Volume 2012, Issue 1 810198

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

Anhui Gu,

Corresponding Author

Anhui Gu

[email protected]

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China shnu.edu.cn

College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China gutcie.cn

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Anhui Gu,

Corresponding Author

Anhui Gu

[email protected]

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China shnu.edu.cn

College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China gutcie.cn

Search for more papers by this author

First published: 10 May 2012

https://doi.org/10.1155/2012/810198

Citations: 5

Academic Editor: Oluwole D. Makinde

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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 𝒪 be an open bounded set of Rⁿ (n ≤ 3) with a locally Lipschitz continuous boundary ∂𝒪. We consider the stochastic three-component reversible Gray-Scott system with multiplicative noise

()

where

, and

are real-valued functions on 𝒪 × [τ, +∞), τ ∈ R; all the parameters are arbitrarily given positive constants; B_t is a Brownian motion and ∘ denotes the Stratonovich sense of the stochastic term. In this work, we consider the homogenous Neumann boundary condition

()

where ∂/∂ν 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 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, “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, ℱ, P) be a probability space, and {θ_t : Ω → Ω, t ∈ R} is a family of measure preserving transformations such that (t, ω) ↦ θ_tω is measurable, θ₀ = id, and θ_t+s = θ_tθ_s for all s, t ∈ R. The flow θ_t together with the probability space (Ω, ℱ, P, (θ_t) _t∈R) is called as a measurable dynamical system.

A random dynamical system (RDS) on a Polish space (X, d) with Borel σ-algebra ℬ over θ on (Ω, ℱ, P) is a measurable map

()

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, θ_sω)φ(s, ω) for all s, t ≥ 0.

An RDS is continuous or differentiable if φ(t, ω) : X → X is continuous or differentiable.

A map B : Ω → 2^X is said to be a closed (compact) random set if B(ω) is closed (compact) for P-a.s. ω ∈ Ω and if ω ↦ d(x, B(ω)) is P-a.s. measurable for all x ∈ X.

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

()

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 L^p(𝒪) will be denoted by

if p ≠ 2, X = (L⁶(𝒪)) ³.

Define the unbounded positive linear operator

()

where

()

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 {e^At, t ≥ 0} on the Hilbert space H. Its spectral set consists of only nonnegative eigenvalues, denoted by μ_i = λ_i + F, i ≥ 0, where λ_i are the corresponding eigenvalues of

satisfying

()

By the fact that H¹(𝒪)↪L⁶(𝒪) is a continuous embedding for n ≤ 3 and by the generalized H

lder inequality, one has

()

Therefore, the nonlinear mapping defined on E,

()

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

. B_t is a one-dimensional two-sided Wiener process on a probability space (Ω, ℱ, P), where

()

the Borel σ-algebra ℱ on Ω is generated by the compact open topology, and P is the corresponding Wiener measure on ℱ. We can define a family of measure-preserving and ergodic transformations (a flow) {θ_t} _t∈R by

()

By means of the change of variables

()

system (1.1) can be written as

()

That is g(t, ·) = (u(t, ·), v(t, ·), w(t, ·)) satisfies

()

with initial condition

()

where

()

Due to the fact that H¹(𝒪)↪L⁶(𝒪) and (2.6), we know that

is locally Lipschitz continuous with respect to g and bounded for every ω ∈ Ω. By the same method in [17, Chapters II and XV], we can prove for P-a.s. every ω ∈ Ω 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 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 ω ∈ Ω, the following statements hold for all τ < T.

(i) If , then lies in
()
(ii) is jointly continuous in t and 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 ω ∈ Ω. 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 𝒦(ω) is said to absorb the set B ⊂ X for an RDS φ if P-a.s. there exists t_B(ω) such that

()

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. ω ∈ Ω, 𝒜(ω) is compact, and for all x ∈ X and P-a.s. the map x ↦ dist (x, 𝒜(ω)) is measurable.
(ii) φ(t, ω)𝒜(ω) = A(θ_tω) for all t ≥ 0 (invariance).
(iii) For all bounded B ⊂ X,
()
where dist (·, ·) denotes the Hausdorff semidistance:
()

Proposition 3.1 (see [14], [15].)Let ϕ be an RDS on a Polish space (X, d) with Borel σ-algebra ℬ over the flow {θ_t} _t∈R on a probability space (Ω, ℱ, P). Suppose there exists a random compact set 𝒦(ω) such that for any bounded nonrandom set B ⊂ X P-a.s

()

Then the set

()

is a unique random attractor for ϕ, where the union is taken over all bounded B ⊂ X and Λ_B(ω) is the omega-limit set of B 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 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 t₀ ≤ t(ω), and for all with , the solution of system (2.16) over [t₀, ∞), 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 Gu(t), v(t), and GW(t), respectively. Then sum up the resulting equalities. By the Neumann boundary condition (1.2), we get

()

where |𝒪| denotes the volume of 𝒪. Set

()

Then (3.12) yields

()

Applying Gronwall′s inequality to (3.14) and then integrating in [t₀, −1], with t₀ ≤ −1 we have

()

Consequently, give B(0, ρ) ⊂ H, P-a.s. there exists t(ω, ρ)≤−1 such that for t₀ ≤ t(ω, ρ) and all g₀ ∈ B(0, ρ),

()

with

()

Indeed, it is enough to choose t(ω, ρ) such that

()

and take into account (3.15) and the fact that P-a.s.

as s → −∞.

If we now return to (3.14) and integrate for t ∈ [−1,0], we have

()

Thus, we can conclude that given B(0, ρ) ⊂ H and P-a.s. there exists t(ω, ρ)≤−1 such that for t₀ ≤ t(ω, ρ) and for all g₀ ∈ B(0, ρ),

()

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 t₀ ≤ t(ω) and for all with , the solution g(t, ω; t₀, α(t₀)g₀) of system (2.16) over [t₀, ∞), with , satisfies the inequality

()

Also, for t ∈ [−1,0] one has

()

Proof. Letting V(t, x) = v(t, x)/G, (3.9)–(3.11) can be written as

()

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

()

By using Young′s inequality, we obtain

()

From (3.27), (3.26) yields

()

that is,

()

By denoting

()

then (3.29) implies that

()

Integrating in [t₀, −1], with t₀ ≤ −1 we have

()

Consequently, given B(0, ρ^′) ⊂ H, P-a.s. there exists t(ω, ρ^′)≤−1 such that for all t₀ ≤ t(ω, ρ^′) and for all g₀ ∈ B(0, ρ^′),

()

with

()

In fact, it is enough to choose t(ω, ρ^′) to satisfy

()

and take into (3.32) and the fact that P-a.s.

as s → −∞. Also, from (3.32) and for t ∈ [−1,0] we get

()

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 t₀ ≤ t(ω) and for all with , there exists a unique solution g(t, ω; t₀, α(t₀)g₀) of system (2.16) over [t₀, ∞), with , and put . Then

()

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

()

that is,

()

Here d = min {d₁, d₂, d₃}, C₂ = (1/d)(8/3 + 2G²), C₃ = (1/d)max {2N², k²}. Then from (3.39) we have

()

Integrating (3.40) in [s, 0], s ∈ [−1,0], we obtain

()

Integrating (3.41) in [−1,0] and by (3.20),

()

It is now straightforward from (3.31) and (3.36) that

()

Consequently, P-a.s. there exists r²(ω) such that given ρ > 0, 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 𝒜(ω).

Proof. This follows from Lemma 3.2 and Lemma 3.4 combined with the embedding of E↪H 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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Random Attractors for Stochastic Three-Component Reversible Gray-Scott System with Multiplicative White Noise

Abstract

1. Introduction

2. Existence and Uniqueness of Solutions

3. Existence of a Random Attractor

Acknowledgments

References

Citing Literature

References

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

Abstract

1. Introduction

2. Existence and Uniqueness of Solutions

3. Existence of a Random Attractor

Acknowledgments

References

Citing Literature

References

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