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

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 𝐑^𝑛(𝑛≤3) with a locally Lipschitz continuous boundary 𝜕𝒪. We consider the stochastic three-component reversible Gray-Scott system with multiplicative noise𝜕𝑢̃𝜕𝑡=𝑑¹▵𝑢̃−(𝐹+𝑘)𝑢̃+𝑢²̃𝑣̃−𝐺𝑢³̃+𝑁𝑤+𝜎𝑢̃∘𝑑𝐵^𝑡𝜕𝑣̃𝑑𝑡,𝜕𝑡=𝑑²▵𝑣̃+𝐹(1−𝑣)̃−𝑢²̃𝑣̃+𝐺𝑢³̃+𝜎𝑣̃∘𝑑𝐵^𝑡𝜕𝑤𝑑𝑡,𝜕𝑡=𝑑³▵𝑤+𝑘𝑢̃−(𝐹+𝑁)𝑤+𝜎𝑤∘𝑑𝐵^𝑡𝑑𝑡,(1.1) where 𝑢̃=𝑢(̃𝑥,𝑡), 𝑣̃=𝑣(̃𝑥,𝑡), and 𝑤=𝑤(𝑥,𝑡) 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𝜕𝑢̃𝜕𝝂(𝑥,𝑡)=𝜕𝜕𝝂𝑣̃(𝑥,𝑡)=𝜕𝜕𝑤𝝂(𝑥,𝑡)=0,𝑥∈𝜕𝒪,(1.2) where 𝜕/𝜕𝝂 is the outward normal derivative, and with an initial condition𝑢̃(𝑥,0)=𝑢₀̃(𝑥),𝑣̃(𝑥,0)=𝑣₀̃(𝑥),𝑤(𝑥,0)=𝑤₀(𝑥),𝑥∈𝒪.(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, “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 map𝜑∶𝐑⁺×Ω×𝑋⟶𝑋,(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,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((𝑢,𝑣))=_𝑛(𝑢,𝑣)=_Ω𝑢𝑣𝑑𝑥,|𝑢|=(𝑢,𝑢)^1/2∀𝑢,𝑣∈𝐿²(𝒪),_𝑖=1𝐷_𝑖𝑢,𝐷_𝑖𝑣+𝐹(𝑢,𝑣),‖𝑢‖=((𝑢,𝑢))^1/2∀𝑢,𝑣∈𝐻¹(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 𝐿^𝑝(𝒪) will be denoted by |⋅|_𝐿^𝑝 if 𝑝≠2,𝑋=(𝐿⁶(𝒪))³.

Define the unbounded positive linear operator𝐴(𝐷(𝐴)⟶𝐻)=Δ−𝐹𝐼,(2.3) whereΔ=⎛⎜⎜⎜⎜⎝𝑑¹0▵𝑑₂0000▵𝑑₃0𝐷(𝐴)=(𝜙,𝜑,𝜓)∈𝐻₂▵⎞⎟⎟⎟⎟⎠,(𝒪)³∶𝜕𝜙𝜕𝝂=𝜕𝜕𝜑𝝂=𝜕𝜕𝜓𝝂=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 {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 Δ satisfying0=𝜆₀<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑖≤⋯𝜆_𝑖⟶∞as𝑖⟶∞.(2.5) By the fact that 𝐻¹(𝒪)↪𝐿⁶(𝒪) is a continuous embedding for 𝑛≤3 and by the generalized Hölder inequality, one has||𝑢²𝑣||≤|𝑢|^2𝐿₆|𝑣|_𝐿⁶,for𝑢,𝑣∈𝐿⁶(𝒪).(2.6) Therefore, the nonlinear mapping defined on 𝐸,𝑓𝑢,̃𝑣,̃𝑤=⎛⎜⎜⎜⎜⎝−𝑘𝑢̃+𝑢²̃𝐹−𝑣𝑢₂̃̃−𝐺𝑢³̃+𝑁𝑤𝑣̃+𝐺𝑢³̃𝑘𝑢̃−𝑁𝑤⎞⎟⎟⎟⎟⎠∶𝐸⟶𝐻,(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 noise𝑑𝑔̃𝑑𝑡=𝐴𝑔̃+𝑓(𝑔̃)+𝜎𝑔̃∘𝑑𝑑𝐵𝑡^𝑡,(2.8) and an initial condition𝑔̃(0)=𝑔̃₀=𝑢₀̃,𝑣₀̃,𝑤₀∈𝐻,(2.9) where 𝑔̃(𝑡)=(𝑢(̃𝑡,⋅),𝑣(̃𝑡,⋅),𝑤(𝑡,⋅)). 𝐵_𝑡 is a one-dimensional two-sided Wiener process on a probability space (Ω,ℱ,𝑃), whereΩ={𝜔∈𝐶(𝐑,𝐑^𝑚)∶𝜔(0)=0},(2.10) 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 by𝜃_𝑡𝜔(⋅)=𝜔(⋅+𝑡)−𝜔(𝑡).(2.11) By means of the change of variables𝑤𝑢𝑣(((𝑡𝑡𝑡)))===𝛼𝛼𝛼(((𝑡𝑡𝑡)))𝑤𝑢𝑣̃(̃((𝑡𝑡𝑡))),,with𝛼(𝑡)=𝑒_{−𝜎𝐵𝑡},(2.12) system (1.1) can be written as 𝜕𝑢𝜕𝑡=𝑑¹𝜕𝑣▵𝑢−(𝐹+𝑘)𝑢+𝛼⁻²(𝑡)𝑢²𝑣−𝐺𝛼⁻²(𝑡)𝑢³+𝑁𝑤,(2.13)𝜕𝑡=𝑑²▵𝑣𝜕+𝑤𝐹(𝛼(𝑡)−𝑣)−𝛼⁻²(𝑡)𝑢²𝑣+𝐺𝛼⁻²(𝑡)𝑢³,(2.14)𝜕𝑡=𝑑³▵𝑤+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)=𝑔̃(0)=𝑔₀=𝑢₀,𝑣₀,𝑤₀∈𝐻,(2.17) where𝑓(𝑔,𝜔)=⎛⎜⎜⎜⎜⎝−𝑘𝑢+𝛼⁻²𝐹−𝛼(₋₂𝑡)𝑢²𝑣−𝐺𝛼⁻²(𝑡)𝑢³+𝑁𝑤(𝑡)𝑢²𝑣+𝐺𝛼⁻²(𝑡)𝑢³𝑘𝑢−𝑁𝑤⎞⎟⎟⎟⎟⎠.(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 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 𝑔̃(0,𝜔)∈𝐻, then 𝑔̃(𝑡,𝜔) lies in 𝐶([𝜏,𝑇);𝐻)∩𝐶¹((𝜏,𝑇);𝐻)∩𝐿²([𝜏,𝑇);𝐸).(2.19)(ii)𝑔̃(𝑡,𝑔̃(0,𝜔)) is jointly continuous in 𝑡 and 𝑔̃(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 𝜔∈Ω. Hence the solution mapping𝑆(𝑡,𝜔)∶𝑔̃(𝜏,𝜔)⟼𝑔̃(𝑡,𝜔)(2.20) generates an RDS. So the transformation𝑆(𝑡,𝜔)∶𝛼⁻¹(𝑡)𝑔̃(𝜏,𝜔)⟼𝛼⁻¹(𝑡)𝑔̃(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 𝐵⊂𝑋 for an RDS 𝜑 if 𝑃-a.s. there exists 𝑡_𝐵(𝜔) such that𝜑𝑡,𝜃_−𝑡𝜔𝐵⊂𝒦(𝜔)∀𝑡≥𝑡_𝐵(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. 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, lim_𝑡→∞dist𝜑𝑡,𝜃_−𝑡𝜔𝐵,𝒜(𝜔)=0,(3.2) where dist(⋅,⋅) denotes the Hausdorff semidistance: dist(𝑌,𝑍)=supinf_{𝑥∈𝑌𝑦∈𝑍}𝑑(𝑥,𝑦),𝑌,𝑍∈𝑋.(3.3)

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 dist𝜑𝑡,𝜃_−𝑡𝜔𝐵,𝒦(𝜔)⟶0as𝑡⟶+∞.(3.4) Then the set 𝒜(𝜔)=_𝐵_⊂𝑋Λ_𝐵(𝜔)(3.5) 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 Λ_𝐵(𝜔)=_𝑠≥0_𝑡≥𝑠𝜙𝑡,𝜃_−𝑡𝜔𝐵.(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 𝑡₀≤𝑡(𝜔), and for all 𝑔̃₀∈𝐻 with |𝑔̃₀|≤𝜌, the solution 𝑔(𝑡,𝜔;𝑡₀,𝛼(𝑡₀)𝑔̃₀) of system (2.16) over [𝑡₀,∞), with 𝑔(𝑡₀)=𝛼(𝑡₀,𝜔)𝑔̃₀, satisfies the inequality ||𝑔−1,𝜔;𝑡₀,𝛼𝑡₀,𝜔𝑔̃₀||²≤𝑟²¹(𝜔).(3.7)

Proof. Define 𝑊(𝑡,𝑥)=𝑁𝑘𝑤(𝑡,𝑥),𝜇=𝑁𝑘.(3.8) Then (2.13)–(2.15) can be written as 𝜕𝑢𝜕𝑡=𝑑¹𝜕𝑣▵𝑢−(𝐹+𝑘)𝑢+𝛼⁻²(𝑡)𝑢²𝑣−𝐺𝛼⁻²(𝑡)𝑢³+𝑘𝑊,(3.9)𝜕𝑡=𝑑²𝜇▵𝜕𝑣𝑊+𝐹(𝛼(𝑡)−𝑣)−𝛼⁻²(𝑡)𝑢²𝑣+𝐺𝛼⁻²(𝑡)𝑢³,(3.10)𝜕𝑡=𝜇𝑑³▵𝑊+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 12𝑑𝑑𝑡𝐺|𝑢|²+|𝑣|²+𝜇𝐺||𝑊||²+𝐺(𝐹+𝑘)|𝑢|²+𝐹|𝑣|²+𝐺(𝜇𝐹+𝑘)||𝑊||²+𝑑₁𝐺‖𝑢‖²+𝑑₂‖𝑣‖²+𝜇𝐺𝑑₃‖𝑊‖²≤≤𝑘𝐺_Ω|𝐹𝑢|𝛼₂(𝑡)𝑣𝑑𝑥+2𝑘𝐺_Ω𝑢𝑊𝑑𝑥−𝛼⁻²(𝑡)_Ω𝐺𝑢²−𝑢𝑣²𝑑𝑥+𝑘𝐺||𝑊||²+𝐹2|𝑣|²+𝐹𝛼²(𝑡)2||𝒪||,(3.12) where |𝒪| denotes the volume of 𝒪. Set 𝑑=min𝑑₁,𝑑₂,𝑑₃,𝐶₁=𝐹||𝒪||min{1,𝐺,𝐺/𝜇}.(3.13) Then (3.12) yields 𝑑𝑑𝑡||𝑔(𝑡)||²+𝑑‖𝑔(𝑡)‖²+𝐹||𝑔(𝑡)||²≤𝐶₁𝛼²(𝑡).(3.14) Applying Gronwall's inequality to (3.14) and then integrating in [𝑡₀,−1], with 𝑡₀≤−1 we have ||𝑔(−1)||²≤||𝛼𝑡₀𝑔₀||²𝑒^{−𝐹(−1−𝑡0)}+𝐶₁^𝑡−1₀≤𝑒_𝐹𝑒^{−𝐹(−1−𝑠)}𝛼²(𝑠)𝑑𝑠𝑒^𝐹𝑡0||𝛼𝑡₀𝑔₀||²+𝐶₁^𝑡−1₀𝑒^𝐹𝑠𝛼²(𝑠)𝑑𝑠.(3.15) 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), ||𝑔−1,𝜔;𝑡₀,𝛼𝑡₀𝑔̃₀||²≤𝑟²¹(𝜔),(3.16) with 𝑟²¹(𝜔)=𝑒^𝐹1+𝐶¹^−1−∞𝑒^𝐹𝑠𝛼²(𝑠)𝑑𝑠.(3.17) Indeed, it is enough to choose 𝑡(𝜔,𝜌) such that 𝑒^𝐹𝑡0𝛼²𝑡₀𝜌²≤1,(3.18) and take into account (3.15) and the fact that 𝑃-a.s. 𝑒^𝐹𝑠𝛼²(𝑠)=𝑒^𝐹𝑠𝑒^{−2𝜎𝑊𝑠}→0 as 𝑠→−∞.

If we now return to (3.14) and integrate for 𝑡∈[−1,0], we have||𝑔(𝑡)||²≤||𝑔(−1)||²𝑒^{−𝐹(𝑡+1)}+𝐶₁^𝑡−1𝑑𝑒^{−𝐹(𝑡−𝑠)}𝛼²(𝑠)𝑑𝑠,₀₋₁‖𝑔(𝑠)‖²𝑑𝑠≤||𝑔(−1)||²+𝐶₁⁰⁻¹𝛼²(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 𝑡₀≤𝑡(𝜔,𝜌) and for all 𝑔₀∈𝐵(0,𝜌),||𝑔𝑡,𝜔;𝑡₀,𝑔₀||²≤𝑒^{−𝐹(𝑡+1)}𝑟²¹(𝜔)+𝐶₁^𝑡−1𝑑𝑒^{−𝐹(𝑡−𝑠)}𝛼²(𝑠)𝑑𝑠,₀₋₁‖‖𝑔(𝑠,𝜔;𝑡₀,𝑔₀)‖‖²𝑑𝑠≤𝑟²¹(𝜔)+𝐶₁⁰⁻¹𝛼²(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 𝑡₀≤𝑡(𝜔) and for all 𝑔̃₀∈𝐻 with |𝑔̃₀|≤𝜌^, the solution 𝑔(𝑡,𝜔;𝑡₀,𝛼(𝑡₀)𝑔₀) of system (2.16) over [𝑡₀,∞), with 𝑔(𝑡₀)=𝛼(𝑡₀,𝜔)𝑔̃₀, satisfies the inequality ||𝑔−1,𝜔;𝑡₀,𝛼𝑡₀,𝜔𝑔̃₀||^6𝑋≤𝑟²³(𝜔).(3.21) Also, for 𝑡∈[−1,0] one has ||𝑔(𝑡)||^6𝑋≤𝑒^{−𝐹(𝑡+1)}𝑟²³(𝜔)+𝐶₄^𝑡−1𝑒^{−𝐹(𝑡−𝜏)}𝛼⁶(𝜏)𝑑𝜏.(3.22)

Proof. Letting 𝑉(𝑡,𝑥)=𝑣(𝑡,𝑥)/𝐺, (3.9)–(3.11) can be written as 𝜕𝑢𝜕𝑡=𝑑¹𝜕𝑉▵𝑢−(𝐹+𝑘)𝑢+𝐺𝛼⁻²(𝑡)𝑢²𝑉−𝐺𝛼⁻²(𝑡)𝑢³+𝑘𝑊,(3.23)𝜕𝑡=𝑑²▵𝑣+𝐹𝐺𝛼(𝑡)−𝐹𝑉−𝛼⁻²𝜇𝜕𝑊(𝑡)𝑢²𝑉+𝛼⁻²(𝑡)𝑢³,(3.24)𝜕𝑡=𝜇𝑑³▵𝑊+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=−(𝐹++𝐺𝑘||𝑉)||^6𝐿_6Ω+𝜇||𝑊||^6𝐿₆+5𝑑¹‖‖𝑢²∇𝑢‖‖²+𝑑₂𝐺‖‖𝑉²∇𝑉‖‖²+𝜇𝑑₃𝐺‖‖𝑊²∇𝑊‖‖²−(𝜇𝐹+𝑘)_Ω𝑢⁶𝑑𝑠+𝐹_Ω𝛼(𝑡)𝑉⁵𝑑𝑥−𝐺𝐹_Ω𝑉⁶𝑑𝑥−𝐺𝛼₋₂𝑊⁶𝑑𝑥+𝑘_Ω𝑢⁵𝑊𝑑𝑥+𝑘_Ω𝑢𝑊⁵𝑑𝑥(𝑡)_Ω𝑢⁸−𝑢⁷𝑉−𝑢³𝑉⁵+𝑢²𝑉⁶𝑑𝑥.(3.26) By using Young's inequality, we obtain −𝑘𝐺𝛼⁻²(𝑡)_Ω𝑢⁸−𝑢⁷𝑉−𝑢³𝑉⁵+𝑢²𝑉⁶𝑑𝑥≤0,_Ω𝑢⁵𝑊𝑑𝑥+𝑘_Ω𝑢𝑊⁵𝑑𝑥≤𝑘|𝑢|^6𝐿₆+𝑘||𝑊||^6𝐿₆.(3.27) From (3.27), (3.26) yields 𝑑𝑑𝑡|𝑢|^6𝐿6+𝐺||𝑉||^6𝐿₆+𝜇||𝑊||^6𝐿₆+𝐹|𝑢|^6𝐿6+𝐺||𝑉||^6𝐿₆+𝜇||𝑊||^6𝐿₆≤𝐹||𝒪||𝐺⁵𝛼⁶(𝑡),(3.28) that is, 𝑑𝑑𝑡|𝑢|^6𝐿6+𝐺⁻⁵|𝑣|^6𝐿₆+𝜇⁻⁵|𝑤|^6𝐿₆+𝐹|𝑢|^6𝐿6+𝐺⁻⁵|𝑣|^6𝐿₆+𝜇⁻⁵|𝑤|^6𝐿₆≤𝐹||𝒪||𝐺⁵𝛼⁶(𝑡).(3.29) By denoting 𝐶₄=𝐹||𝒪||𝐺⁵min1,𝐺⁻⁵,𝜇⁻⁵,(3.30) then (3.29) implies that 𝑑𝑑𝑡||𝑔(𝑡)||^6𝑋+𝐹||𝑔(𝑡)||^6𝑋≤𝐶₄𝛼⁶(𝑡).(3.31) Integrating in [𝑡₀,−1], with 𝑡₀≤−1 we have ||𝑔(−1)||^6𝑋≤||𝛼𝑡₀𝑔₀||^6𝑋𝑒^{−𝐹(−1−𝑡0)}+𝐶₄^𝑡−1₀𝑒^{−𝐹(−1−𝑠)}𝛼⁶(𝑠)𝑑𝑠.(3.32) 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^), ||𝑔−1,𝜔,𝑡₀,𝛼𝑡₀𝑔₀||^6𝑋≤𝑟²³(𝜔),(3.33) with 𝑟²³(𝜔)=𝑒^𝐹1+𝐶⁴^−1−∞𝑒^𝐹𝑠𝛼⁶(𝑠)𝑑𝑠.(3.34) In fact, it is enough to choose 𝑡(𝜔,𝜌^) to satisfy 𝑒^𝐹𝑡0𝛼⁶𝑡₀𝜌^6≤1,(3.35) and take into (3.32) and the fact that 𝑃-a.s. 𝑒^𝐹𝑠𝛼⁶(𝑠)=𝑒^𝐹𝑠𝑒^{−6𝜎𝑊𝑠}→0 as 𝑠→−∞. Also, from (3.32) and for 𝑡∈[−1,0] we get ||𝑔(𝑡)||^6𝑋≤𝑒^{−𝐹(𝑡+1)}𝑟²³(𝜔)+𝐶₄^𝑡−1𝑒^{−𝐹(𝑡−𝜏)}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 𝑡₀≤𝑡(𝜔) and for all 𝑔̃₀∈𝐻 with |𝑔̃₀|≤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)𝑔̃₀, and put 𝑔̃(𝑡,𝜔;𝑡₀,𝑔̃₀)∶=𝛼⁻¹(𝑡,𝜔)𝑔(𝑡,𝜔;𝑡₀,𝑔̃₀). Then ‖‖𝑔̃0,𝜔;𝑡₀,𝑔̃₀‖‖²≤𝑟²²(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 12𝑑𝑑𝑡‖𝑢‖²+‖𝑣‖²+‖𝑤‖²+𝑑₁|▵𝑢|²+𝑑₂|▵𝑣|²+𝑑₃|▵𝑤|²+(𝐹+𝑘)‖𝑢‖²+𝐹‖𝑣‖²+(𝐹+𝑁)‖𝑤‖²=−−𝛼𝐹𝛼⁻²((𝑡𝑡))_ΩΩ𝑢²𝑣▵𝑢𝑑𝑥+𝐺𝛼⁻²(𝑡)_Ω𝑢³▵𝑢𝑑𝑥−𝑁_Ω𝑤▵𝑢𝑑𝑥−𝑘_Ω▵𝑣𝑑𝑥+𝛼⁻²(𝑡)_Ω𝑢²𝑣▵𝑣𝑑𝑥−𝐺𝛼⁻²(𝑡)_Ω𝑢³▵𝑣𝑑𝑥≤𝑑𝑢▵𝑤𝑑𝑥₁2|▵𝑢|²+𝑑²2|▵𝑣|²+𝑑³2|▵𝑤|²+𝛼⁻⁴(𝑡)1𝑑₁+1𝑑₂_Ω+𝑁𝑢⁴𝑣²𝑑𝑥₂𝑑₁_Ω𝑤²𝑑𝑥+𝑘²2𝑑₃_Ω𝑢²𝑑𝑥+𝐺²𝛼⁻⁴(𝑡)𝑑₂_Ω𝑢⁶𝑑𝑥,(3.38) that is, 𝑑𝑑𝑡‖𝑢‖²≤𝛼₋₄+‖𝑣‖²+‖𝑤‖²+𝐹‖𝑢‖²+‖𝑣‖²+‖𝑤‖²(𝑡)43𝑑₁+43𝑑₂+2𝐺²𝑑₂_Ω𝑢⁶+𝑣⁶𝑑𝑥+2𝑁²𝑑₁|𝑤|²+𝑘²𝑑₃|𝑢|²≤𝐶₂𝛼⁻⁴(𝑡)||𝑔||^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 𝑑𝑑𝑡‖𝑔‖²+𝐹‖𝑔‖²≤𝐶₂𝛼⁻⁴(𝑡)||𝑔||^6𝑋+𝐶₃||𝑔||².(3.40) Integrating (3.40) in [𝑠,0], 𝑠∈[−1,0], we obtain ‖𝑔(0)‖²≤‖𝑔(𝑠)‖²+𝐶₂^0𝑠𝛼⁻⁴(𝜏)||𝑔(𝜏)||^6𝑋𝑑𝜏+𝐶₃^0𝑠||𝑔(𝜏)||²𝑑𝜏.(3.41) Integrating (3.41) in [−1,0] and by (3.20), ‖𝑔(0)‖²≤⁰⁻¹‖𝑔(𝜏)‖²𝑑𝜏+𝐶₂⁰⁻¹𝛼⁻⁴(𝜏)||𝑔(𝜏)||^6𝑋𝑑𝜏+𝐶₃⁰⁻¹≤1||𝑔(𝜏)||²𝑑𝜏𝑑+𝐶𝐹³𝑟²¹(𝜔)+𝐶₁⁰⁻¹𝛼²(𝜏)𝑑𝜏+𝐶₂sup_{−1≤𝑡≤0}𝛼⁻⁴(𝑡)⁰⁻¹||𝑔(𝜏)||^6𝑋.(3.42) It is now straightforward from (3.31) and (3.36) that ‖𝑔(0)‖²=‖𝑔̃(0)‖²≤1𝑑+𝐶𝐹³𝑟²¹(𝜔)+𝐶₁⁰⁻¹+𝐶𝛼²(𝜏)𝑑𝜏₂𝐹sup_{−1≤𝑡≤0}𝛼⁻⁴(𝑡)𝑟²³(𝜔)+𝐶₄⁰⁻¹𝛼⁶(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 𝑡₀≤𝑡(̃𝜔) and 𝑔̃₀∈𝐻 with |𝑔̃₀|≤𝜌, ‖‖𝑔̃0,𝜔;𝑡₀,𝑔̃₀‖‖²≤𝑟²²(𝜔),(3.44) where 𝑟²²(𝜔)=𝐶₅+𝐶₆𝑒^𝐹+𝐶₁𝐶₅^−1−∞𝑒^𝐹(𝑠+1)𝛼²(𝑠)𝑑𝑠+𝐶₁𝐶₅⁰⁻¹+𝐶₄𝛼²(𝑠)𝑑𝑠𝐶₆^−1−∞𝑒^𝐹(𝑠+1)𝛼⁶(𝑠)𝑑𝑠+𝐶₄𝐶₆⁰⁻¹𝐶𝛼⁶(𝑠)𝑑𝑠,₅=1𝑑+𝐶𝐹³,𝐶⁶=𝐶²𝐹sup_{−1≤𝑡≤0}𝛼⁻⁴(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 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 𝜕𝑈𝜕𝑡=𝑑¹𝜕𝑉▵𝑈−(𝐹+𝑘)𝑈+2𝑢𝑣𝑒^{2𝜎𝑊𝑡}𝑈+𝑢²𝑉𝑒^{2𝜎𝑊𝑡}−3𝐺𝑒^{2𝜎𝑊𝑡}𝑢²𝑈+𝑁𝑊,𝜕𝑡=𝑑¹▵𝑉𝜕−𝑊𝐹𝑉−2𝑢𝑣𝑒^{2𝜎𝑊𝑡}𝑈−𝑢²𝑉𝑒^{2𝜎𝑊𝑡}+3𝐺𝑢²𝑒^{2𝜎𝑊𝑡}𝑈,𝜕𝑡=𝑑¹▵𝑊+𝑘𝑈−(𝐹+𝑁)𝑊.(3.46) Letting Φ(𝑡)=(𝑈,𝑉,𝑊),𝑞(𝑡)=𝑔̃(𝑡)−𝑔(𝑡)−Φ(𝑡), then 𝑞(𝑡)=𝑞₁(𝑡),𝑞₂(𝑡),𝑞₃(𝑡)=𝑢̃−𝑢−𝑈,𝑣̃−𝑣−𝑉,𝑤−𝑤−𝑊(3.47) satisfies 𝜕𝑞₁𝜕𝑡=𝑑¹+3▵𝐺𝑞𝑢₁₂̃−(𝐹+𝑘)𝑞₁+𝛼(𝑡)𝑢²̃𝑣̃−𝑢²𝑣−2𝑢𝑣̃𝑒̃^{2𝜎𝑊𝑡}(𝑡)𝑈−𝑢²̃𝑒^{2𝜎𝑊𝑡}𝑉𝜕𝑞₂𝑒^{2𝜎𝑊𝑡}𝑈−𝐺𝛼(𝑡)𝑢³̃−𝑢³+𝑁𝑞³,𝜕𝑡=𝑑¹−3▵𝐺𝑞𝑢₂₂̃−𝐹𝑞₂−𝛼(𝑡)𝑢²̃𝑣̃−𝑢²𝑣+2𝑢𝑣̃𝑒̃^{2𝜎𝑊𝑡}𝑈+𝑢²̃𝑒^{2𝜎𝑊𝑡}𝑉𝜕𝑞₃𝑒^{2𝜎𝑊𝑡}𝑈+𝐺𝛼(𝑡)𝑢³̃−𝑢³,𝜕𝑡=𝑑¹▵𝑞₃+𝑘𝑞₁−(𝐹+𝑁)𝑞₃.(3.48) Here, let 𝑔̃(𝑡)=𝑔̃𝑡,𝜔;0,𝑔̃₀=𝑢̃𝑡,𝜔;0,𝑢₀̃,𝑣̃𝑡,𝜔;0,𝑣₀̃,𝑤𝑡,𝜔;0,𝑤₀,𝑔(𝑡)=𝑔𝑡,𝜔;0,𝑔₀=𝑢𝑡,𝜔;0,𝑢₀,𝑣𝑡,𝜔;0,𝑣₀,𝑤𝑡,𝜔;0,𝑤₀(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.