Random Attractors for Stochastic Three-Component Reversible Gray-Scott System on Infinite Lattices

Abstract

We prove the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices.

1. Introduction

Consider the following stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices: 𝑑𝑢_𝑖𝑑𝑡=𝑑₁𝑢_𝑖−1−2𝑢_𝑖+𝑢_𝑖+1−(𝐹+𝑘)𝑢_𝑖+𝑢²_𝑖𝑣_𝑖−𝐺𝑢³_𝑖+𝑁𝑤_𝑖+𝑢_𝑖∘𝑑𝑊,𝑑𝑡𝑑𝑣_𝑖𝑑𝑡=𝑑₂𝑣_𝑖−1−2𝑣_𝑖+𝑣_𝑖+1+𝐹1−𝑣_𝑖−𝑢²_𝑖𝑣_𝑖+𝐺𝑢³_𝑖+𝑣_𝑖∘𝑑𝑊,𝑑𝑡𝑑𝑤_𝑖𝑑𝑡=𝑑₃𝑤_𝑖−1−2𝑤_𝑖+𝑤_𝑖+1+𝑘𝑢_𝑖−(𝐹+𝑁)𝑤_𝑖+𝑤_𝑖∘𝑑𝑊,𝑢(0)𝑑𝑡=𝑢₀=𝑢_{𝑖0𝑖∈ℤ},𝑣(0)=𝑣₀=𝑣_{𝑖0𝑖∈ℤ},𝑤(0)=𝑤₀=𝑤_{𝑖0𝑖∈ℤ}, (1.1) where 𝑖∈ℤ (the set of integers), (𝑢_𝑖)_𝑖∈ℤ∈ℓ²,(𝑣_𝑖)_𝑖∈ℤ∈ℓ², and (𝑤_𝑖)_𝑖∈ℤ∈ℓ²; all the parameters are positive constants; 𝑊 is a Brownian motion on (Ω,ℱ,ℙ) and ∘ denotes the Stratonovich sense of the stochastic term.

System (1.1) can be considered as a discrete model of stochastic three-component reversible Gray-Scott system in which the existence of a random attractor has been established [1]. When there is no stochastic term, system (1.1) can be considered as a discrete analogue of the following three-component reversible Gray-Scott system in 𝐑:𝜕𝑢𝜕𝑡=𝑑₁Δ𝑢−(𝐹+𝑘)𝑢+𝑢²𝑣−𝐺𝑢³+𝑁𝑤,𝜕𝑣𝜕𝑡=𝑑₂Δ𝑣+𝐹(1−𝑣)−𝑢²𝑣+𝐺𝑢³,𝜕𝑤𝜕𝑡=𝑑₃Δ𝑤+𝑘𝑢−(𝐹+𝑁)𝑤, (1.2) which was firstly introduced by Mahara et al. [2], then it was reduced to system (1.2) under some nondimensional transformations in You [3]. Also, the existence of a global attractor for the solution semiflow of (1.2) with Neumann boundary condition on a bounded domain of space dimension 𝑛≤3 was proved in [3].

When 𝐺=0 and 𝑤=0, system (1.2) becomes the two-components Gray-Scott equations which was one of the models signified the seminal work of the Brussell school. The model originated from describing an isothermal, cubic autocatalytic, continuously fed and diffusive reactions of two chemicals (see [4–8]), but neglected the reversible factors. Indeed, the reversibility in the interactions of multispecies is an indispensable factor in many processes in natural and social sciences. If we take the reversibility into account, it yields system (1.2).

Stochastic lattice differential equations have discrete spatial structures and take random influences into account. These random effects are not only introduced to compensate for the defects in some deterministic models, but are also rather intrinsic phenomena. Bates et al. [9] initiated the consideration of stochastic lattice dynamical systems with additive noises and Caraballo and Lu [10] was the first to consider the stochastic lattice dynamical systems with a multiplicative noise, and Han et al. [11] generalized the results of [9, 10] to a more general space. For more details and the quite recent results, we can refer to, for example, [12–15].

Just like the models considered in biology, the discrete time models governed by difference equations are more appropriate than the continuous ones; we can also deal with the chemical and biochemical reactions in the same manner, see, for example, [16, 17] and the references therein. However, very few investigations are on this topic, especially for the stochastic three-component reversible Gray-Scott system on infinite lattices, is widely open, to the best of our knowledge.

The paper is organized as follows. In Section 2, we present some preliminaries and definitions. Section 3 is devoted to the existence of a random attractor.

2. Preliminaries

Let (Ω,ℱ,ℙ) be a probability space, where Ω is a subset of 𝒞₀(𝐑,𝐑)={𝜔∈𝒞(𝐑,𝐑)∶𝜔(0)=0}, which endowed with the compact open topology (see [18]), ℱ is the Borel 𝜎-algebra, and ℙ is the corresponding Wiener measure on Ω. Let 𝜃_𝑡𝜔(⋅)=𝜔(⋅+𝑡)−𝜔(𝑡), 𝑡∈𝐑, then (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈𝐑) is an ergodic metric dynamical system. Throughout the paper, we denote ℓ²=𝑥=𝑥_{𝑖𝑖∈ℤ}∶𝑥_𝑖∈𝐑,_𝑖∈ℤ𝑥²_𝑖<∞, (2.1) the Hilbert space equipped with the usual inner product and norm: 𝑥,𝑦=_𝑖∈ℤ𝑥_𝑖,𝑦_𝑖,‖𝑥‖²=⟨𝑥,𝑥⟩,∀𝑥=𝑥_{𝑖𝑖∈ℤ},𝑦=𝑦_{𝑖𝑖∈ℤ}∈ℓ². (2.2)

For the reader's convenience, we introduce some basic concepts related to random dynamical systems and random attractor, which are taken from [11, 18, 19]. Let (𝐻,‖⋅‖_𝐻) be a separable Hilbert space and (Ω₀,ℱ,ℙ) a probability space.

Definition 2.1. A stochastic process {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω0} is a continuous random dynamical system over (Ω₀,ℱ,ℙ,(𝜃_𝑡)_𝑡∈𝐑) if 𝜑 is (ℬ[0,∞)×ℱ×ℬ(𝐻),ℬ(𝐻))-measurable, and for all 𝜔∈Ω₀,(i)the mapping 𝜑(𝑡,𝜔)∶𝐻↦𝐻, 𝑥↦𝜑(𝑡,𝜔)𝑥 is continuous for every 𝑡≥0,(ii)𝜑(0,𝜔) is the identity on 𝐻,(iii)(cocycle property) 𝜑(𝑠+𝑡,𝜔)=𝜑(𝑡,𝜃_𝑠𝜔)𝜑(𝑠,𝜔) for all 𝑠,𝑡≥0.

Definition 2.2. (i) A set-valued mapping 𝜔↦𝐵(𝜔)⊂𝐻 (we may write it as 𝐵(𝜔) for short) is said to be a random set if the mapping 𝜔↦ dist_𝐻(𝑥,𝐵(𝜔)) is measurable for any 𝑥∈𝐻, where dist_𝐻(𝑥,𝐷) is the distant in 𝐻 between the element 𝑥 and the set 𝐷⊂𝐻.
(ii) A random set 𝐵(𝜔) is said to be bounded if there exist 𝑥₀∈𝐻 and a random variable 𝑟(𝜔)>0 such that 𝐵(𝜔)⊂{𝑥∈𝐻∶‖𝑥−𝑥₀‖_𝐻≤𝑟(𝜔),𝑥₀∈𝐻} for all 𝜔∈Ω₀.
(iii) A random set 𝐵(𝜔) is called a compact random set if 𝐵(𝜔) is compact for all 𝜔∈Ω₀.
(iv) A random bounded set 𝐵(𝜔)⊂𝐻 is called tempered with respect to (𝜃_𝑡)_𝑡∈𝐑 if for a.e. 𝜔∈Ω₀, lim_𝑡→+∞𝑒^−𝛾𝑡sup_{𝑥∈𝐵(𝜃−𝑡𝜔)}‖𝑥‖_𝐻=0  for  all  𝛾>0. A random variable 𝜔↦𝑟(𝜔)∈𝐑 is said to be tempered with respect to (𝜃_𝑡)_𝑡∈𝐑 if for a.e. 𝜔∈Ω₀, lim_𝑡→+∞sup_𝑡∈𝐑𝑒^−𝛾𝑡𝑟(𝜃_−𝑡𝜔)=0  for  all  𝛾>0.
We consider a continuous random dynamical system (RDS) {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω0} over (Ω₀,ℱ,ℙ,(𝜃_𝑡)_𝑡∈𝐑) and 𝒟(𝐻) the set of all tempered random sets of 𝒟(𝐻).

Definition 2.3. A random set 𝒦 is called an absorbing set in 𝒟(𝐻) if for all 𝐵∈𝒟(𝐻) and a.e. 𝜔∈Ω₀ there exists 𝑡_𝐵(𝜔)>0 such that 𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔⊂𝒦(𝜔)∀𝑡≥𝑡_𝐵(𝜔). (2.3)

Definition 2.4. A random set 𝒜 is called a global random 𝒟(𝐻) attractor (pullback 𝒟(𝐻) attractor) for {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω0} if the following hold:(i)𝒜 is a random compact set, that is, 𝜔↦𝑑(𝑥,𝒜(𝜔)) is measurable for every 𝑥∈𝐻 and 𝒜(𝜔)) is compact for a.e. 𝜔∈Ω₀;(ii)𝒜 is strictly invariant, that is, for 𝜔∈Ω₀ and all 𝑡≥0, 𝜑(𝑡,𝜔)𝒜(𝜔)=𝒜(𝜃_𝑡𝜔);(iii)𝒜 attracts all sets in 𝒟(𝐻), that is, for all 𝐵∈𝒟(𝐻) and a.e. 𝜔∈Ω₀, we have lim_𝑡→+∞𝑑𝜑𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔,𝒜(𝜔)=0, (2.4)where 𝑑(𝑋,𝑌)=sup_𝑥∈𝑋inf_𝑦∈𝑌‖𝑥−𝑦‖_𝐻 is the Hausdorff semimetric (𝑋⊆𝐻,𝑌⊆𝐻).

Proposition 2.5 2.5 (see [11]). Suppose that(a)there exists a random bounded absorbing set 𝐾(𝜔)∈𝒟(ℓ²), 𝜔∈Ω₀, such that for any 𝐵(𝜔)∈𝒟(ℓ²) and all 𝜔∈Ω₀, there exists 𝑇(𝜔,𝐵)>0 yielding 𝜑(𝑡,𝜃_−𝑡𝜔,𝐵(𝜃_−𝑡𝜔))⊂𝐾(𝜔) for all 𝑡≥𝑇(𝜔,𝐵);(b)the RDS {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω0} is random asymptotically null on 𝐾(𝜔), that is, for any 𝜖>0, there exist 𝑇(𝜖,𝜔,𝐾)>0 and 𝐼₀(𝜖,𝜔,𝐾)∈ℕ such that sup_{(𝜔)𝑢∈𝐾|𝑖|>𝐼0((𝜔𝜖,𝜔,𝐾))}𝜑𝑡,𝜃_−𝑡𝜔,𝑢𝜃_−𝑡𝜔_𝑖²≤𝜖²,∀𝑡≥𝑇(𝜖,𝜔,𝐾(𝜔)). (2.5)
Then the RDS {𝜑(𝑡,𝜔,⋅)}_{𝑡≥0,𝜔∈Ω0} possesses a unique global random 𝒟(𝐻) attractor given by 𝒜(𝜔)=_{()𝜏≥𝑇𝜔,𝐾𝑡≥𝜏}𝜑𝑡,𝜃_−𝑡𝜔,𝐾𝜃_−𝑡𝜔. (2.6)

3. Existence of a Random Attractor

In this section, we will derive the random attractor of the stochastic three-component reversible Gray-Scott lattice system (1.1) with a multiplicative white noise.

For 𝑥=(𝑥_𝑖)_𝑖∈ℤ, we define 𝐴,𝐵,𝐵^∗ to be linear operators from ℓ² to ℓ² for 𝑖∈ℤ, as follows: (𝐴𝑥)_𝑖=−𝑥_𝑖−1+2𝑥_𝑖−𝑥_𝑖+1,(𝐵𝑥)_𝑖=𝑥_𝑖+1−𝑥_𝑖,𝐵^∗𝑥_𝑖=𝑥_𝑖−1−𝑥_𝑖. (3.1) It is easy to show that 𝐴=𝐵𝐵^∗=𝐵^∗𝐵, (𝐵^∗𝑥,𝑥′)=(𝑥,𝐵𝑥′) for all 𝑥,𝑥′∈ℓ², which implies that (𝐴𝑥,𝑥)≥0.

In the sequel, we rewrite the system (1.1) with initial values (𝑢₀,𝑣₀,𝑤₀)∶=(𝑢_0𝑖,𝑣_0𝑖,𝑤_0𝑖)_𝑖∈ℤ∈ℓ²×ℓ²×ℓ²=𝐻 as the following integral equations in 𝐻 for 𝑡≥0,𝜔∈Ω∶𝑢(𝑡)=𝑢₀+^𝑡₀−𝑑₁𝐴𝑢(𝑠)−(𝐹+𝑘)u(𝑠)+𝑢²(𝑠)𝑣(𝑠)−𝐺𝑢³(𝑠)+𝑁𝑤(𝑠)𝑑𝑠+^𝑡₀𝑢(𝑠)𝑣(𝑡)∘𝑑𝑊,=𝑣₀+^𝑡₀−𝑑₂𝐴𝑣(𝑠)+𝐹(1−𝑣(𝑠))−𝑢²(𝑠)𝑣(𝑠)+𝐺𝑢³(𝑠)𝑑𝑠+^𝑡₀𝑣(𝑠)𝑤(𝑡)∘𝑑𝑊,=𝑤₀+^𝑡₀−𝑑₃𝐴𝑤(𝑠)+𝑘𝑢(𝑠)−(𝐹+𝑁)𝑤(𝑠)𝑑𝑠+^𝑡₀𝑤(𝑠)∘𝑑𝑊, (3.2) where 𝑊 is a two-sided Brownian motion on the same probability space (Ω,ℱ,ℙ). To prove that this system (3.2) generates a random dynamical system, we will transform it into a random differential equation system in 𝐻.

Before performing this transformation, we need to recall some properties of the Ornstein-Uhlenbeck processes. Let 𝑧𝜃_𝑡𝜔=−⁰_−∞𝑒^𝜏𝜃_𝑡𝜔(𝜏)𝑑𝜏,𝑡∈𝐑,𝜔∈Ω. (3.3) We know that 𝑧(𝜃_𝑡𝜔) is an Ornstein-Uhlenbeck process on (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈𝐑) and solves the following one-dimensional stochastic differential equation (see [20] for details): 𝑧=−𝑧𝑑𝑡+𝑑𝑤(𝑡),𝑧(−∞)=0,∀𝑡≥0,𝜔∈Ω, (3.4) where 𝑤(𝑡)(𝜔)=𝑤(𝑡,𝜔)=𝜔(𝑡) for 𝜔∈Ω,𝑡∈𝐑. In fact, we have the following.

Lemma 3.1 3.1 (see [10, 18]). There exists a 𝜃_𝑡-variant set Ω∈ℱ of 𝒞₀(𝐑,𝐑) of full ℙ measure such that, for 𝜔∈Ω, one has(i)the random variable |𝑧(𝜔)| is tempered;(ii)the mapping (𝑡,𝜔)⟼𝑧𝜃_𝑡𝜔=−⁰_−∞𝑒^𝑠𝜔(𝑡+𝑠)𝑑𝑠+𝜔(𝑡) (3.5) is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;(iii)lim_𝑡→±∞𝑧𝜃_𝑡𝜔𝑡=0,lim_𝑡→±∞1𝑡^𝑡₀𝑧𝜃_𝑠𝜔𝑑𝑠=0,lim_𝑡→±∞1𝑡^𝑡₀𝑧𝜃_𝑠𝜔𝑑𝑠=𝔼|𝑧|<∞. (3.6)(3.7)

Obviously, 𝑒^{𝑧(𝜃𝑡𝜔)}𝟏𝐝_𝐻 is clearly a homeomorphism in 𝐻, and the inverse operator 𝑒^{−𝑧(𝜃𝑡𝜔)}𝟏𝐝_𝐻 is well defined. It easily follows from (3.6) that ‖𝑒^{𝑧(𝜃𝑡𝜔)}𝟏𝐝_𝐻‖_𝐻 and ‖𝑒^{−𝑧(𝜃𝑡𝜔)}𝟏𝐝_𝐻‖_𝐻 have subexponential growth as 𝑡→±∞ for all 𝜔∈Ω, which implies that they are tempered. Since the mapping of 𝜃 on Ω has the same properties as the original one if we choose the trace 𝜎-algebra with respect to Ω to be denoted also by ℱ, we can change our metric dynamical system with respect to Ω, and still denoted by the symbols (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈𝐑).

Let 𝑔(𝑡,𝜔)=𝑒^{−𝑧(𝜃𝑡𝜔)}𝟏𝐝_𝐻𝑔(𝑡,𝜔),𝜔∈Ω, (3.8) where 𝑔(𝑡,𝜔)=(𝑢(𝑡,𝜔),𝑣(̃𝑡,𝜔),𝑤(𝑡,𝜔)), and 𝑔(𝑡,𝜔)=(𝑢(𝑡,𝜔),𝑣(𝑡,𝜔),𝑤(𝑡,𝜔)) is a solution of (3.2). Then system (1.1) can be written as the following random system with random coefficients but without white noise: 𝑑𝑢𝑑𝑡=−𝑑₁𝐴𝑢−𝐹+𝑘−𝑧𝜃_𝑡𝜔𝑢+𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣̃−𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³𝑑𝑣̃+𝑁𝑤,𝑑𝑡=−𝑑₂𝐴𝑣̃+𝐹𝑒^{−𝑧(𝜃𝑡𝜔)}−𝐹−𝑧𝜃_𝑡𝜔𝑣̃−𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣̃+𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³,𝑑𝑤𝑑𝑡=−𝑑₃𝐴𝑤+𝑘𝑢−𝐹+𝑁−𝑧𝜃_𝑡𝜔𝑤 (3.9) and an initial condition 𝑔₀(𝜔)=(𝑢(0),𝑣̃(0),𝑤(0))=𝑢₀,𝑣₀̃,𝑤₀∈𝐻. (3.10)

Now we establish the following result.

Theorem 3.2. Let 𝑇>0 and 𝑔₀(𝜔)∈𝐻 be fixed. Then the following properties hold:(i)for every 𝜔∈Ω, system (3.9) admits a unique solution 𝑔(⋅,𝜔,𝑔₀)∈𝒞([0,𝑇],𝐻),(ii)the solution 𝑔(𝑡,𝜔) of system (3.9) depends continuously on the initial data 𝑔₀, that is, for each 𝜔∈Ω, the mapping 𝑔₀∈𝐻↦𝑔(⋅,𝜔,𝑔₀)∈𝒞([0,𝑇],𝐻) is continuous.

Proof. (1) Denote 𝐿𝑔=−𝑑₁𝑤𝑑𝐴+𝐹+𝑘𝑢0𝑁0−₂̃𝑘𝑑𝐴+𝐹𝑣0𝑢0−₃𝑤,Ψ𝐴+𝐹+𝑁𝑔,𝜃_𝑡𝜔=𝑧𝜃_𝑡𝜔𝑢+𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣̃−𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³𝐹𝑒^{−𝑧(𝜃𝑡𝜔)}+𝑧𝜃_𝑡𝜔𝑣̃−𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣̃+𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³𝑧𝜃_𝑡𝜔𝑤. (3.11) Then system (3.9) can be written as 𝑑𝑔𝑔𝑑𝑡=𝐿+Ψ𝑔,𝜃_𝑡𝜔. (3.12) Since 𝑧(𝜃_𝑡𝜔) is continuous with respect to 𝑡, define Ψ=𝑧(𝜔)𝑔,𝜔𝑢+𝑒^2𝑧(𝜔)𝑢²𝑣̃−𝐺𝑒^2𝑧(𝜔)𝑢³𝐹𝑒^{−𝑧(𝜔)}+𝑧(𝜔)𝑣̃−𝑒^2𝑧(𝜔)𝑢²𝑣̃+𝐺𝑒^2𝑧(𝜔)𝑢³𝑧(𝜔)𝑤, (3.13) then Ψ𝑔,𝜔_𝐻≤2𝐹𝑒^{−𝑧(𝜔)}+2|𝑧(𝜔)|+max{1,𝐺}𝑒^2𝑧(𝜔)𝑔_𝐻+3(1+𝐺)𝑒^2𝑧(𝜔)𝑔²_𝐻. (3.14) For any 𝑔=(𝑢,𝑣,𝑤)=(𝑢_𝑖,𝑣_𝑖,𝑤_𝑖)_𝑖∈ℤ, 𝑔=(𝑢,𝑣,̃𝑤)=(𝑢_𝑖,𝑣_𝑖̃,𝑤_𝑖)_𝑖∈ℤ∈𝐻, Ψ𝑔,𝜔−Ψ𝑔,𝜔_𝐻≤2‖𝑧(𝜔)|+3(1+𝐺)𝑒^2𝑧(𝜔)𝑔²_𝐻+𝑔²_𝐻𝑔−𝑔_𝐻. (3.15) For any bounded set 𝐷∈𝐻 with sup_𝑔∈𝐷‖𝑔‖_𝐻≤𝑟, define a random variable 𝜚_𝐷(𝜔) by 𝜚_𝐷(𝜔)=2|𝑧(𝜔)|+3(1+𝐺)𝑒^2𝑧(𝜔)𝑟+max{1,𝐺}𝑒^2𝑧(𝜔)𝑟+2𝐹𝑒^{−𝑧(𝜔)}≥0. (3.16) Then _𝜏^𝜏+1𝜚_𝐷𝜃_𝑡𝜔𝑑𝑡=2_𝜏^𝜏+1𝑧𝜃_𝑡𝜔+3(1+𝐺)𝑒^{2𝑧(𝜃𝑡𝜔)}𝑟+max{1,𝐺}𝑒^{2𝑧(𝜃𝑡𝜔)}𝑟+𝐹𝑒^{−𝑧(𝜃𝑡𝜔)}𝑑𝑡<∞,∀𝜏∈𝐑, (3.17) and, for any 𝑔,𝑔,𝑔∈𝐷, we have Ψ𝑔,𝜔_𝐻≤𝜚_𝐷(𝜔),Ψ𝑔,𝜔−Ψ𝑔,𝜔_𝐻≤𝜚_𝐷(𝜔)𝑔−𝑔_𝐻. (3.18) Then we obtain that 𝐿(𝑔)+Ψ(𝑔,𝜃_𝑡𝜔) is locally Lipschitz in 𝑔 from 𝐻 to 𝐻. By Proposition 2.1.1 in [19], problem (3.12) possesses a unique local solution 𝑔(⋅,𝜔,𝑔₀)∈𝒞¹((0,𝑇_max),𝐻), where (0,𝑇_max) is the maximal interval of existence of the solution of (3.12). Now, we will show that the local solution is a global one. Define 𝑟̃(𝑡)=𝑁𝑘𝑤(𝑡)𝑘,𝜇=𝑁, (3.19) then system (3.9) can be written as 𝑑𝑢𝑑𝑡=−𝑑₁𝐴𝑢−𝐹+𝑘−𝑧𝜃_𝑡𝜔𝑢+𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣̃−𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³̃𝑑𝑣̃+𝑘𝑟,𝑑𝑡=−𝑑₂𝐴𝑣̃+𝐹𝑒^{−𝑧(𝜃𝑡𝜔)}−𝐹−𝑧𝜃_𝑡𝜔𝑣̃−𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣̃+𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³,𝜇𝑑𝑟̃𝑑𝑡=−𝜇𝑑₃𝐴𝑟̃+𝑘𝑢−𝜇𝐹+𝑘−𝑧𝜃_𝑡𝜔𝑟.̃ (3.20) Taking the inner products of (3.20) with 𝐺𝑢(𝑡), 𝑣(̃𝑡) and 𝐺𝑤(𝑡), respectively, and adding up the resulting equalities, we get 12𝑑𝐺‖𝑢‖𝑑𝑡²+‖𝑣‖̃²+𝜇𝐺‖𝑟‖̃²+𝐺𝐹‖𝑢‖²+𝐹2‖𝑣‖̃²+𝜇𝐺𝐹‖𝑟‖̃²≤𝑧𝜃_𝑡𝜔𝐺‖𝑢‖²+‖𝑣‖̃²+𝐺‖𝑟‖̃²+𝐹2𝑒^{−2𝑧(𝜃𝑡𝜔)}, (3.21) which implies that 𝑑𝐺‖𝑢‖𝑑𝑡²+‖𝑣‖̃²+𝐺𝜇‖𝑤‖²+2𝐺𝐹‖𝑢‖²+𝐹‖𝑣‖̃²+2𝐺𝐹𝜇‖𝑤‖²≤2𝑧𝜃_𝑡𝜔𝐺‖𝑢‖²+‖𝑣‖̃²+𝐺𝜇²‖𝑤‖²+𝐹𝑒^{−2𝑧(𝜃𝑡𝜔)}. (3.22)
Setting 𝜆=𝐹min1,2𝐺,2𝐺/𝜇max1,𝐺,𝐺/𝜇,𝐶₁=2max1,𝐺,𝐺/𝜇²min1,𝐺,𝐺/𝜇,𝐶₂=𝐹,min1,𝐺,𝐺/𝜇 (3.23) then (3.22) yields 𝑑𝑔(𝑡𝑑𝑡)²_𝐻+𝜆−𝐶₁𝑧𝜃_𝑡𝜔𝑔(𝑡)²_𝐻≤𝐶₂𝑒^{−2𝑧(𝜃𝑡𝜔)}. (3.24) Applying Gronwall's lemma to (3.24), we obtain that, for 𝑡≥0, 𝑔𝑡,𝜔,𝑔₀(𝜔)²_𝐻≤𝑒^{−𝜆𝑡+𝐶1∫𝑡0𝑧(𝜃𝑠𝜔)𝑑𝑠}𝑔₀(𝜔)²_𝐻+𝐶₂𝑒^{−𝜆𝑡+𝐶1∫𝑡0𝑧(𝜃𝑠𝜔)𝑑𝑠𝑡}₀𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠. (3.25)
Denoting 𝛼(𝜔)=𝐶₁^𝑇₀𝑧𝜃_𝑠𝜔𝛽(𝜔)𝑑𝑠,=𝐶₂max_{[]𝑡∈0,𝑇}𝑒^{−𝜆𝑡+𝐶1∫𝑡0𝑧(𝜃𝑠𝜔)𝑑𝑠𝑡}₀𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠, (3.26) we get 𝑔𝑡,𝜔,𝑔₀(𝜔)²_𝐻≤𝑔₀(𝜔)²_𝐻𝑒^𝛼(𝜔)+𝛽(𝜔), (3.27) which implies that the solution 𝑔 is defined in any interval [0,𝑇].
(2) Let 𝑔₀,𝑔₀∈𝐻 and 𝑋(𝑡)∶=𝑔𝑡,𝜔,𝑔₀,𝑌(𝑡)∶=𝑔𝑡,𝜔,𝑔₀ (3.28) be the corresponding solutions of (3.12). Then, denoting 𝑍(𝑡)=(𝑍₁,𝑍₂,𝑍₃)=𝑋(𝑡)−𝑌(𝑡)=(𝑢−𝑢,𝑣−𝑣,̃𝑤−𝑤), we have 𝑑𝑍₁𝑑𝑡=−𝑑₁𝐴𝑍₁−𝐹+𝑘−𝑧𝜃_𝑡𝜔𝑍₁+𝑁𝑍₃+𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣−𝑢²𝑣̃−𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³−𝑢³,𝑑𝑍₂𝑑𝑡=−𝑑₂𝐴𝑍₂−𝐹−𝑧𝜃_𝑡𝜔𝑍₂−𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣−𝑢²𝑣̃+𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³−𝑢³,𝑑𝑍₃𝑑𝑡=−𝑑₃𝐴𝑍₃+𝑘𝑍₁−𝐹+𝑁−𝑧𝜃_𝑡𝜔𝑍₃. (3.29) Taking the inner product of (3.29) with 𝑍₁,𝑍₂ and 𝑍₃, respectively, it yields 𝑑𝑍𝑑𝑡¹²+𝑍₂²+𝑍₃²≤2𝑧𝜃_𝑡𝜔𝑍₁²+𝑍₂²+𝑍₃²+𝑁2𝑍₁²+𝑘2𝑍₃²+2𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣−𝑢²𝑣,̃𝑍₁−𝑍₂−2𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³−𝑢³,𝑍₁−𝑍₂. (3.30) Due to (3.27), we have 𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣−𝑢²𝑣,̃𝑍₁−𝑍₂−𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³−𝑢³,𝑍₁−𝑍₂≤2(1+𝐺)𝑒^{2𝑧(𝜃𝑡𝜔)}𝑔²_𝐻+𝑔²_𝐻𝑍₁²+𝑍₂²≤4(1+𝐺)𝑒^{2𝑧(𝜃𝑡𝜔)}𝑔²_𝐻𝑍₁²+𝑍₂²+𝑍₃². (3.31) Denoting 𝛿=(1/2)max{𝑁,𝑘}, we obtain that 𝑑𝑍𝑑𝑡¹²+𝑍₂²+𝑍₃²≤𝛿+𝑧𝜃_𝑡𝜔+8(1+𝐺)𝑒^{2𝑧(𝜃𝑡𝜔)}𝑔²_𝐻𝑍₁²+𝑍₂²+𝑍₃². (3.32) By Gronwall's lemma, for 𝑡∈[0,𝑇], we have 𝑍₁²+𝑍₂²+𝑍₃²≤𝑍₁(0)²+𝑍₂(0)²+‖𝑍₃(0)𝑒^{𝛿′𝑡+8(1+𝐺)∫𝑡0𝑒2𝑧(𝜃𝑠𝜔)‖}𝑔^(𝑠,𝜔,𝑔₀^(𝜔))‖2_𝐻^𝑑𝑠, (3.33) where 𝛿′=𝛿+max_{𝑡∈[0,𝑇]}|𝑧(𝜃_𝑡𝜔)|. According to (3.25), for 𝑡∈[0,𝑇], ^𝑡₀𝑒^{2𝑧(𝜃𝑠𝜔)}𝑔𝑠,𝜔,𝑔₀(𝜔)²_𝐻≤𝑔𝑑𝑠₀(𝜔)²_𝐻^𝑡₀𝑒^{2𝑧(𝜃𝑠𝜔)+𝐶1∫𝑠0𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠+𝐶₂^𝑡₀𝑒^{2𝑧(𝜃𝑠𝜔)+𝐶1∫𝑠0𝑧(𝜃𝜏𝜔)𝑑𝜏𝑠}₀𝑒^{−2𝑧(𝜃𝜏𝜔)−𝐶1∫𝑡0𝑧(𝜃𝜍𝜔)𝑑𝜍}𝑔𝑑𝜏𝑑𝑠∶=₀(𝜔)²_𝐻^𝑡₀𝜉𝜃_𝑠𝜔𝑑𝑠+𝐶₂^𝑡₀𝜉𝜃_𝑠𝜔^𝑠₀𝜉⁻¹𝜃_𝜏𝜔𝑑𝜏𝑑𝑠, (3.34) where 𝜉(𝜃_𝑡𝜔)=𝑒^{2𝑧(𝜃𝑡𝜔)+𝐶1∫𝑡0𝑧(𝜃𝜏𝜔)𝑑𝜏}. Obviously, 𝑡↦ln𝜉(𝜃_𝑡𝜔) is continuous ℙ-a.s. Also, we have lim_𝑡→+∞(ln𝜉(𝜃_𝑡𝜔)/𝑡)=0, which implies that 𝜉(𝜃_𝑡𝜔) is a tempered random variable. Then by Proposition 4.3.3, [18], for given 𝜀>0, there is an 𝜀-slowly varying random variable 𝑅(𝜔) for which 𝑒^{−𝜀|𝑡|}𝑅(𝜔)≤𝜉𝜃_𝑡𝜔≤𝑒^𝜀|𝑡|𝑅(𝜔),∀𝑡∈𝐑,𝜔∈Ω, (3.35) where 𝑅(𝜔),𝜔∈Ω satisfies 𝑒^{−𝜀|𝑡|}𝑅(𝜔)≤𝑅𝜃_𝑡𝜔≤𝑒^𝜀|𝑡|𝑅(𝜔)∀𝑡∈𝐑,𝜔∈Ω. (3.36) Combining with (3.34) and (3.35), we easily conclude that for 𝑡∈[0,𝑇], ^𝑡₀𝑒^{2𝑧(𝜃𝑠𝜔)}𝑔𝑠,𝜔,𝑔₀(𝜔)²_𝐻𝑔𝑑𝑠≤⁰(𝜔)²_𝐻𝜀𝑅(𝜔)𝑒^𝜀𝑇+𝐶₂2𝜀²𝑒^2𝜀𝑇, (3.37) which implies sup_{[]𝑡∈0,𝑇}‖𝑋(𝑡)−𝑌(𝑡)‖²≤𝑢₀−𝑢₀²+𝑣₀−𝑣̃₀²+𝑤₀−𝑤₀²𝑒^{𝛿′′𝑇}, (3.38) where 𝛿^′′=𝛿^′+8(1+G)((‖𝑔₀(𝜔)‖²_𝐻/𝜀𝑇)𝑅(𝜔)𝑒^𝜀𝑇+(𝐶₂/2𝜀²𝑇)𝑒^2𝜀𝑇)<∞. If 𝑢₀=𝑢₀,𝑣₀=𝑣₀̃ and 𝑤₀=𝑤₀, then the above inequality shows the uniqueness and continuous dependence on the initial data of the solution of (3.12). So the both results of the theorem hold.

Theorem 3.3. System (3.12) generates a continuous random dynamical system (𝜑(𝑡))_𝑡≥0 over (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈𝐑), where 𝜑(𝑡,𝜔,𝑔₀)=𝑔(𝑡,𝜔,𝑔₀) for 𝑔₀∈𝐻, 𝑡≥0 and for all 𝜔∈Ω. Moreover, if one defines 𝜑 by 𝜓𝑡,𝜔,𝑔₀=𝑒^{𝑧(𝜃𝑡𝜔)}𝜑𝑡,𝜔,𝑒^{−𝑧(𝜔)}𝑔₀ (3.39) for 𝑔₀∈𝐻, 𝑡≥0 and for all 𝜔∈Ω, then 𝜓 is another random dynamical system for which the process (𝜔,𝑡)↦𝜓(𝑡,𝜔,𝑔₀) solves (3.2) for any initial condition 𝑔₀∈𝐻.

Proof. The fact that 𝜑 is continuous random dynamical system follows from Theorem 3.2. The measurability of 𝜓 follows from the properties of the transformation (see [18, 19]). It follows directly the other statements.

Note that the two random dynamical systems are equivalent. It is easy to check that 𝜓 has a random attractor provided 𝜑 possesses a random attractor. Then, we only need to consider the random dynamical system 𝜑.

Now, we are in the position to study the existence of tempered random bounded absorbing set and global random attractor for the RDS 𝜑 in 𝐻.

Lemma 3.4. There exists a random bounded ball 𝐾(𝜔)∈𝒟(𝐻) centered at 0 with random radius 𝜌(𝜔)>0 such that 𝐾(𝜔)∈𝒟(𝐻) is a random absorbing set for 𝜑 in 𝒟(𝐻); that is, for any 𝐵(𝜔)∈𝒟(𝐻) and 𝜔∈Ω, there exists 𝑇_𝐵(𝜔)>0 yielding 𝜑(𝑡,𝜃_−𝑡𝜔,𝐵(𝜃_−𝑡𝜔))⊂𝐾(𝜔), for all 𝑡≥𝑇_𝐵(𝜔).

Proof. By substituting 𝜔 by 𝜃_−𝑡𝜔 in (3.25), we have 𝑔𝑡,𝜃_−𝑡𝜔,𝑔₀𝜃_−𝑡𝜔²_𝐻≤𝑒^{−𝜆𝑡+𝐶1∫𝑡0𝑧(𝜃𝑠−𝑡𝜔)𝑑𝑠}𝑔₀𝜃_−𝑡𝜔²_𝐻+𝐶₂𝑒^{−𝜆𝑡+𝐶1∫𝑡0𝑧(𝜃𝑠−𝑡𝜔)𝑑𝑠𝑡}₀𝑒^{−2𝑧(𝜃𝑠−𝑡𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏−𝑡𝜔)𝑑𝜏}𝑑𝑠≤𝑒^{−𝜆𝑡+𝐶1∫𝑡0𝑧(𝜃𝑠−𝑡𝜔)𝑑𝑠}𝑔₀𝜃_−𝑡𝜔²_𝐻+𝐶₂⁰_−𝑡𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆(𝑠−𝑡)+𝐶1∫𝑡𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠≤𝑒^{−𝜆𝑡+𝐶1∫0−𝑡𝑧(𝜃𝑠𝜔)𝑑𝑠}𝑔₀𝜃_−𝑡𝜔²_𝐻+𝐶₂⁰_−𝑡𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠≤𝑒^{−𝜆𝑡+𝐶1∫0−𝑡𝑧(𝜃𝑠𝜔)𝑑𝑠}𝑔₀𝜃_−𝑡𝜔²_𝐻+𝐶₂⁰_−∞𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠. (3.40) By the properties of the Ornstein-Uhlenbeck process, ⁰_−∞𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠<+∞. (3.41) Notice that {𝐵(𝜔)}∈𝒟(𝐻) is tempered, then for 𝑔₀(𝜔)∈𝐵(𝜃_−𝑡𝜔), lim_𝑡→+∞𝑒^{−𝜆𝑡+𝐶1∫0−𝑡𝑧(𝜃𝑠𝜔)𝑑𝑠}𝑔₀𝜃_−𝑡𝜔²_𝐻=0. (3.42) We can choose 𝜌²(𝜔)=1+𝐶₂⁰_−∞𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠, (3.43) then 𝐾(𝜔) is a random absorbing set for 𝜑 in 𝒟(𝐻), and 𝐾(𝜔)∈𝒟(𝐻). Here, we remain only to check that lim_𝑡→+∞𝑒^−𝛾𝑡𝜌²𝜃_−𝑡𝜔=0. (3.44) Indeed, obviously we have 𝑒^−𝛾𝑡𝜌²𝜃_−𝑡𝜔=𝑒^−𝛾𝑡+𝐶₂𝑒^−𝛾𝑡0_−∞𝑒^{−2𝑧(𝜃𝑠−𝑡𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏−𝑡𝜔)𝑑𝜏}𝑑𝑠=𝑒^−𝛾𝑡_{→0as𝑡→+∞}+𝐶₂𝑒^{−𝛾𝑡−𝑡}_−∞𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆𝑠+𝐶1∫𝑠−𝑡𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠_{→0as𝑡→+∞}. (3.45)

Lemma 3.5. The RDS {𝜑(𝑡,𝜔,⋅)}_{𝑡≥0,𝜔∈Ω} generated by (3.9) is random asymptotically null on 𝐾(𝜔); that is, for any 𝜖>0, there exist 𝑇(𝜖,𝜔,𝐾)>0, ∀𝑡≥𝑇(𝜖,𝜔,𝐾(𝜔)), and 𝑀(𝜖,𝜔,𝐾(𝜔))∈ℕ such that sup_{𝜑0∈𝐾(𝜔)|𝑖|((𝜔>𝑀𝜖,𝜔,𝐾))}𝜑𝑡,𝜃_−𝑡𝜔,𝜑₀𝜃_−𝑡𝜔_𝑖²_𝐻≤𝜖². (3.46)

Proof. Choose a smooth cut-off 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⁺.
Set 𝑥=(𝜌(|𝑖|/𝑀)𝑢_𝑖)_𝑖∈ℤ,𝑦=(𝜌(|𝑖|/𝑀)𝑣_𝑖̃)_𝑖∈ℤ, and 𝑧=(𝜌(|𝑖|/𝑀)𝑤_𝑖)_𝑖∈ℤ. By taking the inner product of (3.20) with 𝐺𝑥, 𝑦, and 𝐺𝑧, respectively, we get 𝐺2𝑑𝑑𝑡_𝑖∈ℤ𝜌|𝑖|𝑀𝑢_𝑖²=−𝐺𝑑₁(𝐴𝑢,𝑥)−𝐺𝐹+𝑘−𝑧𝜃_𝑡𝜔_𝑖∈ℤ𝜌|𝑖|𝑀𝑢_𝑖²+𝑘𝐺(𝑟,̃𝑥)+𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣,̃𝑥−𝐺²𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³,1,𝑥2𝑑𝑑𝑡_𝑖∈ℤ𝜌|𝑖|𝑀𝑣̃_𝑖²=−𝑑₂𝐴𝑣,̃𝑦−𝐹−𝑧𝜃_𝑡𝜔_𝑖∈ℤ𝜌|𝑖|𝑀𝑣̃_𝑖²−𝐺𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢²𝑣,̃𝑦+𝐺²𝑒^{2𝑧(𝜃𝑡𝜔)}𝑢³,𝑦+𝑒^{−𝑧(𝜃𝑡𝜔)},𝑦,𝜇𝐺2𝑑𝑑𝑡_𝑖∈ℤ𝜌|𝑖|𝑀𝑟̃_𝑖²=−𝜇𝐺𝑑₃(𝐴𝑟,̃𝑧)−𝐺𝜇𝐹+𝑘−𝑧𝜃_𝑡𝜔_𝑖∈ℤ𝜌|𝑖|𝑀𝑟̃_𝑖²+𝑘𝐺(𝑢,𝑧). (3.47) Due to [9, 10], we have −(𝐴)≤‖𝑢‖𝑢,𝑥2𝑐²𝑀𝐴̃≤‖𝑣‖̃,−𝑣,𝑦2𝑐²𝑀(𝐴̃)≤‖𝑟‖̃,−𝑟,𝑧2𝑐²𝑀. (3.48) Combining with (3.47) to (3.48), we obtain 12𝑑𝑑𝑡_𝑖∈ℤ𝜌|𝑖|𝑀𝐺𝑢_𝑖²+𝑣̃_𝑖²+𝜇𝐺𝑟̃_𝑖²+𝐺𝐹−𝑧𝜃_𝑡𝜔_𝑖∈ℤ𝜌|𝑖|𝑀𝑢_𝑖²+𝐹2−𝑧𝜃_𝑡𝜔_𝑖∈ℤ𝜌|𝑖|𝑀𝑣̃_𝑖²+𝐺𝜇𝐹−𝑧𝜃_𝑡𝜔_𝑖∈ℤ𝜌|𝑖|𝑀𝑟̃_𝑖²≤2𝑐𝑀𝐺𝑑₁‖𝑢‖²+𝑑₂‖𝑣‖̃²+𝜇𝐺𝑑₃‖𝑟‖̃²+𝐹2𝑒^{−2𝑧(𝜃𝑡𝜔)}, (3.49) that is, 𝑑𝑑𝑡_𝑖∈ℤ𝜌|𝑖|𝑀𝐺𝑢_𝑖²+𝑣̃_𝑖²+𝐺𝜇𝑤_𝑖²+2𝐺𝐹_𝑖∈ℤ𝜌|𝑖|𝑀𝑢_𝑖²+𝐹_𝑖∈ℤ𝜌|𝑖|𝑀𝑣̃_𝑖²+2𝐺𝐹𝜇_𝑖∈ℤ𝜌|𝑖|𝑀𝑤_𝑖²≤2𝑧𝜃_𝑡𝜔_𝑖∈ℤ𝜌|𝑖|𝑀𝐺𝑢_𝑖²+𝑣̃_𝑖²+𝐺𝜇²𝑤_𝑖²+4𝑐𝑀𝐺𝑑₁‖𝑢‖²+𝑑₂‖𝑣‖̃²+𝐺𝜇𝑑₃‖𝑤‖²+𝐹𝑒^{−2𝑧(𝜃𝑡𝜔)}. (3.50)
Denote 𝐶₃=2𝑐max𝑑₁,𝑑₂,𝑑₃𝐶₁, (3.51) then (3.50) yields 𝑑𝑑𝑡_𝑖∈ℤ𝜌|𝑖|𝑀𝑔_𝑖²_𝐻+𝜆−𝐶₁𝑧𝜃_𝑡𝜔_𝑖∈ℤ𝜌|𝑖|𝑀𝑔_𝑖²_𝐻≤𝐶₃𝑀𝑔²_𝐻+𝐶₂𝑒^{−2𝑧(𝜃𝑡𝜔)}. (3.52) By using Gronwall's lemma, for 𝑡≥𝑇_𝐾=𝑇_𝐾(𝜔), we have _𝑖∈ℤ𝜌|𝑖|𝑀𝑔_𝑖𝑡,𝜔,𝑔₀(𝜔)²_𝐻≤𝑒^{−𝜆(𝑡−𝑇𝐾)+𝐶1∫𝑡𝑇𝑘𝑧(𝜃𝑠𝜔)𝑑𝑠}_𝑖∈ℤ𝜌|𝑖|𝑀𝑔_𝑖𝑇_𝑘,𝜔,𝑔₀(𝜔)²_𝐻+𝐶₃𝑀^𝑡_𝑇𝑘𝑒^{−𝜆(𝑡−𝑠)+𝐶1∫𝑡𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑔𝑠,𝜔,𝑔₀(𝜔)²_𝐻𝑑𝑠+𝐶₂^𝑡_𝑇𝑘𝑒^{−𝜆(𝑡−𝑠)−2𝑧(𝜃𝑠𝜔)+𝐶1∫𝑡𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠. (3.53) Replace 𝜔 by 𝜃_−𝑡𝜔. We then estimate each term on the right-hand side of (3.53). From (3.25) with 𝑡 replaced by 𝑇_𝐾 and 𝜔 by 𝜃_−𝑡𝜔, respectively, it then follows that 𝑒^{−𝜆(𝑡−𝑇𝐾)+𝐶1∫𝑡𝑇𝑘𝑧(𝜃𝑠−𝑡𝜔)𝑑𝑠}_𝑖∈ℤ𝜌|𝑖|𝑀𝑔_𝑖𝑇_𝑘,𝜃_−𝑡𝜔,𝑔₀𝜃_−𝑡𝜔²_𝐻≤𝑒^{−𝜆𝑡+𝐶1∫𝑡0𝑧(𝜃𝑠−𝑡𝜔)𝑑𝑠}𝑔₀(𝜔)²_𝐻+𝐶₂^𝑇𝐾₀𝑒^{−2𝑧(𝜃𝑠−𝑡𝜔)+𝜆(𝑠−𝑡)+𝐶1∫𝑡𝑠𝑧(𝜃𝜏−𝑡𝜔)𝑑𝜏}𝑑𝑠≤𝑒^{−𝜆𝑡+𝐶1∫0−𝑡𝑧(𝜃𝑠𝜔)𝑑𝑠}𝑔₀(𝜔)²_𝐻+𝐶₂^{𝑇𝐾−𝑡}_−𝑡𝑒^{−2𝑧(𝜃𝑠𝜔)+𝜆𝑠+𝐶1∫0𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝑑𝑠. (3.54) Hence, by using (3.6), there is a 𝑇₁(𝜖,𝜔,𝐾(𝜔))>𝑇_𝐾(𝜔), such that if 𝑡>𝑇₁(𝜖,𝜔,𝐾(𝜔)), 𝑒^{−𝜆(𝑡−𝑇𝐾)+𝐶1∫𝑡𝑇𝑘𝑧(𝜃𝑠−𝑡𝜔)𝑑𝑠}_𝑖∈ℤ𝜌|𝑖|𝑀𝑔_𝑖𝑇_𝑘,𝜃_−𝑡𝜔,𝑔₀𝜃_−𝑡𝜔²_𝐻<𝜖²3. (3.55) Next, we estimate 𝐶₃𝑀^𝑡_𝑇𝑘𝑒^{−𝜆(𝑡−𝑠)+𝐶1∫𝑡𝑠𝑧(𝜃𝜏−𝑡𝜔)𝑑𝜏}𝑔𝑠,𝜃_−𝑡𝜔,𝑔₀𝜃_−𝑡𝜔²_𝐻≤𝐶𝑑𝑠₃𝑀𝑡−𝑇_𝐾𝑒^{−𝜆𝑡+𝐶1∫𝑡0𝑧(𝜃𝑟−𝑡𝜔)𝑑𝑟}𝑔₀(𝜔)²_𝐻+𝐶₂𝐶₃𝑀^𝑡_𝑇𝐾^𝑠₀𝑒^{−2𝑧(𝜃𝜏−𝑡𝜔)−𝜆(𝑡−𝜏)+𝐶1∫𝑡𝜏𝑧(𝜃𝑟−𝑡𝜔)𝑑𝑟}≤𝐶𝑑𝜏𝑑𝑠₃𝑀𝑡−𝑇_𝐾𝑒^{−𝜆𝑡+𝐶1∫0−𝑡𝑧(𝜃𝑟𝜔)𝑑𝑟}𝑔₀(𝜔)²_𝐻+𝐶₂𝐶₃𝑀^𝑡_𝑇𝐾^s−𝑡_−𝑡𝑒^{−2𝑧(𝜃𝜏𝜔)+𝜆𝜏+𝐶1∫0𝜏𝑧(𝜃𝑟𝜔)𝑑𝑟}≤𝐶𝑑𝜏𝑑𝑠₃𝑀𝑡−𝑇_𝐾𝑒^{−𝜆𝑡+𝐶1∫0−𝑡𝑧(𝜃𝑟𝜔)𝑑𝑟}𝑔₀(𝜔)²_𝐻+𝐶₂𝐶₃𝑀𝑡−𝑇_𝐾⁰_−𝑡𝑒^{−2𝑧(𝜃𝜏𝜔)+𝜆𝜏+𝐶1∫0𝜏𝑧(𝜃𝑟𝜔)𝑑𝑟}𝑑𝜏. (3.56) By using (3.7), there exist 𝑇₂(𝜖,𝜔,𝐾(𝜔))>𝑇_𝐾(𝜔) and 𝑀₁(𝜖,𝜔,𝐾(𝜔))>0, such that if 𝑡>𝑇₂(𝜖,𝜔,𝐾(𝜔)) and 𝑀>𝑀₁(𝜖,𝜔,𝐾(𝜔)), then 𝐶₃𝑀^𝑡_𝑇𝑘𝑒^{−𝜆(𝑡−𝑠)+𝐶1∫𝑡𝑠𝑧(𝜃𝜏−𝑡𝜔)𝑑𝜏}𝑔𝑠,𝜃_−𝑡𝜔,𝑔₀𝜃_−𝑡𝜔²_𝐻𝜖𝑑𝑠≤²3. (3.57) By using (3.7) again, there exists 𝑇₃(𝜖,𝜔,𝐾(𝜔))>0, such that if 𝑇>𝑇₃(𝜖,𝜔,𝐾(𝜔)), we have 𝐶₂^𝑡_𝑇𝑘𝑒^{−𝜆(𝑡−𝑠)−2𝑧(𝜃𝑠𝜔)+𝐶1∫𝑡𝑠𝑧(𝜃𝜏𝜔)𝑑𝜏}𝜖𝑑𝑠≤²3. (3.58) Therefore, by letting 𝑇(𝜖,𝜔,𝐾(𝜔))=max𝑇₁(𝜖,𝜔,𝐾(𝜔)),𝑇₂(𝜖,𝜔,𝐾(𝜔)),𝑇₃((𝜔,𝑀((𝜔𝜖,𝜔,𝐾))𝜖,𝜔,𝐾))=𝑀₁(𝜖,𝜔,𝐾(𝜔)), (3.59) we obtain, for 𝑡>𝑇(𝜖,𝜔,𝐾(𝜔)) and 𝑀>𝑀(𝜖,𝜔,𝐾(𝜔)), _{|𝑖|≥2𝑀}𝑔_𝑖𝑡,𝜃_−𝑡𝜔,𝑔₀𝜃_−𝑡𝜔²_𝐻≤_𝑖∈ℤ𝜌|𝑖|𝑀𝑔_𝑖𝑡,𝜃_−𝑡𝜔,𝑔₀𝜃_−𝑡𝜔²_𝐻≤𝜖², (3.60) which implies _{((𝜔|𝑖|>𝑀𝜖,𝜔,𝐾))}𝜑𝑡,𝜃_−𝑡𝜔,𝜑₀𝜃_−𝑡𝜔_𝑖²_𝐻≤𝜖²,∀𝑡≥𝑇(𝜖,𝜔,𝐾(𝜔)). (3.61) The proof is completed.

Now, we have the main result.

Theorem 3.6. The random dynamical system 𝜑 generated by system (3.9) has a unique global random attractor.

Proof. It is obvious from Lemmas 3.4, and 3.5, and Proposition 2.5.

Acknowledgments

This work was supported by National Natural Science Foundation of China under Grant 11071165 and Guangxi Provincial Department of Research Project under Grant 201010LX166.