Uniform attractors of stochastic two-compartment Gray-Scott system with multiplicative noise

We first show that the stochastic two-compartment Gray-Scott system generates a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic two-compartment Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.

In [8], Crauel, Kloeden and Yang firstly introduced the concept of cocycle attractors for NRDS. Then Wang explained cocycle attractors in detail in [21,22]. It is noted that the attraction universes are non-autonomous in [21,22]. For other different kinds of attractors, such as random attractors, pullback attractors, global attractors and so on, we can see [3,6,7,14,18,20,23,24]. In [13], Flandoli and Schmalfuss studied random attractors of stochastic Navier-Stokes equation. Uniform attractors of non-autonomous three-component reversible Gray-Scott system were considered in [15]. In [16], Jia, Gao and Ding studied random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise. Cui and Langa investigated uniform attractors for non-autonomous random dynamical systems in [11].
This paper is organized as follows. In Section 2, we will introduce some concepts about uniform and cocycle attractors. In Section 3, we will provide some basic settings of stochastic two-compartment Gray-Scott system. In Section 4, we give some uniform estimates of solutions. In Section 5, the existence of uniform and cocycle attractors is provided.
The following notations will be used throughout the paper. Denote by · and (·, ·) the norm and inner product in L 2 (O) or [L 2 (O)] 4 . We use · L p and · H 1 to denote the norm in L p (O) and H 1 (O).
Using Poincaré's inequality, there exists a constant γ > 0 satisfying We know that H 1 0 (O) → L 6 (O) for n ≤ 3. There is a constant η > 0 satisfying the following embedding inequality 2. Preliminaries. In this part, some concepts about uniform attractors and cocycle attractors are provided, see [9,11,4]. Let φ be a NRDS, let D be an autonomous universe and let (X, d) be a Polish metric space. For non-empty sets in X, the Hausdorff semi-metric is defined by Let (X, d X ) be the extended space of Σ × X, i.e., χ ∈ X if and only if χ = {σ} × {x} for some σ ∈ Σ and x ∈ X. For any metric space M , let B(M ) be the Borel sigma-algebra of M . Let (Σ, d Σ ) be a compact Polish space which is invariant For each Ξ ⊂ Σ and ω ∈ Ω, define the following omega limit set of any B ∈ D, Normally, it is not difficult to find that W(ω, Ξ, B) = ∪ σ∈Ξ W(ω, σ, B). In reality, Definition 2.1. For any s ∈ R, define an operator A σ(s) (·) : G 1 → G 0 , where G 1 , G 0 are Banach spaces. σ(s), s ∈ R reflects the dependence on time. The function σ(s) is called the symbol of system (1). Definition 2.2. The set Σ is called the symbol space of system (1) if Σ contains all σ(·) and Property 2.3. {θ t } t∈R is the smooth translation operator satisfying (1) θ 0 = identity operator on Σ; (2) θ s • θ t = θ t+s , ∀t, s ∈ R; (3) (t, σ) → θ t σ is continuous.
Similarly, the continuity of φ in X can be defined.
Proposition 2.11. If a random set A is uniformly D-pullback attracting under NRDS φ, then A is forward uniformly attracting in probability, i.e., Theorem 2.13. Assume that NRDS φ is continuous in Σ and X, and each Ξ ∈ Σ is dense. If φ possesses a compact uniformly D-attracting set K and a closed uniformly D-absorbing set B ∈ D, then φ possesses a unique random uniform attractor A ∈ D,
Proposition 2.14. Assume that NRDS φ is continuous in Σ and X, and U is a random uniform attractor. If φ has a D-random uniform attractor A, and U uniformly attracts deterministic compact sets, then 3. NRDS generated by stochastic two-compartment Gray-Scott system.
holds uniformly in g 0 ∈ D and t ≥ T .
There is a random variable ρ 3 (ω) such that, for any D ∈ D and ω ∈ Ω, we can find a time T = T (D, ω) > 1 such that, for any σ ∈ Σ, holds uniformly in g 0 ∈ D and t ≥ T .
There is a random variable ρ 4 (ω) such that, for any D ∈ D and ω ∈ Ω, we can find a time T = T (D, ω) > 1 such that, for any σ ∈ Σ, holds uniformly in g 0 ∈ D and t ≥ T .
Applying (8) and Lemma 4.5, the solutions of (1) can be estimated uniformly.
Next, the existence of D-cocycle and D-uniform attractors is proved.
By Theorem 2.9, Theorem 2.13, Proposition 2.11 and Proposition 2.14, Theorem 5.1 can be written as follows. (91) A is forward-attracting in probability. Furthermore, A is upper semi-continuous in symbols.