Random Attractors of Stochastic Three-Component Reversible Gray-Scott System on Unbounded Domains

and Applied Analysis 3 2. Preliminaries In this section, we recall some basic concepts related to random attractors for random dynamical systems. We refer the reader to 1, 23, 24 for more details. Let X, ‖ · ‖X be a separable Hilbert space with Borel σ-algebra B X , and let Ω,F,P be a probability space. Definition 2.1. Ω,F,P, θt t∈R is called a metric dynamical system if θ : R×Ω → Ω is B R × F,F -measurable and θ0 is the identity on Ω, θs t θtθs for all s, t ∈ R and θtP P for all t ∈ R. Definition 2.2. A stochastic process {φ t, ω }t≥0, ω∈Ω is a continuous random dynamical system RDS over Ω,F,P, θt t∈R if φ is B 0,∞ × F × B X ,B X -measurable, and for all ω ∈ Ω, i the mapping φ t, ω : X → X, x → φ t, ω x is continuous for every t ≥ 0; ii φ 0, ω is the identity on X; iii cocycle property φ s t, ω φ t, θsω φ s,ω for all s, t ≥ 0. Definition 2.3. i A set-valued mapping ω → B ω ⊂ X we may write it as B ω for short is said to be a random set if the mapping ω → distX x, B ω is measurable for any x ∈ X. ii A random set B ω is said to be bounded if there exist x0 ∈ X and a random variable r ω > 0 such that B ω ⊂ {x ∈ X : ‖x − x0‖X ≤ r ω , x0 ∈ X} for all ω ∈ Ω. iii A random set B ω is called a compact random set if B ω is compact for allω ∈ Ω. iv A random bounded set B ω ⊂ X is called tempered with respect to θt t∈R if for a.e. ω ∈ Ω limt→ ∞esupx∈B θ−tω ‖x‖X 0 for all γ > 0. A random variable ω → r ω ∈ R is said to be tempered with respect to θt t∈R if for a.e. ω ∈ Ω, limt→ ∞esupt∈R|r θ−tω | 0 for all γ > 0. We consider a continuous RDS {φ t, ω }t≥0,ω∈Ω over Ω,F,P, θt t∈R and D the set of all tempered random sets of X. Definition 2.4. A random set K ω is called an absorbing set in D if for all B ∈ D and a.e. ω ∈ Ω there exist tB ω > 0 such that φ t, θ−tω B θ−tω ⊂ K ω ∀ t ≥ tB ω . 2.1 Definition 2.5. A random setA is called a globalD-random attractor orD-pullback attractor for {φ t, ω }t≥0, ω∈Ω if the following hold: i A is a random compact set, that is ω → d x,A ω is measurable for every x ∈ X andA ω is compact for a.e. ω ∈ Ω; ii A is strictly invariant, that is, forω ∈ Ω and all t ≥ 0 one has φ t, ω A ω A θtω ; iii A attracts all sets in D, that is, for all B ∈ D and a.e. ω ∈ Ωwe have lim t→ ∞ d ( φ t, θ−tω B θ−tω , A ω ) 0, 2.2 where d Y,Z supy∈Y infz∈Z‖y − z‖X is the Hausdorff semimetric Y ⊆ X,Z ⊆ X . 4 Abstract and Applied Analysis Proposition 2.6 see 24 . Let K ω ∈ D be a random absorbing set for the continuous RDS φ t t≥0, which is closed and satisfies for a.e. ω ∈ Ω the following asymptotic compactness condition: each sequence xn ∈ φ tn, θ−tnω,K θ−tnω with tn → ∞ has a convergent subsequence in X. Then, the cocycle φ has a unique global random attractor. A ω ⋂


Introduction
Consider the asymptotic behavior of solutions of the following stochastic three-component reversible Gray-Scott system with multiplicative noise defined in the entire space R n ×R n ×R n : where all the parameters are given positive constants; f i i 1, 2, 3 are nonlinear functions satisfying certain conditions; B t is a two-sided real-valued Wiener process on a probability space Ω, F, P , Ω {ω ∈ C R, R : ω 0 0}, the Borel σ-algebra F on Ω is generated by the compact open topology see 1 , and P is the corresponding Wiener measure on F; • denotes the Stratonovich sense of the stochastic term.
Historically, when w 0, G 0, f 1 f 3 0, f 2 F and there are no random terms σ 0 , system 1.1 reduces to the two-component Gray-Scott system which signified one of the Brussels schools led by the renowned physical chemist and Nobel Prize laureate 1977 , Ilya Prigogine, which originated from describing an isothermal, cubic autocatalytic, continuously fed, and diffusive reactions of two chemicals see 2-6 . The three-component reversible Gray-Scott model was firstly introduced by Mahara et al., which is based on the scheme of two reversible chemical or biochemical reactions 7 . Then in 8 , You took some nondimensional transformations, the three-component reversible system was reduced to the system 1.1 without random forces. In 8 , You considered the existence of global attractor for the system 1.1 with Neumann boundary condition on a bounded domain of space dimension n ≤ 3 by the method of the rescaling and grouping estimation.
Stochastic differential equations of this type arise from many chemical or biochemical systems when random spatiotemporal forcing is taken into consideration. These random perturbations are intrinsic effects in a variety of settings and spatial scales. They may be most obviously influential at the microscopic and smaller scales, but indirectly they play an important role in macroscopic phenomena. Recently, Gu 9 gave the existence of a compact random attractor for stochastic three-component reversible Gray-Scott system with multiplicative white noise in a bounded domain of R n n ≤ 3 when f 1 f 3 0, f 2 F in system 1.1 . As pointed in 10 , the discussion of the same or similar coupled reactiondiffusion systems on a higher dimensional domain with the space dimension n > 3 and on an unbounded domain is still open to the best of our knowledge. Here, we intend to investigate the dynamical behavior of the system 1.1 on unbounded domains and give a partly answer to the problems proposed in 10 . It is worth mentioning that Sobolev embedding is not compact on domains of infinite volume. This introduces a major obstacle for proving the existence of random attractors for partial differential equations on unbounded domains. For some deterministic equations, the difficulty caused by the unboundedness of domains can be overcome by the energy equation approach which developed by Ball in 11, 12 and used by many authors see, e.g., 13-15 . In this paper, we will use the uniform estimates on the far-field values of solutions to circumvent the difficulty caused by the unboundedness of the domain. This idea was developed in 16 to prove the asymptotic compactness of solutions for autonomous parabolic equations on R n , and later extended to nonautonomous equations see, e.g., 17-19 and stochastic equations see, e.g., 20-22 . Here, we will use the method of tail-estimates to investigate the asymptotic behavior of system 1.1 with initial data 1.2 .
The paper is organized as follows. In the next section, we recall the fundamental concepts and results for pullback random attractors for random dynamical systems. In Section 3, we define a cocycle for the stochastic three-component reversible Gray-Scott system on R n × R n × R n . Section 4 is devoted to deriving the uniform estimates of solutions for large space and time variables. In the last section, we give the asymptotic compactness of the solution by using uniform estimates on the tails of solutions and then prove the existence of a pullback random attractor.
The following notations will be used throughout the paper. We denote by · and , ·, the norm and inner product in L 2 R n or H L 2 R n 3 . Let U L 6 R n 3 ; E H 1 R n 3 ; · L 6 and · U denote the norm in L 6 R n and U.

Preliminaries
In this section, we recall some basic concepts related to random attractors for random dynamical systems. We refer the reader to 1, 23, 24 for more details.
Let X, · X be a separable Hilbert space with Borel σ-algebra B X , and let Ω, F, P be a probability space.
Definition 2.1. Ω, F, P, θ t t∈R is called a metric dynamical system if θ : R × Ω → Ω is B R × F, F -measurable and θ 0 is the identity on Ω, θ s t θ t θ s for all s, t ∈ R and θ t P P for all t ∈ R.
Definition 2.2. A stochastic process {ϕ t, ω } t≥0, ω∈Ω is a continuous random dynamical system RDS over Ω, F, P, θ t t∈R if ϕ is B 0, ∞ × F × B X , B X -measurable, and for all ω ∈ Ω, i the mapping ϕ t, ω : X → X, x → ϕ t, ω x is continuous for every t ≥ 0; ii ϕ 0, ω is the identity on X; iii cocycle property ϕ s t, ω ϕ t, θ s ω ϕ s, ω for all s, t ≥ 0. Definition 2.3. i A set-valued mapping ω → B ω ⊂ X we may write it as B ω for short is said to be a random set if the mapping ω → dist X x, B ω is measurable for any x ∈ X.
ii A random set B ω is said to be bounded if there exist x 0 ∈ X and a random variable We consider a continuous RDS {ϕ t, ω } t≥0,ω∈Ω over Ω, F, P, θ t t∈R and D the set of all tempered random sets of X.
ii A is strictly invariant, that is, for ω ∈ Ω and all t ≥ 0 one has ϕ t, ω A ω A θ t ω ; iii A attracts all sets in D, that is, for all B ∈ D and a.e. ω ∈ Ω we have

Proposition 2.6 see 24 .
Let K ω ∈ D be a random absorbing set for the continuous RDS ϕ t t≥0 , which is closed and satisfies for a.e. ω ∈ Ω the following asymptotic compactness condition: each sequence x n ∈ ϕ t n , θ −t n ω, K θ −t n ω with t n → ∞ has a convergent subsequence in X. Then, the cocycle ϕ has a unique global random attractor.
A ω

RDS Generated by Stochastic Three-Component Reversible Gray-Scott System
In this section, we will give the basic setting of system 1.1 and show that it generates a random dynamical system. Let Ω, F, P be a probability space as in Section 1. Define θ t t∈R on Ω via θ t ω · ω · t − ω t , t ∈ R, then Ω, F, P, θ t t∈R is an ergodic metric dynamical system see 1, 23 .
Denote g u, v, w T , system 1.1 with initial data 1.2 can be rewritten as and f x f 1 x , f 2 x , f 3 x T , here T denotes the transposition. For our purpose, it is convenient to transform the problem 3.1 into a deterministic system with a random parameter and then show that it generates a random dynamical system.
Before performing this transformation, we need to recall some properties of the Ornstein-Uhlenbeck processes. Let We know that z θ t ω is an Ornstein-Uhlenbeck process on Ω, F, P, θ t t∈R and solves the following one-dimensional stochastic differential equation see 25 for details : Abstract and Applied Analysis 5 where B t ω B t, ω ω t for ω ∈ Ω, t ∈ R. In fact, from 1, 26 , we know that the random variable z ω is tempered, and there is a θ t -invariant set Ω ⊂ Ω of full P measure such that for ω ∈ Ω, t → z θ t ω is continuous in t; furthermore, 3.6 then system 3.1 can be written as We will consider 3.8 for ω ∈ Ω and still use Ω instead of Ω from now on. As in the case of a bounded domain with Dirichlet boundary conditions which are studied in 27 , for Λ : E ∩ U → H is locally Lipschitz continuous and f ∈ H ∩ U, by a Galerkin method, one can show that for P-a.e. ω ∈ Ω and for all g 0 ∈ H, 3.8 has a unique solution g ·, ω, g 0 ∈ C 0, ∞ , H ∩ L 2 0, T ; E with g 0, ω, g 0 g 0 for every T > 0. Similarly to 28 , because of the continuous nonlinearity Λ, one may take the domain to be a sequence of balls with radius approaching ∞ to deduce the existence of a weak solution to 3.8 on R n × R n × R n . Furthermore, one may get that g t, ω, g 0 is unique and continuous with respect 6 Abstract and Applied Analysis to g 0 in H for all t ≥ 0. Then, 3.8 generates a continuous random dynamical system ϕ t t≥0 over Ω, F, P, θ t t∈R according to the conditions i -iii in Definition 2.2, where ϕ t, ω, g 0 g t, ω, g 0 , ∀ t, ω, g 0 ∈ R × Ω × H.

3.10
We now define a mapping φ : R × Ω × H → H by φ t, ω, g 0 g t, ω, g 0 g t, ω, e −σz ω g 0 e σz θ t ω , 3.11 Then φ is a continuous random dynamical system associated with problem 3.1 on R n × R n × R n .
We remark that the two random dynamical systems are conjugated to each other; thus, the inverse transformation of ϕ is a solution of the original system. For more details on the conjugate theory of stochastic and random differential equations, we can refer to 29 . Thus, in the following sections, we only need to consider the existence of a random attractor of ϕ.

Uniform Estimates of Solutions
In this section, we establish the uniform estimates on the solution of the stochastic Gray-Scott system on R n × R n × R n when t → ∞ in order to derive the existence of a bounded random absorbing set and the asymptotic compactness of the random dynamical system associated with the problem. Particularly when time is sufficiently large, we will show that the tails of the solutions for large space variables are uniformly small.
We always assume that D is the collection of all tempered subsets of H with respect to Ω, F, P, θ t t∈R . The next lemma implied that ϕ has a random absorbing set in D.
Lemma 4.1. Assume that f ∈ H. Then, there exists a random ball {A ω } ∈ D centered at 0 with radius R ω > 0 such that {A ω } is a random absorbing set for ϕ in D, that is, for any {B ω } ∈ D and P-a.e. ω ∈ Ω, there is T B ω > 0 such that 4.2 then the system 3.8 becomes

4.8
By substituting ω with θ −t ω in 4.8 , it yields that Notice that B ω ∈ D is tempered, then for any then {A ω } ∈ D is a random absorbing set for ϕ in D, which completes the proof.

Lemma 4.2.
Assume that f ∈ H. Then there exists a tempered random variable R ω > 0 such that for any B ω ∈ D and g 0 ω ∈ B ω , there exists a T B ω > 0 such that the solution ϕ of 3.8 satisfies for P-a.e. ω ∈ Ω, for all t ≥ T B ω , Proof. By replacing t and ω with T and θ −t ω in 4.8 , for T > 0, we get Multiplying both sides of 4.14 with e 2σ t T z θ s−t ω ds−F t− T , we have for t ≥ T

4.18
By substituting t and T for t 1 and t in 4.18 , we find that e −2σz θ s ω 2σ 0 s z θ τ ω dτ Fs ds.

4.19
We know for s ∈ t, t 1 ,

4.20
Due to 3.5 and the temperedness of g 0 ω , there exists a T B ω > 0 such that for t ≥ T B ω , and by 4.19 and 4.20 , we get

4.21
By 3.5 again, we have R ω is tempered and the proof is completed.
We need the proposition to prove the next result.

Proposition 4.3.
Assume that f ∈ H ∩ U. Then, there exists a tempered random variable R ω > 0 such that for any B ω ∈ D and g 0 ω ∈ B ω , there exists a T B ω > 0 such that the solution ϕ of 3.8 satisfies for P-a.e. ω ∈ Ω for all t ≥ T B ω Proof. Let V t, x v t, x /G, then 4.3 -4.5 can be written as

4.23
Taking the inner products of 4.23 with u 5 t , GV 5 t , and W 5 t , and summing up the resulting equalities, we get

4.29
The last but one line in 4.29 due to B ω ∈ D is tempered, then for any g 0 The proof is completed.
Proof. Taking the inner products of 4.3 -4.5 with −Δu, −Δv, −ΔW, respectively, and summing up the three resulting equalities, we have

4.38
Note that f ∈ H ∩ U. It is easy to see that R ω is tempered. This completes the proof.

4.39
Proof. Choose a smooth cutoff function satisfying 0 ≤ ρ s ≤ 1 for s ∈ R and ρ s 0 for 0 ≤ s ≤ 1, ρ s 1 for s ≥ 2. Suppose there exists a constant c such that |ρ s | ≤ c for s ∈ R . Taking the inner product of 4.3 , 4.4 , and 4.5 with Gρ |x| 2 /K 2 u, ρ |x| 2 /K 2 v and Gρ |x| 2 /K 2 W in L 2 R n , respectively, we get

4.40
Adding up the three equalities, we have