Random Attractors for the Stochastic Discrete Long Wave-Short Wave Resonance Equations

Abstract

We prove the existence of the random attractor for the stochastic discrete long wave-short wave resonance equations in an infinite lattice. We prove the asymptotic compactness of the random dynamical system and obtain the random attractor.

1. Introduction

There has been considerable progress in the study of infinite-dimensinal dynamical systems in the past few decades (see [1–5]). Recently, the dynamics of infinite lattice systems has attracted a great deal of attention from mathematicians and physicists; see [6–11] and the references therein. Various properties of solutions for lattice dynamical systems (LDSs) have been extensively investigated. For example, the long-time behavior of LDSs was studied in [5, 10]. Lattice dynamical systems play an important role in their potential application such as biology, chemical reaction, pattern recognition and image processing, electrical engineering, and laser systems. However, a system in reality is usually affected by external perturbations within many cases that are of great uncertainty or random influence. These random effects are introduced not only to compensate for the defects in some deterministic models but also to explain the intrinsic phenomena. Therefore, there is much work concerning stochastic lattice dynamical systems. The study of random attractors gained considerable attention during the past decades; see [12] for a comprehensive survey. Bates et al. [13] first investigated the existence of global random attractor for a kind of first-order dynamic systems driven by white noise on lattice ℤ; then, Lv and Sun [14] extended the results of Bates to the dimensional lattices. Stochastic complex Ginzburg-Landau equations, FitzHugh-Nagumo equation, and KGS equations in an infinite lattice are studied by Lv and Sun [15], Huang [16], and Yan et al. [17], respectively.

The long wave-short wave (LS) resonance system is an important model in nonlinear science. Long wave-short wave resonance equations arise in the study of the interaction of surface waves with both gravity and capillary modes present and also in the analysis of internal waves as well as Rossby waves as in [18]. In the plasma physics they describe the resonance of the high-frequency electron plasma oscillation and the associated low-frequency ion density perturbation in [19].

Due to their rich physical and mathematical properties the long wave-short wave resonance equations have drawn much attention of many physicists and mathematicians. The LS system is as follows: 𝑖𝑢_𝑡+𝑢_𝑥𝑥−𝑢𝑣+𝑖𝛼𝑢=𝑓(𝑥,𝑡),𝑣_𝑡+𝛾|𝑢|²_𝑥+𝛽𝑣=𝑔(𝑥,𝑡),𝑥∈ℝ,𝑡≥0, (1.1) where 𝑢 denotes a complex-valued vector and 𝑣 represents a real-valued function; 𝑓(𝑥,𝑡) and 𝑔(𝑥,𝑡) are given complex- and real-valued functions, respectively. The constants 𝛼,𝛽 are positive, and 𝛾∈ℝ⧵{0} is real. For the LS equations, Boling Guo obtained the global solution in [20]. The existence of global attractor was studied in [21–23].

Throughout this paper, we set 𝕃²=𝑢=𝑢_{𝑛𝑛∈ℤ},𝑢_𝑛∈ℂ∶_𝑛∈ℤ𝑢_𝑛²<∞,ℓ²=𝑣=𝑣_{𝑛𝑛∈ℤ},𝑣_𝑛∈ℝ∶_𝑛∈ℤ𝑣²_𝑛<∞. (1.2) For brevity, we use 𝐻 to denote Hilbert space 𝕃² or ℓ² and equip 𝐻 with the inner product and norm as (𝑢,𝑣)=_𝑛∈ℤ𝑢_𝑛𝑣_𝑛,‖𝑢‖²=(𝑢,𝑢)=_𝑛∈ℤ|𝑢|²,∀𝑢,𝑣∈𝐻, (1.3) where 𝑣_𝑛 denotes the conjugate of 𝑣_𝑛.

In this paper, we consider the following stochastic discrete LS equations 𝑖𝑑𝑢_𝑛−()𝑑𝑡𝐴𝑢_𝑛−𝑣_𝑛𝑢_𝑛+𝑖𝛼𝑢_𝑛=𝑓_𝑛(𝑡)+𝑎_𝑛𝑢_𝑛𝑑𝑤¹_𝑛𝑑𝑡,𝑛∈ℤ,𝑡>0,𝑑𝑣_𝑛𝐵|𝑢|𝑑𝑡+𝛾²_𝑛+𝛽𝑣_𝑛=𝑔_𝑛(𝑡)+𝑏_𝑛𝑑𝑤²_𝑛𝑑𝑡,𝑛∈ℤ,𝑡>0, (1.4)(1.5) with the initial conditions𝑢_𝑛(0)=𝑢_𝑛0,𝑣_𝑛(0)=𝑣_𝑛0,𝑛∈ℤ, (1.6) where |𝑢|²=(|𝑢_𝑛|²)_𝑛∈ℤ, 𝑢_𝑛(𝑡)∈ℂ, 𝑣_𝑛(𝑡)∈ℝ (ℂ,ℝ are the sets of complex and real numbers, resp.), 𝑎=(𝑎_𝑛)_𝑛∈ℤ∈ℓ² and 𝑏=(𝑏_𝑛)_𝑛∈ℤ∈ℓ², 𝑓(𝑡)=(𝑓_𝑛(𝑡))_𝑛∈ℤ, 𝑔(𝑡)=(𝑔_𝑛(𝑡))_𝑛∈ℤ∈𝐶_𝐵(ℝ,ℓ²), the space of bounded continuous functions from ℝ into ℓ². {𝑤¹_𝑛(𝑡)∶𝑛∈ℤ} and {𝑤²_𝑛(𝑡)∶𝑛∈ℤ} are two independent two-side real-valued standard Wiener process. ℤ is the integer set, 𝑖 is the unit of the imaginary numbers such that 𝑖²=−1, and 𝐴,𝐵 are linear operators defined, respectively, by(𝐴𝑢)_𝑛=𝑢_𝑛+1+𝑢_𝑛−1−2𝑢_𝑛,𝑛∈ℤ,∀𝑢=𝑢_{𝑛𝑛∈ℤ},(𝐵𝑢)_𝑛=𝑢_𝑛+1−𝑢_𝑛,𝑛∈ℤ,∀𝑢=𝑢_{𝑛𝑛∈ℤ}. (1.7) In addition, simple computation shows that, for 𝑢=(𝑢_𝑛)_𝑛∈ℤ∈𝐻, there holds‖𝐵𝑢‖²=_𝑛∈ℤ𝑢_𝑛+1−𝑢_𝑛²≤2_𝑛∈ℤ𝑢_𝑛+1²+𝑢_𝑛²=4‖𝑢‖². (1.8)

This paper is organized as follows. In the next section, we recall some basic concepts and already know results to random dynamical system and random attractors. In Section 3, we prove the existence of the global random attractor for stochastic LS lattice dynamical systems (1.4)–(1.6).

2. Preliminaries

In this section, we first introduce the definitions of the random dynamical systems and random attractor, which are taken from [13]. Let (𝐻,‖⋅‖_𝐻) be a Hilbert space and (Ω,ℱ,ℙ) a probability space.

Definition 2.1. (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) are called metric dynamical systems; if 𝜃∶ℝ×Ω→Ω is (𝔹(ℝ)×ℱ,ℱ)-measurable, 𝜃₀=𝕀, 𝜃_𝑡+𝑠=𝜃_𝑡∘𝜃_𝑠 for all 𝑡,𝑠∈ℝ, and 𝜃_𝑡ℙ=ℙ for all 𝑡∈ℝ.

Definition 2.2. A stochastic process 𝜙(𝑡,𝜔) is called a continuous random dynamical system (RDS) over (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) if 𝜙 is (𝔹(ℝ⁺)×ℱ×𝔹(𝐻),𝔹(𝐻))-measurable, and for all 𝜔∈Ω(i)the mapping 𝜙∶ℝ⁺×Ω×𝐻→𝐻 is continuous;(ii)𝜙(0,𝜔)=𝕀on𝐻; (iii)𝜙(𝑡+𝑠,𝜔,𝜒)=𝜙(𝑡,𝜃_𝑠𝜔,𝜙(𝑠,𝜔,𝜒)) for all 𝑡, 𝑠≥0 and 𝜒∈𝐻 (cocycle property).

Definition 2.3. A random bounded set 𝐵(𝜔)⊂𝐻 is called tempered with respect to (𝜃_𝑡)_𝑡∈ℝ if for a.e. 𝜔∈Ω and all 𝜖>0lim_𝑡→∞𝑒^−𝜖𝑡𝑑𝐵𝜃_−𝑡𝜔=0, (2.1) where 𝑑(𝐵)=sup_𝜒∈𝐵‖𝜒‖_𝐻.

Consider a continuous random dynamical system 𝜙(𝑡,𝑤) over (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ), and let 𝔻 be the collection of all tempered random sets of 𝐻.

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 𝑡_𝐵(𝜔)>0 such that 𝜙𝑡,𝜃_−𝑡𝜔,𝐵𝜃_−𝑡𝜔⊂𝕂(𝜔),𝑡≥𝑡_𝐵(𝜔). (2.2)

Definition 2.5. A random set 𝔸(𝜔) is a random 𝔻-attractor for RDS 𝜙 if (i)𝔸(𝜔) is a random compact set, that is, 𝜔→𝑑(𝜒,𝔸(𝜔)) is measurable for every 𝜒∈𝐻 and 𝔸(𝜔) is compact for a.e. 𝜔∈Ω;(ii)𝔸(𝜔) is strictly invariant, that is, 𝜙(𝑡,𝜔,𝔸(𝜔))=𝔸(𝜃_𝑡𝜔),forall𝑡≥0 and for a.e. 𝜔∈Ω;(iii)𝔸(𝜔) attracts all sets in 𝔻, that is, for all 𝐵∈𝔻 and a.e. 𝜔∈Ω we have lim_𝑡→∞𝑑𝜙𝑡,𝜃_−𝑡𝜔,𝐵𝜃_−𝑡𝜔𝔸(𝜔)=0, (2.3) where 𝑑(𝑋,𝑌)=sup_𝜒∈𝑋inf_𝑦∈𝑌‖𝜒−𝑦‖_𝐻,𝑋,𝑌⊂𝐻.

The collection 𝔻 is called the domain of attraction of 𝔸.

Definition 2.6. Let 𝜙 be an RDS on Hilbert space 𝐻. 𝜙 is called asymptotically compact if, for any bounded sequence {𝜒_𝑛}⊂𝐻 and 𝑡_𝑛→∞, the set {𝜙(𝑡_𝑛,𝜃_−𝑡𝑛𝜔,𝜒_𝑛)} is precompact in 𝐻, for any 𝜔∈Ω.

From [13], we have the following result.

Proposition 2.7. Let 𝕂∈𝔻 be an absorbing set for an asymptotically compact continuous RDS 𝜙. Then 𝜙 has a unique global random 𝔻-attractor 𝔸(𝜔)=_{𝜅≥𝑡𝕂(𝜔)𝑡≥𝜅}𝜙𝑡,𝜃_−𝑡𝜔,𝐾𝜃_−𝑡𝜔 (2.4) which is compact in 𝐻.

Let 𝑊¹(𝑡)=∑_𝑛∈ℤ𝑎_𝑛𝑤¹_𝑛(𝑡)𝑒_𝑛 and 𝑊²(𝑡)=∑_𝑛∈ℤ𝑏_𝑛𝑤²_𝑛(𝑡)𝑒_𝑛, where (𝑎_𝑛)_𝑛∈ℤ,(𝑏_𝑛)_𝑛∈ℤ∈ℓ². Here {𝑒_𝑛} denotes the standard complete orthonormal system in ℓ², which means that the 𝑛th component of 𝑒_𝑛 is 1 and all other elements are 0. Then 𝑊¹(⋅) and 𝑊²(⋅) are ℓ²-valued 𝑄-Wiener processes. It is obvious that 𝐸𝑊¹(𝑡)=𝐸𝑊²(𝑡)=0. For details we refer to [24].

Now, we abstract (1.4)–(1.6) as stochastic ordinary differential equations with respect to time 𝑡 in 𝐸=𝕃²×ℓ². Let 𝑎=(𝑎_𝑛)_𝑛∈ℤ,𝑏=(𝑏_𝑛)_𝑛∈ℤ,𝑓=(𝑓_𝑛)_𝑛∈ℤ, and 𝑔=(𝑔_𝑛)_𝑛∈ℤ. Then (1.4)–(1.6) can be written as the following integral equations:𝑢(𝑡)=𝑢₀+^𝑡₀−𝑖𝑢(𝑠)𝑣(𝑠)−𝛼𝑢(𝑠)−𝑖𝐴𝑢(𝑠)−𝑖𝑓(𝑠)𝑑𝑠−𝑖^𝑡₀𝑢(𝑠)𝑑𝑊¹,𝑣(𝑡)=𝑣₀+^𝑡₀−𝛽𝑣(𝑠)−𝛾𝐵|𝑢|²+𝑔(𝑠)𝑑𝑠+𝑊². (2.5)(2.6)

Remark 2.8. The special form of multiplicative noise in (2.5) is more suitable than the white noise “𝑎𝑑𝑊” and the additive noise “∑^𝑛_𝑗=1𝑎_𝑗𝑑𝑊^𝑗”, because it is more approximative to the perturbations of the short wave for this model.

For our purpose we introduce the probability space as Ω=𝜔∈𝐶ℝ,ℓ²∶𝜔(0)=0 (2.7) endowed with the compact open topology [12]. ℙ is the corresponding Wiener measure, and ℱ is the ℙ-completion of the Borel 𝜎-algebra on Ω.

Let 𝜃_𝑡𝜔(⋅)=𝜔(⋅+𝑡)−𝜔(𝑡),𝑡∈ℝ. Then (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) is a metric dynamical system with the filtration ℱ_𝑡∶=∨_𝑠≤𝑡ℱ^𝑡_𝑠,𝑡∈ℝ, where ℱ^𝑡_𝑠=𝜎{𝑊(𝑡₂)−𝑊(𝑡₁)∶𝑠≤𝑡₁≤𝑡₂≤𝑡} is the smallest 𝜎-algebra generated by the random variable 𝑊(𝑡₂)−𝑊(𝑡₁) for all 𝑡₁,𝑡₂ such that 𝑠≤𝑡₁≤𝑡₂≤𝑡; see [12] for more details.

3. The Existence of a Random Attractor

In this section, we study the dynamics of solutions for the stochastic LS (1.4)–(1.6). Then we apply Proposition 2.7 to prove the existence of a global random attractor for stochastic lattice LS equations.

Before proving the existence of global solution for (2.5)-(2.6), we need the following a priori estimate.

Lemma 3.1. Suppose that 𝑓(𝑡)=(𝑓_𝑛(𝑡))_𝑛∈ℤ∈𝐶_𝐵(ℝ,ℓ²). Then, the solution of (1.4)–(1.6) satisfies 𝑢𝑡,𝜔;𝑢₀²≤𝑒^−𝛼𝑡𝑢₀²+1𝛼𝑓²,𝑡≥0, (3.1) for all 𝜔∈Ω and ‖𝑓‖=sup_𝑡∈ℝ|𝑓(𝑡)|².

Proof. We write (1.4) in the form of vector as 𝑖𝑢_𝑡−𝐴𝑢−𝑣𝑢+𝑖𝛼𝑢=𝑓+𝑢𝑊¹_𝑡,𝑡>0. (3.2) Taking the imaginary part of the inner product of (3.2) with 𝑢, we obtain 12𝑑‖𝑢‖𝑑𝑡²+𝛼‖𝑢‖²≤𝛼=−Im𝑓,𝑢2‖𝑢‖²+1𝑓2𝛼². (3.3) So we have 𝑑𝑢𝑑𝑡𝑡,𝜔;𝑢₀²+𝛼𝑢𝑡,𝜔;𝑢₀²≤1𝛼𝑓². (3.4) By the Gronwall inequality, we get 𝑢𝑡,𝜔;𝑢₀²≤𝑒^−𝛼𝑡𝑢₀²+1𝛼𝑓². (3.5) Thus, we derive (3.1).

By Lemma 3.1, we know that ‖𝑢‖² is bounded in any bounded subset of [0,∞), that is, ‖𝑢‖²≤𝑒^−𝛼𝑡‖𝑢₀‖²+(1/𝛼)‖𝑓‖²,0≤𝑡≤𝑇, for any fixed constant 𝑇>0.

In order to show the existence of global solutions of (2.5)-(2.6), we first change (2.5)-(2.6) into deterministic equations. First, due to special linear multiplicative noise, (2.5) can be reduced to an equation with random coefficients by a suitable change of variable. Consider the process 𝑧(𝑡)=𝑒^{𝑖𝑊1(𝑡)}, which satisfies the stochastic differential equation (𝑡)1𝑑𝑧=−2𝑧(𝑡)𝑑𝑡+𝑖𝑧(𝑡)𝑑𝑊¹. (3.6) The process 𝑢=𝑧(𝑡)𝑢(𝑡) follows the random differential equation, 𝑖𝑑𝑢1𝑑𝑡−𝐴𝑢+𝑖𝛼+2𝑢−𝑣𝑢−𝑓(𝑡)𝑧(𝑡)=0. (3.7)

We denote 𝑣=𝑣(𝑡)−𝑊²(𝑡), then (2.5)-(2.6) can be changed into the following equations:𝑢(𝑡)=𝑢₀+^𝑡₀𝑢(𝑠)𝑣(𝑠)−1−𝑖𝛼+2𝑢(𝑠)𝑢(𝑠)(𝑠)𝑧(𝑠𝑣(𝑡)=𝑣−𝑖𝐴−𝑖𝑓)𝑑𝑠,₀+^𝑡₀−𝛽𝑣(𝑠)−𝛾𝐵|𝑢|²+𝑔(𝑠)−𝛽𝑊²(𝑠)𝑑𝑠. (3.8)

Remark 3.2. For the general multiplicative noise, we can also choose a suitable process and a change of variable to convert the stochastic equations into deterministic equations.

For each fixed 𝜔∈Ω, (3.8) are deterministic equations, and we have the following result.

Theorem 3.3. For any 𝑇>0, (2.5)-(2.6) are well posed and admit a unique solution (𝑢(𝑡),𝑣(𝑡))∈𝕃²(Ω;𝐶([0,𝑇];𝐸)). Moreover, the solution of (2.5)-(2.6) depends continuously on the initial data (𝑢₀,𝑣₀).

Proof. By standard existence theorem for ODEs, it follows that (3.8) possess a local solution (𝑢(𝑡),𝑣(𝑡))∈𝐶(0,𝑇;𝐸), where [0,𝑇_max] is the maximal interval of existence of the solution of (3.8). Now, we prove that this local solution is a global solution. Let 𝜔∈Ω; from (3.8) it follows that ‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²=𝑢₀²+𝑣₀²1−2𝛼+2^𝑡₀‖𝑢(𝑠)‖²𝑑𝑠−2^𝑡₀Re𝑖𝑓𝑧,𝑢(𝑠)𝑑𝑠−2𝛽^𝑡₀‖𝑣(𝑠)‖²𝑑𝑠−2𝛾^𝑡₀𝐵|𝑢|²,𝑣(𝑠)𝑑𝑠+2^𝑡₀𝑔(𝑠),𝑣(𝑠)𝑑𝑠−2𝛽^𝑡₀𝑊²(𝑠),𝑣(𝑠)𝑑𝑠. (3.9)
By the Young inequality and (1.8), direct computation shows that 𝑢≤2−2Re𝑖𝑓𝑧,𝛼𝑓²‖𝑧‖²+𝛼2‖𝑢‖²≤1𝛼𝑓⁴+𝑊¹⁴+𝛼2‖𝑢‖²,−2𝛾𝐵|𝑢|²,𝑣≤2𝛾𝐵|𝑢|²‖𝑣‖≤𝛾𝜀‖𝑣‖²+4𝛾𝜀‖𝑢‖⁴,2𝑔,𝑣≤𝛾𝜀‖𝑣‖²+1𝛾𝜀𝑔²,−2𝛽𝑊²,𝑣≤2𝛽𝑊²‖𝑣‖≤𝛾𝜀‖𝑣‖²+𝛽𝛾𝜀𝑊²². (3.10)
Combining the above inequalities with Lemma 3.1, we obtain ‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²≤𝑢₀²+𝑣₀²−𝐶₀^𝑡₀‖𝑢(𝑠)‖²+‖𝑣(𝑠)‖²𝑑𝑠+𝐶₁^𝑡₀𝑓²+𝑔²+𝑊²²+𝑊¹⁴≤𝑢𝑑𝑠₀₂+𝑣₀²+𝐶₁^𝑡₀𝑓²+𝑔²+𝑊²²+𝑊¹⁴𝑑𝑠, (3.11) where 𝐶₀,𝐶₁ are constants depending on 𝛼,𝛽,𝛾, and 𝜀 is a sufficiently small positive number. By the Gaussian property of 𝑊¹ and 𝑊², (3.11) implied that (3.8) admit a global solution (𝑢(𝑡),𝑣(𝑡))∈𝕃²(Ω;𝐶([0,𝑇];𝐸)). The proof of the lemma is completed.

From the definition (𝜃_𝑡)_𝑡∈ℝ, we know 𝑊(𝑡+ℎ,𝜔)=𝑊𝑡,𝜃_ℎ𝜔+𝑊(ℎ,𝜔),∀𝑡,ℎ∈ℝ, (3.12) and combining the above theorem we have the following result.

Theorem 3.4. System (2.5)-(2.6) generates a continuous random dynamical system (𝜙(𝑡,𝜃_−𝑡𝜔))_𝑡≥0 over (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ).

The proof is similar to that of Theorem  3.2 in [13], so we omit it.

Now, we prove the existence of a random attractor for system (2.5)-(2.6). By Proposition 2.7, we first prove that RDS 𝜙 possesses a bounded absorbing set 𝕂(𝜔). We introduce an Ornstein-Uhlenbeck process in ℓ² on the metric dynamical system (Ω,ℱ,ℙ,𝜃_𝑡) given by Wiener process: 𝑧¹𝜃_𝑡𝜔=−𝜈⁰_−∞𝑒^𝜈ℎ𝜃_𝑡𝜔(ℎ)𝑑ℎ,𝑡∈ℝ,𝑧²𝜃_𝑡𝜔=−𝜇⁰_−∞𝑒^𝜇ℎ𝜃_𝑡𝜔(ℎ)𝑑ℎ,𝑡∈ℝ, (3.13) where 𝜈 and 𝜇 are positive. The above integral exists in the sense of for any path 𝜔 with a subexponential growth and 𝑧¹,𝑧² solve the following Itô equations: 𝑑𝑧¹+𝜈𝑧¹𝑑𝑡=𝑑𝑊¹(𝑡),𝑑𝑧²+𝜇𝑧²𝑑𝑡=𝑑𝑊²(𝑡). (3.14)

In fact, the mapping 𝑡→𝑧^𝑖(𝜃_𝑡𝜔),𝑖=1,2, is the 𝑂−𝑈 process. Furthermore, there exists a 𝜃_𝑡 invariant set Ω′⊂Ω of full ℙ measure such that (1)the mapping 𝑡→𝑧^𝑖(𝜃_𝑡𝜔),𝑖=1,2, is continuous for each 𝜔∈Ω^′,(2)the random variables ‖𝑧^𝑖(𝜔)‖,𝑖=1,2, are tempered.

Lemma 3.5. There exists a 𝜃_𝑡 invariant set Ω′⊂Ω of full ℙ measure and an absorbing random set 𝕂(𝜔),𝜔∈Ω′, for the random dynamical system (𝜙(𝑡,𝜃_−𝑡𝜔))_𝑡≥0.

Proof. We use the estimates in Theorem 3.3. By (3.11), we have 𝑑‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²𝑑𝑡≤−𝐶₀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²+𝐶₁𝜌𝜃_𝑡𝜔, (3.15) where 𝜌(𝜃_𝑡𝜔)=‖𝑓‖²+‖𝑔‖²+‖𝑊¹‖⁴+‖𝑊²‖².
By the Gronwall inequality, we have ‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²≤𝑢₀²+𝑣₀²𝑒^−𝐶0𝑡+^𝑡₀𝑒^{−𝐶0(𝑡−𝑠)}𝜌𝜃_𝑠𝜔𝑑𝑠. (3.16) Replace 𝜔 by 𝜃_−𝑡𝜔 in the above inequality to construct the radius of the absorbing set and define 𝜚²(𝜔)=4lim_𝑡→∞^𝑡₀𝑒^{−𝐶0(𝑡−𝑠)}𝜌𝜃_𝑠−𝑡𝜔𝑑𝑠=4lim_𝑡→∞⁰_−𝑡𝑒^−𝐶0𝑠𝜌𝜃_𝑠𝜔𝑑𝑠. (3.17) Define 𝑅²(𝜔)=𝜚²(𝜔)+4𝛼𝑓²𝑊¹⁴+𝑊²². (3.18) Then, 𝕂(𝜔)≜𝕂(0,𝑅(𝜔)) is a tempered ball by the property of 𝑊¹,𝑊², and, for any 𝐵∈𝔻,𝜔∈Ω. Here, 𝔻 denotes the collection of all tempered random sets of Hilbert space 𝐻. The proof of the lemma is completed.

Lemma 3.6. Let (𝑢₀,𝑣₀)∈𝕂(𝜔), the absorbing set given in Lemma 3.5. Then, for every 𝜀>0 and ℙ-a.e. 𝜔∈Ω, there exist 𝑇(𝜀,𝜔)>0 and 𝑁(𝜀,𝜔)>0 such that the solution (𝑢,𝑣) of system (2.5)-(2.6) satisfies _{|𝑛|>𝑁(𝜀,𝜔)}𝑢_𝑛𝑡,𝜃_−𝑡𝜔²+𝑣_𝑛𝑡,𝜃_−𝑡𝜔²≤𝜀,∀𝑡≥𝑇(𝜀,𝜔). (3.19)

Proof. Let 𝜂(𝑥)∈𝐶(ℝ⁺,[0,1]) be a cut-off function satisfying 𝜂(𝑥)=[),[],1,∀𝑥∈2,+∞0,∀𝑥∈0,1 (3.20) and |𝜂′(𝑥)|≤𝜂₀ (a positive constant).
Taking the inner product of (3.8) with (𝜂(|𝑛|/𝑀)𝑢_𝑛)_𝑛∈ℤ and (𝜂(|𝑛|/𝑀)𝑣_𝑛)_𝑛∈ℤ, respectively, we get 𝑑𝑑𝑡_𝑛∈ℤ𝜂|𝑛|𝑀𝑢_𝑛²1=−2𝛼+2_𝑛∈ℤ𝜂|𝑛|𝑀𝑢_𝑛²−2Im_𝑛∈ℤ𝜂|𝑛|𝑀𝑓_𝑛𝑧_𝑛,𝑢_𝑛,𝑑𝑑𝑡_𝑛∈ℤ𝜂|𝑛|𝑀𝑣_𝑛²=−2𝛽_𝑛∈ℤ𝜂|𝑛|𝑀𝑣_𝑛²−2𝛾_𝑛∈ℤ𝜂|𝑛|𝑀𝐵|𝑢|²_𝑛,𝑣_𝑛+2_𝑛∈ℤ𝜂|𝑛|𝑀𝑔_𝑛,𝑣_𝑛−2𝛽_𝑛∈ℤ𝜂|𝑛|𝑀𝑊²_𝑛,𝑣_𝑛. (3.21)
We also use the estimates in Theorem 3.3. Similar to (3.11), it follows that _𝑛∈ℤ𝜂|𝑛|𝑀𝑢_𝑛²+𝑣_𝑛²≤𝑒^{−𝐶0(𝑡−𝑇𝑘)}_𝑛∈ℤ𝜂|𝑛|𝑀𝑢_𝑛𝑇_𝑘,𝜔²+_𝑛∈ℤ𝜂|𝑛|𝑀𝑣_𝑛𝑇_𝑘,𝜔²+𝐶₁^𝑡_𝑇𝑘𝑒^{𝐶0(𝑠−𝑡)}_{|𝑛|≥𝑀}𝑧¹_𝑛𝜃_𝑠𝜔²+𝑧²_𝑛𝜃_𝑠𝜔²+𝑓_𝑛²+𝑔_𝑛²𝑑𝑠. (3.22)
Replace 𝜔 by 𝜃_−𝑡𝜔 in (3.22). Then, we estimate each term on the right-hand of (3.22); it follows that 𝑒^{−𝐶0(𝑡−𝑇𝑘)}_𝑛∈ℤ𝜂|𝑛|𝑀𝑢_𝑛𝑇_𝑘,𝜃_−𝑡𝜔²+𝑣_𝑛𝑇_𝑘,𝜃_−𝑡𝜔²≤𝑒^{−𝐶0(𝑡−𝑇𝑘)}𝑢₀𝜃_−𝑡𝜔²+𝑣₀𝜃_−𝑡𝜔²𝑒^{−𝐶0𝑇𝑘}+^𝑇𝑘₀𝑒^{−𝐶0(𝑇𝑘−𝑠)}𝑧¹𝜃_𝑠−𝑡𝜔⁴+𝑧²𝜃_𝑠−𝑡𝜔²𝑑𝑠+𝑒^{−𝐶0(𝑡−𝑇𝑘)𝑇𝑘}₀𝑒^{−𝐶0(𝑇𝑘−𝑠)}𝑓²+𝑔²𝑑𝑠. (3.23) Since ‖𝑧^𝑖(𝜔)‖,𝑖=1,2, are tempered and 𝑧^𝑖(𝜃_𝑡𝜔),𝑖=1,2, are continuous in 𝑡, there is a tempered function 𝑟(𝜔)>0 such that 𝑧¹(𝜃_𝑡𝜔)⁴+𝑧²(𝜃_𝑡𝜔)²≤𝑟𝜃_𝑡𝜔. (3.24) Combining (3.23) with (3.24), there is a 𝑇₁(𝜀,𝜔)>𝑇_𝑘 such that 𝑒^{−𝐶0(𝑡−𝑇𝑘)}_𝑛∈ℤ𝜂|𝑛|𝑀𝑢_𝑛𝑇_𝑘,𝜃_−𝑡𝜔²+𝑣_𝑛𝑇_𝑘,𝜃_−𝑡𝜔²≤𝜀3. (3.25) Next, we estimate 𝐶₁^𝑡_𝑇𝑘𝑒^{𝐶0(𝑠−𝑡)}_{|𝑛|≥𝑀}𝑧¹_𝑛(𝜃_𝑠−𝑡𝜔)⁴+𝑧²_𝑛𝜃_𝑠−𝑡𝜔²𝑑𝑠. (3.26) Let 𝑇^∗≥(1/𝐶₀)ln(6𝐶₁𝑟(𝜔)/𝐶₀𝜀) and 𝑁₁(𝜀,𝜔) be fixed positive constants. Then, for 𝑡>𝑇^∗+𝑇_𝑘 and 𝑀>𝑁₁(𝜀,𝜔), by the Lebesgue theorem, we have 𝐶₁^𝑡_𝑇𝑘𝑒^{𝐶0(𝑠−𝑡)}_{|𝑛|≥𝑀}𝑧¹_𝑛𝜃_𝑠−𝑡𝜔⁴+𝑧²_𝑛𝜃_𝑠−𝑡𝜔²𝑑𝑠=𝐶₁⁰_−𝑇∗𝑒^𝐶0𝜉_{|𝑛|≥𝑀}𝑧¹_𝑛𝜃_𝜉𝜔⁴+𝑧²_𝑛𝜃_𝜉𝜔²𝑑𝑠+𝐶₁^−𝑇∗_{𝑇𝑘−𝑡}𝑒^𝐶0𝜉_{|𝑛|≥𝑀}𝑧¹_𝑛𝜃_𝜉𝜔⁴+𝑧²_𝑛𝜃_𝜉𝜔²𝑑𝑠≤𝐶₁⁰_−𝑇∗𝑒^𝐶0𝜉_{|𝑛|≥𝑀}𝑧¹_𝑛𝜃_𝜉𝜔⁴+𝑧²_𝑛𝜃_𝜉𝜔²𝐶𝑑𝑠+¹𝐶₀𝑟(𝜔)𝑒^𝐶0𝑇∗≤𝜀6+𝜀6=𝜀3. (3.27)
Since 𝑓(𝑡)∈𝐶_𝐵(ℝ,ℓ²) and 𝑔(𝑡)∈𝐶_𝐵(ℝ,ℓ²), there exists 𝑁₂(𝜀,𝜔) such that for 𝑀>𝑁₂(𝜀,𝜔)𝐶₁^𝑡_𝑇𝑘𝑒^{𝐶0(𝑠−𝑡)}_{|𝑛|≥𝑀}𝑓_𝑛²+𝑔_𝑛²𝜀𝑑𝑠≤3. (3.28)
Therefore, let 𝑇(𝜀,𝜔)=max𝑇₁(𝜀,𝜔),𝑇^∗(𝜀,𝜔),𝑁(𝜀,𝜔)=max𝑁₁(𝜀,𝜔),𝑁₂(𝜀,𝜔). (3.29)
Then, for 𝑡>𝑇(𝜀,𝜔) and 𝑀>𝑁(𝜀,𝜔), we obtain _|𝑛|>𝑀𝑢_𝑛𝑡,𝜃_𝑡𝜔²+𝑣_𝑛𝑡,𝜃_𝑡𝜔²≤𝜀. (3.30) Direct computation shows that ‖𝑢‖²+‖𝑣‖²≤2‖𝑢‖²+‖𝑣‖²+4𝑧²𝜃_𝑡𝜔². (3.31) Therefore, we obtain _|𝑛|>𝑀𝑢_𝑛𝑡,𝜃_𝑡𝜔²+𝑣_𝑛𝑡,𝜃_𝑡𝜔²≤𝜀. (3.32) The proof of the lemma is completed.

Lemma 3.7. The random dynamical systems (𝜙(𝑡,𝜃_−𝑡𝜔))_𝑡≥0 are asymptotically compact.

Proof. We use the method of [25]. Let 𝜔∈Ω. Consider a sequence (𝑡_𝑛)_𝑛∈ℕ with 𝑡_𝑛→∞ as 𝑛→∞. Since 𝕂(𝜔) is a bounded absorbing set, for large 𝑛, (𝑢_𝑛,𝑣_𝑛)=𝜙(𝑡_𝑛,𝜃_−𝑡𝑛𝜔)(𝑢₀,𝑣₀)∈𝕂(𝜔), where (𝑢₀,𝑣₀)∈𝕂(𝜔). Then, there exist (𝑢,𝑣)∈𝐸 and a sequence (𝑢_𝑛,𝑣_𝑛) (denoted by itself) such that 𝑢_𝑛,𝑣_𝑛⇀(𝑢,𝑣)weaklyin𝐸. (3.33)
Next, we show that the above weak convergence is actually strong convergence in 𝐸.
From Lemma 3.6, for any 𝜀>0, there exist positive constants 𝑁₃(𝜀,𝜔) and 𝑀₁ such that, for 𝑛>𝑀₁, _𝑖>𝑁3𝑢_𝑖𝑛𝑡_𝑛,𝜃_𝑡𝑛𝜔²+𝑣_𝑖𝑛𝑡_𝑛,𝜃_𝑡𝑛𝜔²≤𝜀6. (3.34)
Since (𝑢,𝑣)∈𝐸, there exists 𝑁₄(𝜀,𝜔)>0 such that _{|𝑖|≥𝑁4}𝑢_𝑖²+𝑣_𝑖²≤𝜀6. (3.35)
Let 𝑁(𝜀,𝜔)=max{𝑁₃(𝜀,𝜔),𝑁₄(𝜀,𝜔)}, then, from (3.33), there exists 𝑀₂>0 such that, for 𝑛>𝑀₂, _|𝑖|≤𝑁𝑢_𝑖𝑛−𝑢²+𝑣_𝑖𝑛−𝑣²≤𝜀3. (3.36)
By (3.34)–(3.36), we obtain that, for 𝑛>𝑀=max{𝑀₁,𝑀₂}, 𝑢_𝑛(𝑡_𝑛,𝜃_−𝑡𝑛𝜔)−𝑢²+𝑣_𝑛(𝑡_𝑛,𝜃_−𝑡𝑛𝜔)−𝑣²=_|𝑖|≤𝑁𝑢_𝑖𝑛𝑡_𝑛,𝜃_−𝑡𝑛𝜔−𝑢_𝑖²+𝑣_𝑖𝑛𝑡_𝑛,𝜃_−𝑡𝑛𝜔−𝑣_𝑖²+_|𝑖|>𝑁𝑢_𝑖𝑛𝑡_𝑛,𝜃_−𝑡𝑛𝜔−𝑢_𝑖²+𝑣_𝑖𝑛𝑡_𝑛,𝜃_−𝑡𝑛𝜔−𝑣_𝑖²≤_|𝑖|≤𝑁𝑢_𝑖𝑛𝑡_𝑛,𝜃_−𝑡𝑛𝜔−𝑢_𝑖²+𝑣_𝑖𝑛𝑡_𝑛,𝜃_−𝑡𝑛𝜔−𝑣_𝑖²+2_|𝑖|>𝑁𝑢_𝑖𝑛𝑡_𝑛,𝜃_−𝑡𝑛𝜔²+𝑣_𝑖𝑛𝑡_𝑛,𝜃_−𝑡𝑛𝜔²+2_|𝑖|>𝑁𝑢_𝑖²+𝑣_𝑖²≤𝜀3+2𝜀6+2𝜀6≤𝜀. (3.37) The proof of the lemma is completed.

Now, combining Lemmas 3.5, 3.7 with Proposition 2.7, we can easily obtain the following result.

Theorem 3.8. The random dynamical systems (𝜙(𝑡,𝜃_−𝑡𝜔))_𝑡≥0 possess a global random attractor in 𝐸.

Acknowledgments

The authors are grateful to the anonymous referees for their careful reading of the paper and precious comments. The authors also thank the editors for their kind help. They were supported by the NSF of China (nos. 11071162, 11001116), the NSF of Shandong Province, the Project of Shandong Province Higher Educational Science and Technology Program (nos. J10LA09, J11LA03), and the Project of Discipline Construction Foundation in Ludong University.