum up the resulting equalities. By the Neumann boundary condition, we get
By using Young's inequality, we obtain
From (3.27), (3.26) yields
that is,
By denoting
then (3.29) implies that
Integrating in , with we have
Consequently, given , -a.s. there exists such that for all and for all ,
with
In fact, it is enough to choose to satisfy
and take into (3.32) and the fact that -a.s. as . Also, from (3.32) and for we get

Lemma 3.4. *There exists a random variable , depending on , , , , , and , such that for all there exists such that the following holds -a.s. For all and for all with , there exists a unique solution of system (2.16) over , with , and put . Then
*

*Proof. *To get a bound in , we multiply (2.13)–(2.15) by , , and , respectively. Add up the three equalities, and due to the Neumann boundary condition, we have
that is,
Here , , . Then from (3.39) we have
Integrating (3.40) in , , we obtain
Integrating (3.41) in and by (3.20),
It is now straightforward from (3.31) and (3.36) that
Consequently, -a.s. there exists such that given , there exists such that for all and with ,
where

Thus, we can have the main result.

Theorem 3.5. *The RDS has a nonempty compact random attractor .*

*Proof. *This follows from Lemma 3.2 and Lemma 3.4 combined with the embedding of 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
Letting , then
satisfies
Here, let
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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