Uniformly Random Attractor for the Three-Dimensional Stochastic Nonautonomous Camassa-Holm Equations

Abstract

We consider the uniformly random attractor for the three-dimensional stochastic nonautonomous Camassa-Holm equations in the periodic box [0,𝑙]³ in this paper. We associate with the concepts of uniform attractor and random attractor and produce the concept of uniformly random attractor for a process. Then we establish the existence of the uniformly random attractor in 𝐷(𝐴^1/2) and 𝐷(𝐴) for the equations.

1. Introduction

The Camassa-Holm equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥=0 (1) models the unidirectional propagation of shallow water waves over a flat bottom [1–4]. It has been paid a large number of attentions due to its rich nonlinear phenomenology. It is completely integrable [1], and it has stable solutions [5]. It possesses the peakons 𝑢=𝑒^{|𝑥−𝑐𝑡|} which has been proved stable [6, 7]. It has been shown that (1) is locally well-posed for initial data 𝑢₀∈𝐻^𝑠(ℝ)(𝑠>3/2) [8, 9]. There are a rich variety of global solutions and blow-up solutions obtained in [8–10]. The global existence of weak solutions, conservative solutions, and disispative solutions was investigated in [6, 11, 12].

Following the Camassa-Holm (1), some generalized types of the equation have been deeply considered by many authors, for instance [13–18]. The authors in [19] considered the three-dimensional Camassa-Holm equations subject to periodic boundary conditions: 𝜕𝛼𝜕𝑡²⁰𝑢−𝛼²₁Δ𝑢−𝜈Δ𝛼²₀𝑢−𝛼²₁×𝛼Δ𝑢−𝑢∇ײ₀𝑢−𝛼²₁+1Δ𝑢𝜌₀(𝑥),𝑢()∇𝑝=𝑓∇⋅𝑢=0,𝑥,0=𝑢₀(𝑥). (2) They established the global regularity of solutions of the equation and provided the estimates for the Hausdorff and fractal dimensions of the global attractor.

In [20] the authors analyzed the effects produced by stochastic perturbations in the deterministic version of the three-dimensional Lagrangian averaged Navier-Stokes-𝛼 model: 𝜕()((+()𝜕𝑡𝑢−𝛼Δ𝑢+𝜈𝐴𝑢−𝛼Δ𝐴𝑢))𝑢⋅∇)(𝑢−𝛼Δ𝑢−𝛼∇𝑢^∗()()𝑊̇(𝑡),(),(),𝑢(0)⋅Δ𝑢+∇𝑝=𝐹𝑡,𝑢+𝐺𝑡,𝑢∇⋅𝑢=0,in𝐷×0,+∞𝑢=0,𝐴𝑢=0,on𝜕𝐷×0,+∞=𝑢₀, (3) that is, the persistence of exponential stability as well as possible stabilization effects produced by the noise.

In [21] the authors proved the existence of the pullback and forward attractors for three-dimensional Lagrangian averaged Navier-Stokes-𝛼 model with delay: 𝜕()((+()𝜕𝑡𝑢−𝛼Δ𝑢+𝜈𝐴𝑢−𝛼Δ𝐴𝑢))𝑢⋅∇)(𝑢−𝛼Δ𝑢−𝛼∇𝑢^∗(𝑡)⋅Δ𝑢+∇𝑝=𝑓+𝐹𝑡,𝑢_𝑡,(),(),𝑢(𝑡)()(),𝑢(𝜏)∇⋅𝑢=0,in𝐷×𝜏,+∞𝑢=0,𝐴𝑢=0,on𝜕𝐷×𝜏,+∞=𝜙𝑡−𝜏,in𝜏−ℎ,𝜏=𝑢₀. (4)

In [22] the author investigated the existence of finite dimensional uniform attractor for three-dimensional nonautonomous Camassa-Holm equations with periodic boundary conditions: 𝜕𝛼𝜕𝑡²⁰𝑢−𝛼²₁Δ𝑢−𝜈Δ𝛼²₀𝑢−𝛼²₁𝛼Δ𝑢−𝑢×∇ײ₀𝑢−𝛼²₁+1Δ𝑢𝜌₀(),𝑢()∇𝑝=𝑓𝑥,𝑡∇⋅𝑢=0,𝑥,𝜏=𝑢_𝜏(𝑥). (5)

In [23, 24] the author studied the existence of uniform attractor and convergence of the attractor as 𝜀→0⁺ for a nonautonomous three-dimensional Lagrangian averaged Navier-Stokes-𝛼 model with singularly oscillating external force: 𝜕()((+()𝜕𝑡𝑢−𝛼Δ𝑢+𝜈𝐴𝑢−𝛼Δ𝐴𝑢))𝑢⋅∇)(𝑢−𝛼Δ𝑢−𝛼∇𝑢^∗⋅Δ𝑢+∇𝑝=𝑔^𝜀,(),(),𝑢(𝜏)∇⋅𝑢=0,in𝐷×𝜏,+∞𝑢=0,𝐴𝑢=0,on𝜕𝐷×𝜏,+∞=𝑢_𝜏, (6) where 𝑔^𝜀=𝑔𝑥,𝑦,𝑧,𝑡₀+1𝑥,𝑦,𝑧,𝑡𝜀^𝜌𝑔₁(][),𝑔𝑥,𝑦,𝑧,𝑡,𝜀∈0,1,𝜌∈0,1₀𝑥,𝑦,𝑧,𝑡,𝜀=0. (7)

Motivated by all their works, we initial our work to investigate the equations perturbed by an additive noise. We consider the following viscous version of three-dimensional stochastic nonautonomous Camassa-Holm equation in the periodic box [0,𝑙]³: 𝜕𝛼𝜕𝑡²⁰𝑢−𝛼²₁Δ𝑢−𝜈Δ𝛼²₀𝑢−𝛼²₁𝛼Δ𝑢−𝑢×∇ײ₀𝑢−𝛼²₁+1Δ𝑢𝜌₀()∇𝑝=𝑓𝑥,𝑡+𝑄𝑊_𝑡,𝑢()∇⋅𝑢=0,𝑥,𝜏=𝑢_𝜏(𝑥), (8) where 𝑝/𝜌₀=𝜋/𝜌₀+𝛼²₀|𝑢|²−𝛼²₁(𝑢⋅Δ𝑢) is the modified pressure, while 𝜋 is the pressure, 𝜈>0 is the constant viscosity, and 𝜌₀>0 is a constant density. The function 𝑓 is a given body forcing, and 𝛼₀>0, 𝛼₁≥0 are scale parameters. 𝑊(𝑡) are two-sided real-valued Wiener processes on a probability space which will be specified later. 𝑄∶ℝ^𝑛→𝐿²([0,𝑙]³)³ is a bounded linear operator. Also observing that at the limiting case 𝛼₀=1, 𝛼₁=0, we obtain the three-dimensional stochastic Navier-Stokes with periodic boundary conditions.

Attractor is an important concept to describe the long-time behavior of solutions for a system in mathematical physics [25–27]. The notion of uniform attractor parallelling to that of the global autonomous systems has been systematically considered in [26]. In the approach presented in [27], to construct the uniform attractor, instead of the associated process {𝒰_𝜎(𝑡,𝜏),𝑡≥𝜏,𝜏∈ℝ}, one should consider a family of processes {𝒰_𝜎(𝑡,𝑠)}, 𝜎∈Σ, in some Banach space 𝐸 where the functional parameter 𝜎₀(𝑠), 𝑠∈ℝ, is called the symbol and Σ is the symbol space including 𝜎₀(𝑠). The approach implies that the structure of uniform attractor is described by the representation as a union of sections of all kernels of the family of processes. The kernel is the set of all complete trajectories of a process.

While in the real world, a system is usually uncertain due to some external noise, which is random. The random effects are considered not only as compensations for the defects in some deterministic models but rather essential phenomena [28–32]. In order to capture the essential dynamics of random dynamical systems with wide fluctuations, the concept of pullback random attractor was introduced in [29, 33, 34], as an extension to stochastic systems of the theory of attractors for deterministic systems in [25, 27, 35–43]. A pullback random attractor 𝒜(𝜔) which can be constructed by a closed random absorbing set 𝒦(𝜔) for an asymptotically compact stochastic mapping 𝒮(𝑡,𝑠,𝜔) is given by 𝒜(𝜔)=_{𝑠≥𝜏𝑡≥𝑠}𝒮(𝑡,𝜏,𝜃_𝜏−𝑡𝜔,𝒦(𝜃_𝜏−𝑡𝜔)), (9) where 𝜃_𝑡 is the metric process on probability space. The existence of random attractors for stochastic dynamical systems has been investigated extensively by many authors [29, 33, 34, 44–49]. In our paper, we associate with the concepts of uniform attractor and random attractor together and give the concept of uniformly random attractor. Then we consider (8) in an appropriate space and show that there is a uniformly (with respect to 𝑓) random attractor 𝒜(𝜔) which all solutions approach as 𝑡→∞. To our best knowledge, the long-time dynamical behavior of the three-dimensional stochastic nonautonomous Camassa-Holm equations has not been discussed, and we believe that it is a significant work to obtain a uniformly (with respect to 𝑓) random attractor for the system.

The paper is organized as follows. In Section 2, we present the abstract results describing the uniformly random attractor and some relevant definitions. In Section 3, we give some functional settings which are foundations for us to obtain the existence of uniformly (with respect to 𝑓) random attractor of (8). In Section 4, we convert (8) with an additive noise to deterministic equations with random parameters and define a process corresponding to the equations. In Section 5, we obtain the existence of uniformly (with respect to 𝑓) random attractors for (8) on the basis of the above preparations.

2. Abstract Results

In this section, we associate with the concepts of uniform attractor and random attractor and obtain the notion of uniformly random attractor. Let (𝐸,‖⋅‖_𝐸) be a separable Hilbert space with the Borel 𝜎-algebra ℬ(𝐸), and let (Ω,ℱ,ℙ) be a probability space.

Definition 1. (𝜃_𝑡)_𝑡∈ℝ is called a measurable flow on probability space (Ω,ℱ) if 𝜃∶ℝ×Ω→Ω is (ℝ×ℱ,ℱ)-measurable, 𝜃₀ is the identity on Ω, 𝜃_𝑠+𝑡=𝜃_𝑡∘𝜃_𝑠 for all 𝑠, 𝑡∈ℝ, and 𝜃_𝑡ℙ=ℙ for all 𝑡∈ℝ.

Definition 2. A random bounded set {𝐵(𝜔)}_𝜔∈Ω of 𝐸, is called tempered with respect to (𝜃_𝑡)_𝑡∈ℝ if for ℙ a.e. 𝜔∈Ω, lim_𝑡→∞𝑒^−𝛽𝑡𝑑𝐵𝜃_−𝑡𝜔=0∀𝛽>0, (10) where 𝑑(𝐵)=sup_𝑥∈𝐸‖𝑥‖_𝐸.

Definition 3. A random set {𝒦(𝜔)}_𝜔∈Ω is called an absorbing set of a stochastic mapping 𝒮(𝑡,𝑠,𝜔) in 𝐸 if for every random bounded set 𝐵 and ℙ a.e. 𝜔∈Ω, there exists 𝑡_𝐵(𝜔)>0 such that 𝒮𝑡,𝑠,𝜃_𝑠−𝑡𝜔𝐵𝜃_𝑠−𝑡(𝜔)⊆𝒦(𝜔)∀𝑡≥𝑡_𝐵(𝜔). (11)

Let {𝒰(𝑡,𝜏,𝜔)}={𝒰(𝑡,𝜏,𝜔),𝑡≥𝜏,𝑡,𝜏∈ℝ,𝜔∈Ω} be a three-parameter family of mappings acting on 𝐸: 𝒰(𝑡,𝜏,𝜔)∶𝐸⟶𝐸,𝑡≥𝜏,𝑡,𝜏∈ℝ,𝜔∈Ω. (12)

Definition 4. A three-parameter family of random mappings {𝒰(𝑡,𝜏,𝜔)} is said to be a process in 𝐸 if it is (ℬ(ℝ⁺)×ℬ(ℝ⁺)×ℱ×ℬ(𝐸),ℬ(𝐸))-measurable and for ℙ a.e. 𝜔∈Ω it satisfies 𝒰𝑡,𝑠,𝜃_𝑠−𝜏𝜔()(),𝒰()∘𝒰𝑠,𝜏,𝜔=𝒰𝑡,𝜏,𝜔∀𝑡≥𝑠≥𝜏,𝜏∈ℝ,𝜏,𝜏,𝜔=𝐼𝑑,𝜏∈ℝ. (13)

Definition 5. A family of processes 𝒰_𝜎(𝑡,𝜏,𝜔), 𝜎∈Σ, acting in 𝐸 is said to be (𝐸×Σ,𝐸)-continuous, if, for ℙ a.e. 𝜔∈Ω and fixed 𝑡, 𝜏, 𝑡≥𝜏, the random mapping (𝑢,𝜎)↦𝒰_𝜎(𝑡,𝜏,𝜔)𝑢 is continuous from 𝐸×Σ into 𝐸.

Definition 6. A random curve 𝑢(𝑠,𝜔), 𝑠∈ℝ, is said to be a complete trajectory of the process {𝒰(𝑡,𝜏,𝜔)} if for ℙ a.e. 𝜔∈Ω{𝒰(𝑡,𝜏,𝜔)}𝑢(𝜏,𝜔)=𝑢(𝑡,𝜔),∀𝑡≥𝜏,𝑡,𝜏∈ℝ. (14)

Definition 7. The random kernel 𝒩(𝜔) of the process {𝒰(𝑡,𝜏,𝜔)} consists of all bounded complete trajectories of the process {𝒰(𝑡,𝜏,𝜔)}: 𝒩(𝜔)={𝑢()()‖𝑢(⋅,𝜔∣𝑢⋅,𝜔satisfiesDefinition6and𝑠,𝜔)‖_𝐸≤𝑀_𝑢(𝜔)for𝑠∈ℝ. (15)

Definition 8. The random set 𝒩(𝑠,𝜔)={𝑢(𝑠,𝜔)∣𝑢(⋅,𝜔)∈𝒩(𝜔)}⊆𝐸 (16) is said to be the random kernel section at time 𝑡=𝑠, 𝑠∈ℝ.

Let 𝐵_𝑡(𝜔)=⋃_𝜎∈Σ⋃_𝑠≥𝑡𝒰_𝜎(𝑠,𝑡,𝜃_𝑡−𝑠𝜔)𝐵(𝜃_𝑡−𝑠𝜔), where 𝐵(𝜔) is a random set. Denote that 𝐵(𝜔) is closure of the set 𝐵(𝜔) and ℝ_𝜏={𝑡∈ℝ∣𝑡≥𝜏}.

Definition 9. A random set 𝒲_𝜏,Σ(𝐵)(𝜔)=⋂_𝑡≥𝜏𝐵_𝑡(𝜔) is called the uniformly (with respect to 𝜎∈Σ) random (pullback) omega-limit set of 𝐵(𝜔) which can be characterized as follows, analogously to that for semigroups, 𝑦∈𝒲_𝜏,Σ(𝐵)(𝜔)⟺therearesequences𝑥_𝑛⊂𝐵𝜃_{𝜏−𝑡𝑛}𝜔,𝜎_𝑛⊂Σ,𝑡_𝑛⊂ℝ_𝜏,suchthat𝑡_𝑛𝒰⟶+∞and_𝜎𝑛𝑡_𝑛,𝜏,𝜃_{𝜏−𝑡𝑛}𝜔𝑥_𝑛⟶𝑦(𝑛⟶∞). (17)

Let 𝐵(𝜔)∈ℬ(𝐸), and its Kuratowski measure of noncompactness 𝜅(𝐵) is defined by 𝜅(𝐵){.=inf𝛿>0∣𝐵admitsafinitcoveringbysetsofdiameter≤𝛿 (18)

Definition 10. A family of processes {𝒰_𝜎(𝑡,𝜏,𝜔)}, 𝜎∈Σ, is said to be uniformly (with respect to 𝜎∈Σ) random (pullback) omega-limit compact if, for ℙ a.e. 𝜔∈Ω and any 𝜏∈ℝ, the set 𝐵_𝑡(𝜔) is bounded for every 𝑡 and lim_𝑡→∞𝜅(𝐵_𝑡(𝜔))=0.

We now present a method to verify the uniformly (with respect to 𝜎∈Σ) random (pullback) omega-limit compactness.

Definition 11. A family of processes {𝒰_𝜎(𝑡,𝜏,𝜔)}, 𝜎∈Σ, is said to satisfy uniformly (with respect to 𝜎∈Σ) condition (C) if, for ℙ a.e. 𝜔∈Ω and any fixed 𝜏∈ℝ, tempered set 𝐵(𝜔)∈ℬ(𝐸), 𝜀>0, there exist 𝑡₀=𝑡(𝜏,𝐵,𝜀,𝜔)≥𝜏 and a finite dimensional subspace 𝐸₁ of 𝐸 such that(i)Π(⋃_𝜎∈Σ⋃_𝑡≥𝑡0𝒰_𝜎(𝑡,𝜏,𝜃_𝜏−𝑡𝜔)𝐵(𝜃_𝜏−𝑡𝜔)) is bounded, and(ii)‖(𝐼−Π)(⋃_𝜎∈Σ⋃_𝑡≥𝑡0𝒰_𝜎(𝑡,𝜏,𝜃_𝜏−𝑡𝜔)𝑥)‖≤𝜀, ∀𝑥∈𝐵(𝜃_𝜏−𝑡𝜔),where Π∶𝐸→𝐸₁ is a bounded projector.

Therefore, we have the following results.

Theorem 12. Let Σ be a metric space, and let 𝑇(𝑡) be a continuous invariant semigroup 𝑇(𝑡)Σ=Σ on Σ. A family of processes {𝒰(𝑡,𝜏,𝜔)}, 𝜎∈Σ, acting in 𝐸 is (𝐸×Σ,𝐸)-continuous (weakly) and possesses the compact uniformly (with respect to 𝜎∈Σ) random attractor 𝒜_Σ(𝜔) satisfying 𝒜_Σ(𝜔)=𝒲_0,Σ𝐵₀(𝜔)=𝒲_𝜏,Σ𝐵₀(𝜔)=_𝜎∈Σ𝒩_𝜎(0,𝜔),∀𝜏∈ℝ, (19) if it(i)has a bounded uniformly (with respect to 𝜎∈Σ) random absorbing set 𝐵₀(𝜔), and(ii)satisfies uniformly (with respect to 𝜎∈Σ) condition (C).

Moreover, if 𝐸 is a uniformly convex Banach space, then the converse is true.

3. Functional Setting

We consider the probability space (Ω,ℱ,ℙ) where Ω=𝜔=𝜔₁,𝜔₂,…,𝜔_𝑛^𝑇∈𝐶ℝ,ℝ^𝑛∶𝜔(0)=0. (20)ℱ is the Borel 𝜎-algebra induced by the compact-open topology of Ω and ℙ the corresponding Wiener measure on (Ω,ℱ). Then we will identify 𝑊(𝑡) with 𝑊(𝑡)=𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑛(𝑡)^𝑇=𝜔(𝑡)for𝑡∈ℝ. (21) Define the time shift by 𝜃_𝑡𝜔(⋅)=𝜔(⋅+𝑡)−𝜔(𝑡),𝜔∈Ω,𝑡∈ℝ. (22) Then {𝜃_𝑡}_𝑡∈ℝ is a family of measure preserving transformations on probability space (Ω,ℱ,ℙ) in Definition 1.

Next we define a symbol space Σ(𝜎₀) for (8). We assume that the function 𝑓(⋅,𝑡)=∶𝑓(𝑡)∈𝐿²_loc(ℝ;𝐸) is translation bounded. That is, for 𝑓(𝑠)∈𝐿²_𝑏(ℝ;𝐸), we have 𝑓²_𝐿2𝑏=𝑓²_{𝐿2𝑏(ℝ;𝐸)}=sup_{𝑡∈ℝ𝑡}^𝑡+1𝑓(𝑠)²_𝐸𝑑𝑠<∞. (23)

Definition 13 (cf. [41]). A function 𝜑∈𝐿²_loc(ℝ;𝐸) is said to be normal if, for any 𝜀>0, there exists 𝛿>0 such that sup_{𝑡∈ℝ𝑡}^𝑡+𝛿𝜑(𝑠)²_𝐸𝑑𝑠≤𝜀. (24)

We denote by 𝐿²_𝑛(ℝ;𝐸) the set of all normal functions in 𝐿²_loc(ℝ;𝐸). Obviously, 𝐿²_𝑛(ℝ;𝐸)⊂𝐿²_𝑏(ℝ;𝐸), and it is proved in [41] that 𝐿²_𝑛(ℝ;𝐸) is a closed subset of 𝐿²_𝑏(ℝ;𝐸). Let a fixed symbol 𝜎₀(𝑠)=𝑓₀(𝑠)=𝑓₀(⋅,𝑠) be normal functions in 𝐿²_loc(ℝ;𝐸). That is, the family of translation {𝑓₀(𝑠+𝜂),𝜂∈ℝ} forms a normal function set in 𝐿²_loc([𝑇₁,𝑇₂];𝐸), where [𝑇₁,𝑇₂] is an arbitrary interval of the time axis ℝ. Therefore, Σ𝜎₀=Σ𝑓₀=𝑓₀𝑥,𝑠+𝜂∣𝜂∈ℝ_{𝐿2loc(ℝ;𝐸)}. (25)

After integrating (8), one can easily see that 𝑑𝑑𝑡_[0,𝑙]³𝛼²₀𝑢−𝛼²₁Δ𝑢𝑑𝑥=_[0,𝑙]3𝑓𝑑𝑥+_[0,𝑙]3𝑄𝑊_𝑡𝑑𝑥. (26) On the other hand, because of the spatial periodicity of solution, we have ∫_[0,𝑙]3Δ𝑢𝑑𝑥=0. Then we have 𝑑𝑑𝑡_[0,𝑙]³𝛼²₀𝑢𝑑𝑥=_[0,𝑙]3𝑓𝑑𝑥+_[0,𝑙]3𝑄𝑊_𝑡𝑑𝑥. (27) That is, the mean of solution is invariant provided that the means of the forcing term and the perturbing term are zero. In this paper, we will consider the forcing term, perturbing term and initial values with spatial means that are zero. That is, we assume ∫_[0,𝑙]3𝑓𝑑𝑥=0, ∫_[0,𝑙]3𝑄𝑊_𝑡𝑑𝑥=0, and ∫_[0,𝑙]3𝑢_𝜏𝑑𝑥=0.

Next we introduce some essential functional spaces.(i)We denote by 𝒱={𝜑∶𝜑 a vector-valued trigonometric polynomial defined on [0,𝑙]³, such that ∇⋅𝜑=0 and ∫_[0,𝑙]3𝜑(𝑥)𝑑𝑥=0}, and we let 𝐻 and 𝑉 be the closure of 𝒱 in 𝐿²([0,𝑙]³)³ and in 𝐻¹([0,𝑙]³)³, respectively. We can observe that 𝐻^⟂, the orthogonal complement of 𝐻 in 𝐿²([0,𝑙]³)³, is {∇𝑝∶𝑝∈𝐻¹([0,𝑙]³)} (cf. [38, 41]).(ii)We denote by 𝑃∶𝐿²([0,𝑙]³)³→𝐻 the 𝐿² orthogonal projection, usually referred to as Helmholtz-Leray projector, and by 𝐴=−𝑃Δ the Stokes operator with domain 𝐷(𝐴)=𝐻²([0,𝑙]³)³⋂𝑉. Note that, in the case of periodic boundary condition, 𝐴=−Δ|_𝐷(𝐴) is a self-adjoint positive operator with compact inverse. Hence, the space 𝐻 has an orthogonal basis {𝑒_𝑗}^∞_𝑗 of eigenfunctions of 𝐴, that is, 𝐴𝑒_𝑗=𝜆_𝑗𝑒_𝑗, with 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑗𝜆≤⋯,_𝑗→+∞,as𝑗→∞. (28) In fact, these eigenvalues have the form |𝑘|²4𝜋/𝐿² with 𝑘∈ℤ³/{0}.(iii)We denote by (⋅,⋅) the 𝐿²([0,𝑙]³)³ inner product and by ‖⋅‖₂ the corresponding 𝐿²([0,𝑙]³)³ norm. By virtue of Poincaré inequality one can show that there is a constant 𝑐>0, such that 𝑐‖𝐴𝑢‖₂≤‖𝑢‖_𝐻²≤𝑐⁻¹‖𝐴𝑢‖₂(𝐴),𝑐𝐴forevery𝑢∈𝐷_1/2𝑢₂≤‖𝑢‖_𝐻¹≤𝑐⁻¹𝐴^1/2𝑢₂forevery𝑢∈𝑉. (29) Moreover, one can show that 𝑉=𝐷(𝐴^1/2) (cf. [38, 42]). We denote by 𝑉^′ the dual of 𝑉. Hereafter 𝑐 will denote a generic scale invariant positive constant which is independent of the physical parameters in the equation.(iv)Following the notation for the Navier-Stokes equations we denote yhat 𝐵(𝑢,𝑣)=𝑃[(𝑢⋅∇)𝑣], and we set 𝐵(𝑣)𝑢=𝐵(𝑢,𝑣) for every 𝑢, 𝑣∈𝑉. That is, for every fixed 𝑣∈𝑉, 𝐵(𝑣) is a linear operator acting on 𝑢. Note that (𝐵(𝑢,𝑣),𝑤)=−(𝐵(𝑢,𝑤),𝑣)forevery𝑢,𝑣,𝑤∈𝑉. (30)We also denote that 𝐵(𝑢,𝑣)=−𝑃(𝑢×(∇×𝑣)) for every 𝑢,𝑣∈𝑉. Using the identity (𝑏⋅∇)𝑎+³_𝑗=1𝑎_𝑗∇𝑏_𝑗=−𝑏×(∇×𝑎)+∇(𝑎⋅𝑏), (31) one can easily show that 𝐵()=(𝐵())−(𝐵())=𝐵(𝑣)𝑢,𝑣,𝑤𝑢,𝑣,𝑤𝑤,𝑣,𝑢𝑢−𝐵^∗(𝑣)𝑢,𝑤 (32) for every 𝑢, 𝑣, 𝑤∈𝑉, where 𝐵^∗(𝑣) denotes the adjoint operator of the linear operator 𝐵(𝑣) defined above. As a result we have 𝐵(𝑢,𝑣)=𝐵(𝑣)−𝐵^∗(𝑣)𝑢forevery𝑢,𝑣∈𝑉. (33)

The next lemma will present some properties of the bilinear operator 𝐵.

Lemma 14 (cf. [19]). The operator 𝐵 can be extended continuously from 𝑉×𝑉 with values in 𝑉^′, and it satisfies 𝐵(𝑢,𝑣),𝑤_𝑉′≤𝑐‖𝑢‖₂^1/2𝐴^1/2𝑢₂^1/2𝐴^1/2𝑣₂𝐴^1/2𝑤₂,𝐵(𝑢,𝑣),𝑤_𝑉′≤𝑐𝐴^1/2𝑢₂𝐴^1/2𝑣₂‖𝑤‖₂^1/2𝐴^1/2𝑤₂^1/2 (34) for every 𝑢, 𝑣, 𝑤∈𝑉. Moreover, 𝐵(𝑢,𝑣),𝑤_𝑉′=−𝐵(𝑤,𝑣),𝑢_𝑉′,forevery𝑢,𝑣,𝑤∈𝑉 (35) and in particular 𝐵(𝑢,𝑣),𝑢_𝑉′≡0forevery𝑢,𝑣∈𝑉. (36) Furthermore, one has 𝐵(𝑢,𝑣),𝑤_{𝐷(𝐴)′}≤𝑐‖𝑢‖₂𝐴^1/2𝑣₂𝐴^1/2𝑤₂^1/2‖𝐴𝑤‖₂^1/2 (37) for every 𝑢∈𝐻, 𝑣∈𝑉, and 𝑤∈𝐷(𝐴) and by symmetry one has 𝐵(𝑢,𝑣),𝑤_{𝐷(𝐴)′}≤𝑐𝐴^1/2𝑢₂^1/2‖𝐴𝑢‖₂^1/2𝐴^1/2𝑣₂‖𝑤‖₂ (38) for every 𝑢∈𝐷(𝐴), 𝑣∈𝑉, and 𝑤∈𝐻. Also 𝐵(𝑢,𝑣),𝑤_{𝐷(𝐴)′}≤𝑐‖𝑢‖₂^1/2𝐴^1/2𝑢₂^1/2‖𝑣‖₂‖𝐴𝑤‖₂+‖𝑣‖₂𝐴^1/2𝑢₂𝐴^1/2𝑤₂^1/2‖𝐴𝑤‖₂^1/2 (39) for every 𝑢∈𝑉, 𝑣∈𝐻, and 𝑤∈𝐷(𝐴). In addition, 𝐵(𝑢,𝑣),𝑤_{𝐷(𝐴)′}≤𝑐𝐴^1/2𝑢₂^1/2‖𝐴𝑢‖₂^1/2‖𝑣‖₂𝐴^1/2𝑤₂+‖𝑣‖₂‖𝐴𝑢‖₂‖𝑤‖₂^1/2𝐴^1/2𝑤₂^1/2 (40) for every 𝑢∈𝐷(𝐴), 𝑣∈𝐻, and 𝑤∈𝑉.

4. Stochastic Nonautonomous Camassa-Holm Equations

We now apply the result in Section 2 to the stochastic nonautonomous Camassa-Holm equations. To associate a family of processes 𝒰_𝜎(𝑡,𝑠,𝜔) with the stochastic equations over (Σ,𝑇(𝑡)) and (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ), we need to convert the stochastic equations with a random additive term into deterministic equations with a random parameter.

First we define the bounded linear operator 𝑄 in (8) as follows: 𝑄𝑥=^𝑛_𝑗=1𝑥_𝑗ℎ_𝑗,𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛^𝑇∈ℝ^𝑛, (41) where ℎ_𝑗∈𝐷(𝐴), 𝑗=1,2,…,𝑛.

Given 𝑗=1,2,…,𝑛, consider the Ornstein-Uhlenbeck equation: 𝑑𝑦_𝑗+𝜇𝑦_𝑗𝑑𝑡=𝑑𝑤_𝑗(𝑡). (42) One can easily check that a solution to (42) is given by 𝑦_𝑗(𝑡)=𝑦_𝑗𝜃_𝑡𝜔_𝑗=−𝜇⁰_−∞𝑒^𝜇𝜏𝜃_𝑡𝜔_𝑗(𝜏)𝑑𝜏,𝑡∈ℝ. (43) It is known that the random variable |𝑦_𝑗(𝜔_𝑗)| is tempered and that 𝑦_𝑗(𝜃_𝑡𝜔_𝑗) is ℙ a.e is continuous. Now we put 𝑧(𝜃_𝑡𝜔)=(𝛼²₀+𝛼²₁𝐴)⁻¹∑^𝑚_𝑗=1ℎ_𝑗𝑦_𝑗(𝜃_𝑡𝜔_𝑗). By (42) we have 𝑑𝛼²₀𝑧𝜃_𝑡𝜔+𝛼²₁𝐴𝑧𝜃_𝑡𝜔+𝜇𝛼²₀𝑧𝜃_𝑡𝜔𝑑𝑡+𝛼²₁𝐴𝑧𝜃_𝑡𝜔=𝑑𝑡_𝑛𝑗=1ℎ_𝑗𝑑𝑤_𝑗(𝑡). (44) Employing Cauchy-Schwarz’s inequality, we get 𝑧𝜃_𝑡𝜔²₂+𝐴^1/2𝑧𝜃_𝑡𝜔²₂+𝐴𝑧𝜃_𝑡𝜔²₂≤𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗², (45)

To show that (8) corresponds to a process {𝒰_𝜎(𝑡,𝜏,𝜔)}, we let 𝑣=𝑢−𝑧(𝜃_𝑡𝜔), where 𝑢 is a solution of (8). Then for 𝑣(𝑡,𝜔), we have 𝜕𝛼𝜕𝑡²⁰𝑣+𝛼²₁𝛼𝐴𝑣+𝜈𝐴²₀𝑣+𝛼²₁𝐴𝑣+𝛼²₀𝑧𝜃_𝑡𝜔+𝛼²₁𝐴𝑧𝜃_𝑡𝜔−𝑣+𝑧𝜃_𝑡𝜔×∇×𝛼²₀𝑣+𝛼²₁𝐴𝑣+𝛼²₀𝑧𝜃_𝑡𝜔+𝛼²₁𝐴𝑧𝜃_𝑡𝜔+1𝜌₀()∇𝑝=𝑓𝑥,𝑡+𝜇𝛼²₀𝑧𝜃_𝑡𝜔+𝜇𝛼²₁𝐴𝑧𝜃_𝑡𝜔 (46) defined in the periodic box [0,𝑙]³, satisfying 𝑣()∇⋅𝑣=0,𝑥,𝜏,𝜔=𝑣_𝜏(𝑥,𝜔)=𝑢_𝜏(𝑥)−𝑧𝜃_𝜏𝜔∈𝐻. (47) We apply 𝑃 to (46) and use the notation in Section 3 to obtain the equivalent system of equations 𝜕𝛼𝜕𝑡²⁰𝑣+𝛼²₁𝛼𝐴𝑣+𝜈𝐴²₀𝑣+𝛼²₁𝐴𝑣+𝛼²₀𝑧𝜃_𝑡𝜔+𝛼²₁𝐴𝑧𝜃_𝑡𝜔+𝐵𝑣+𝑧𝜃_𝑡𝜔,𝛼²₀𝑣+𝛼²₁𝐴𝑣+𝛼²₀𝑧𝜃_𝑡𝜔+𝛼²₁𝐴𝑧𝜃_𝑡𝜔=𝑃𝑓(𝑥,𝑡)+𝜇𝛼²₀𝑧𝜃_𝑡𝜔+𝜇𝛼²₁𝐴𝑧𝜃_𝑡𝜔 (48) satisfying the initial condition 𝑣(𝑥,𝜏,𝜔)=𝑣_𝜏(𝑥,𝜔). (49) By a Galerkin method as in [19], it can be proved that if 𝑓(𝑥,𝑡)∈𝐿²_loc((0,𝑇);𝐻) and 𝑣_𝜏∈𝑉, for ℙ a.e. 𝜔∈Ω, (48) has a unique solution satisfying for any 𝑇>𝜏𝑣𝑡,𝜔,𝑣_𝜏∈𝐶([𝜏,𝑇);𝑉)𝐿²([𝜏,𝑇);𝐷(𝐴)),𝑑𝑣𝑑𝑡∈𝐿²([𝜏,𝑇);𝐻) (50) and such that, for almost all 𝑡∈[𝜏,𝑇) and for any 𝑤∈𝐷(𝐴), 𝑑𝛼𝑑𝑡²⁰𝑣+𝛼²₁𝐴𝑣,𝑤_{𝐷(𝐴)′}+𝜈𝐴𝛼²₀𝑣+𝛼²₁𝐴𝑣+𝛼²₀𝑧𝜃_𝑡𝜔+𝛼²₁𝐴𝑧𝜃_𝑡𝜔,𝑤_{𝐷(𝐴)′}+𝐵𝑣+𝑧𝜃_𝑡𝜔,𝛼²₀𝑣+𝛼²₁𝐴𝑣+𝛼²₀𝑧(𝜃_𝑡𝜔)+𝛼²₁𝐴𝑧(𝜃_𝑡𝜔),𝑤_{𝐷(𝐴)′}=𝑃𝑓(𝑥,𝑡)+𝜇𝛼²₀𝑧𝜃_𝑡𝜔+𝜇𝛼²₁𝐴𝑧𝜃_𝑡𝜔,𝑤. (51)

Now for any 𝑓(𝑥,𝑡)∈Σ(𝑓₀), (48) with 𝑓 instead of 𝑓₀ possesses a corresponding process {𝒰_𝑓(𝑡,𝜏,𝜔)} acting on 𝑉. It is analogous to the proof in [27] to prove that, for ℙ a.e 𝜔∈Ω, the family {𝒰_𝑓(𝑡,𝜏,𝜔)∣𝑓∈Σ(𝑓₀)} of processes is (𝑉×Σ(𝑓₀);𝑉)-continuous. Let 𝒩_𝑓(𝜔)=𝑣_𝑓(𝑥,𝑡,𝜔)for𝑡∈ℝ∣𝑣_𝑓()()𝑣𝑥,𝑡,𝜔issolutionof48satisfying_𝑓(𝑥,𝑡,𝜔)_𝐻≤𝑀_𝑓(𝜔)∀𝑡∈ℝ (52) be the so-called kernel of the process {𝒰_𝑓(𝑡,𝜏,𝜔)}.

5. Uniformly Random Attractor for Stochastic Nonautonomous Camassa-Holm Equation

In [19], the authors have shown that the semigroup corresponding to the autonomous system possesses a global attractor. In [21–24], the authors have proved that the deterministic version of nonautonomous system has a uniform attractor. The main objective of this section is to obtain the existence of uniformly (with respect to 𝑓∈Σ(𝑓₀)) random attractor for the stochastic nonautonomous Camassa-Holm equations in 𝑉 and 𝐷(𝐴).

Lemma 15. Let {𝐵(𝜔)}_𝜔∈Ω be tempered and 𝑣_𝜏(𝜔)∈𝐵(𝜔). Then the process {𝒰_𝑓(𝑡,𝜏,𝜔)} corresponding to (48) possesses a uniformly (with respect to 𝑓∈Σ(𝑓₀)) random absorbing set 𝒦₀(𝜔) in 𝑉.

Proof. Letting 𝑤=𝑣 in (51), we have 12𝑑𝛼𝑑𝑡²⁰‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝜈𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂=−𝜈^𝑛_𝑗=1𝐴ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗−𝐵𝜃,𝑣𝑣+𝑧_𝑡𝜔,𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗+()+,𝑣𝑃𝑓𝑥,𝑡,𝑣^𝑛_𝑗=1𝜇ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝑣. (53) Now we estimate the second term of (53) on the right-hand side. Applying Lemma 14 we have 𝐵𝑣+𝑧𝜃_𝑡𝜔,𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗=𝐵𝑧𝜃,𝑣_𝑡𝜔,𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗𝐴,𝑣≤𝑐^1/2𝑧(𝜃_𝑡𝜔)₂^1/2𝐴𝑧(𝜃_𝑡𝜔)₂^1/2×𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗2𝐴^1/2𝑣₂+𝑐𝐴𝑧𝜃_𝑡𝜔₂𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗2×‖𝑣‖₂^1/2𝐴^1/2𝑣₂^1/2≤𝑐𝐴^1/2𝑧(𝜃_𝑡𝜔)₂‖𝑣‖²₂+𝑐𝐴𝑧(𝜃_𝑡𝜔)₂𝐴^1/2𝑣²₂+𝜈𝛼²₁4‖𝐴𝑣‖²₂+𝑐𝐴^1/2𝑧(𝜃_𝑡𝜔)₂𝐴𝑧(𝜃_𝑡𝜔)₂𝐴^1/2𝑣²₂+𝑐𝐴^1/2𝑧(𝜃_𝑡𝜔)₂𝐴𝑧(𝜃_𝑡𝜔)₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝐴^1/2𝑣²₂+𝑐𝐴𝑧𝜃_𝑡𝜔₂‖𝑣‖²₂+𝜈𝛼²₁4‖𝐴𝑣‖²₂+𝑐𝐴𝑧(𝜃_𝑡𝜔)²₂‖𝑣‖²₂+𝑐𝐴𝑧(𝜃_𝑡𝜔)²₂𝐴^1/2𝑣²₂+𝑐𝐴𝑧(𝜃_𝑡𝜔)²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²≤𝜈𝛼²₁2‖𝐴𝑣‖²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗². (54) By Poincaré’s inequality, we have 𝜈^𝑛_𝑗=1𝐴ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝑣≤𝜈𝛼²₀6𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²,𝑃𝑓(𝑥,𝑡),𝑣≤𝑓_𝑉′‖𝑣‖₂≤𝜈𝛼²₀6𝐴^1/2𝑣²₂+𝑐𝑓²_𝑉′,_𝑛𝑗=1𝜇ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝑣≤𝜈𝛼²₀6𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗². (55) Associating with the above inequalities and employing Poincaré’s inequality, we have 𝑑𝛼𝑑𝑡²⁰‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝜈𝜆₁𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂≤𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²+𝑐𝑓²_𝑉′. (56) Applying Gronwall’s lemma, we have 𝛼²₀‖𝑣(𝑡,𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝑡,𝜔)²₂≤𝑒^{𝐽1(𝜏,𝑡,𝜔)}𝛼²₀𝑣_𝜏(𝜔)²₂+𝛼²₁𝐴^1/2𝑣_𝜏(𝜔)²₂+^𝑡_𝜏𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑠𝜔_𝑗²+𝑐𝑓(𝑠)²_𝑉′𝑒^{𝐽1(𝑠,𝑡,𝜔)}𝑑𝑠, (57) where 𝐽₁(𝜏,𝑡,𝜔)=−𝜈𝜆₁𝑡+𝜈𝜆₁𝜏+𝑐^𝑛_𝑗=1^𝑡_𝜏𝑦_𝑗𝜃_𝑠𝜔_𝑗𝑑𝑠+𝑐^𝑛_𝑗=1^𝑡_𝜏𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠. (58) Replacing 𝜔 by 𝜃_−𝑡𝜔 in (57) and (58), we have 𝛼²₀𝑣𝑡,𝜃_−𝑡𝜔²₂+𝛼²₁𝐴^1/2𝑣(𝑡,𝜃_−𝑡𝜔)²₂≤𝑒^{𝐽1(𝜏−𝑡,0,𝜔)}𝛼²₀𝑣_𝜏𝜃_−𝑡𝜔²₂+𝛼²₁𝐴^1/2𝑣_𝜏𝜃_−𝑡𝜔²₂+⁰_𝜏−𝑡𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑠𝜔_𝑗²+𝑐𝑓(𝑠+𝑡)²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}𝑑𝑠. (59) Note that {|𝑦_𝑗(𝜃_𝑡𝜔_𝑗)|}^𝑛_𝑗=1 are stationary and ergodic (cf. [24]), then it follows from ergodic theorem that lim_𝑡→∞1𝑡⁰_−𝑡𝑦_𝑗𝜃_𝜏𝜔_𝑗𝑑𝜏=𝐸𝑦_𝑗𝜔_𝑗. (60) On the other hand, we have lim_𝑡→∞1𝑡⁰_−𝑡𝑦_𝑗𝜃_𝜏𝜔_𝑗𝑑𝜏=𝐸𝑦_𝑗𝜔_𝑗=𝑐√𝜇,lim_𝑡→∞1𝑡⁰_−𝑡𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏=𝐸𝑦_𝑗𝜔_𝑗²=𝑐𝜇. (61) Employing (61), we have lim_{𝑠→−∞}𝐽₁(𝑠,0,𝜔)𝑠=lim_{𝑠→−∞}𝜈𝜆₁+𝑐^𝑚_𝑗=11𝑠⁰_𝑠𝑦_𝑗𝜃_𝜏𝜔_𝑗𝑑𝜏+𝑐^𝑚_𝑗=11𝑠⁰_𝑠𝑦_𝑗𝜃_𝜏𝜔_𝑗²=𝑑𝜏𝜈𝜆₁−𝑐√𝜇−𝑐𝜇. (62) So for ℙ a.e. 𝜔∈Ω, there are 𝑇₀(𝜔)>0 and 𝜇₀>0 such that, for 𝑠≥𝑇₁(𝜔) and 𝜇≥𝜇₀, 𝐽₁(𝑠,0,𝜔)𝑠=𝜈𝜆₁+𝑐^𝑚_𝑗=11𝑠⁰_𝑠𝑦_𝑗𝜃_𝜏𝜔_𝑗𝑑𝜏+𝑐∑^𝑚_𝑗=11𝑠∫⁰_𝑠𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝐽𝑑𝜏<0,₁(𝑠,0,𝜔)⟶−∞,as𝑠⟶−∞. (63) Since {|𝑦_𝑗(𝜔_𝑗)|}^𝑛_𝑗=1 are tempered, the integral 𝑟₁(𝜔)=^0𝑛_−∞𝑗=1𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑒^{𝐽1(𝑠,0,𝜔)}𝑑𝑠 (64) is convergent. For any 𝑡>𝜏, there exists 𝐾∈ℕ such that −𝐾<𝜏−𝑡, and we have ⁰_𝜏−𝑡𝑓(𝑠+𝑡)²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}=𝑑𝑠_0𝜏−𝑡𝑓₀(𝑠+𝑡+𝜂)²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}≤𝑑𝑠_0−𝐾𝑓₀(𝑠+𝑡+𝜂)²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}𝑑𝑠. (65) Then by piecewise integration, we have ⁰_−𝐾𝑓₀𝑠+𝑡+𝜂²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}=𝑑𝑠_{𝐾𝑗=1−𝑛+𝑗−𝑛+𝑗−1}𝑓₀𝑠+𝑡+𝜂²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}𝑑𝑠. (66) By differential mean value theorem, for each 1≤𝑗≤𝐾, there exists 𝑠_𝑗∈[−𝑛+𝑗−1,−𝑛+𝑗], such that ^𝐾_𝑗=1^−𝑛+𝑗_{−𝑛+𝑗−1}𝑓₀𝑠+𝑡+𝜂²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}=𝑑𝑠_𝐾𝑗=1𝑒^{𝐽1(𝑠𝑗,0,𝜔)−𝑛+𝑗}_{−𝑛+𝑗−1}𝑓₀(𝑠+𝑡+𝜂)²_𝑉′𝑑𝑠. (67) By the property of normal function, there exists 𝑀_𝑓0, which just depends on 𝑓₀, such that sup_{𝑡∈ℝ𝑡}^𝑡+1𝑓₀(𝑠)²_𝑉′𝑑𝑠<𝑀_𝑓0. (68) It is easy to check that the series ∑^∞_𝑗=1𝑒^{𝐽1(𝑠𝑗,0,𝜔)} is convergent. Associating with (65)–(68), for any 𝑡>𝜏, we have ⁰_𝜏−𝑡𝑓(𝑠+𝑡)²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}𝑑𝑠≤𝑟₂(𝜔), (69) where 𝑟₂(𝜔)=𝑀_𝑓0^∞_𝑗=1𝑒^{𝐽1(𝑠𝑗,0,𝜔)}. (70) Therefore (64) and (69) imply that, for 𝑡>𝜏, the second term on the right-hand side of (59) can be bounded by ⁰_𝜏−𝑡𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑠𝜔_𝑗²+𝑐𝑓(𝑠+𝑡)²_𝑉′𝑒^{𝐽1(𝑠,0,𝜔)}𝑑𝑠≤𝑟₃(𝜔), (71) where 𝑟₃(𝜔)=𝑐𝑟₁(𝜔)+𝑐𝑟₂(𝜔). For 𝑣_𝜏(𝜔)∈𝐵(𝜔) being tempered, there exists 𝑇₁(𝜔)>0 such that, when 𝑡≥𝑇₂(𝜔), 𝑒^{𝐽1(𝜏−𝑡,0,𝜔)}𝛼²₀𝑣_𝜏𝜃_−𝑡𝜔²₂+𝛼²₁𝐴^1/2𝑣_𝜏𝜃_−𝑡𝜔²₂<𝑟₃(𝜔). (72) Letting 𝑇₃(𝜔)=max{𝑇₁(𝜔),𝑇₂(𝜔)} and 𝑟₄(𝜔)=2𝑟₃(𝜔), when 𝑡≥𝑇₃(𝜔), we have 𝛼²₀𝑣(𝑡,𝜃_−𝑡𝜔)²₂+𝛼²₁𝐴^1/2𝑣(𝑡,𝜃_−𝑡𝜔)²₂≤𝑟₄(𝜔),∀𝑡≥𝑇₃(𝜔),𝑓∈Σ𝑓₀. (73)
We define 𝒦₁(𝜔)=𝑣(𝜔)∈𝑉∣𝛼²₀‖𝑣(𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝜔)²₂≤𝑟₄(𝜔). (74) In conclusion, 𝒦₁(𝜔) is a uniformly random absorbing set for {𝒰(𝑡,𝜏,𝜔)} in 𝑉, which complete the proof.

Lemma 16. Let {𝐵(𝜔)}_𝜔∈Ω be tempered and 𝑣_𝜏(𝜔)∈𝐵(𝜔). Then the process {𝒰_𝑓(𝑡,𝜏,𝜔)} corresponding to (48) possesses a uniformly (with respect to 𝑓∈Σ(𝑓₀)) random absorbing set 𝒦₂(𝜔) in 𝐷(𝐴).

Proof. Integrating (56) into [𝑡,𝑡+1] where 𝑡≥𝑇₃(𝜔), we have 𝛼²₀‖𝑣(𝑡+1,𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝑡+1,𝜔)²₂+𝜈_𝑡^𝑡+1𝛼²₀𝐴^1/2𝑣(𝑠,𝜔)²₂+𝛼²₁‖𝐴𝑣(𝑠,𝜔)‖²₂≤𝛼𝑑𝑠₂₀‖𝑣(𝑡,𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝑡,𝜔)²₂+𝑐^𝑛_𝑗=1𝑡^𝑡+1𝑦_𝑗𝜃_𝑠𝜔_𝑗×𝛼²₀‖𝑣(𝑠,𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝑠,𝜔)²₂𝑑𝑠+𝑐^𝑛_𝑗=1𝑡^𝑡+1𝑦_𝑗𝜃_𝑠𝜔_𝑗²×𝛼²₀‖𝑣(𝑠,𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝑠,𝜔)²₂𝑑𝑠+𝑐^𝑛_𝑗=1𝑡^𝑡+1𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠. (75) Replacing 𝜔 by 𝜃_−𝑡−1𝜔 in (75), we have 𝜈_𝑡^𝑡+1𝛼²₀𝐴^1/2𝑣𝑠,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴𝑣𝑠,𝜃_−𝑡−1𝜔²₂𝑑𝑠≤𝑐^𝑛_𝑗=1𝑡^𝑡+1𝑦_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗𝛼²₀𝑣𝑠,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴^1/2𝑣𝑠,𝜃_−𝑡−1𝜔²₂𝑑𝑠+𝑐^𝑛_𝑗=1𝑡^𝑡+1𝑦_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗²𝛼²₀𝑣𝑠,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴^1/2𝑣𝑠,𝜃_−𝑡−1𝜔²₂𝑑𝑠+𝑐^𝑛_𝑗=1𝑡^𝑡+1𝑦_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗²+𝛼𝑑𝑠₂₀𝑣𝑡,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴^1/2𝑣𝑡,𝜃_−𝑡−1𝜔²₂. (76) As the previous consideration in Lemma 15, for 𝑠∈[𝑡,𝑡+1) where 𝑡≥𝑇₂(𝜔), we have 𝛼²₀𝑣(𝑠,𝜃_−𝑡−1𝜔)²₂+𝛼²₁𝐴^1/2𝑣𝑠,𝜃_−𝑡−1𝜔²₂≤𝑟₅(𝜔), (77) where 𝑟₅(𝜔)=𝑒^𝜈𝜆1𝑟₄(𝜔).
Associating (76) with (77), we have _𝑡^𝑡+1𝛼²₀𝐴^1/2𝑣𝑠,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴𝑣𝑠,𝜃_−𝑡−1𝜔²₂𝑑𝑠≤𝑟₆(𝜔), (78) where 𝑟₆(𝜔)=𝑐𝑟₅(𝜔)^𝑛_𝑗=1⁰₋₁𝑦_𝑗𝜃_𝑠𝜔_𝑗𝑑𝑠+𝑐𝑟₅(𝜔)^𝑛_𝑗=1⁰₋₁𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠+𝑐^𝑛_𝑗=1⁰₋₁𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠+𝑟₅(𝜔). (79)
Now we let 𝑤=𝐴𝑣 in (51), and we have 12𝑑𝛼𝑑𝑡²⁰𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂+𝜈𝛼²₀‖𝐴𝑣‖²₂+𝛼²₁𝐴^3/2𝑣²₂=−𝜈^𝑛_𝑗=1𝐴ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗−𝐵𝜃,𝐴𝑣𝑣+𝑧_𝑡𝜔,𝛼²₀𝑣+𝛼²₁+𝐴𝑣_𝑛𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗+()+,𝐴𝑣𝑃𝑓𝑥,𝑡,𝐴𝑣^𝑛_𝑗=1𝜇ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝐴𝑣. (80) Employing Lemma 14, the bilinear term in (80) can be bounded by 𝐵𝑣+𝑧𝜃_𝑡𝜔,𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗𝐴,𝐴𝑣≤𝑐^1/2𝑣+𝐴^1/2𝑧𝜃_𝑡𝜔₂^1/2𝐴𝑣+𝐴𝑧(𝜃_𝑡𝜔)₂^1/2×𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗2𝐴^3/2𝑣₂+𝑐𝐴𝑣+𝐴𝑧𝜃_𝑡𝜔₂𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗2×‖𝐴𝑣‖₂^1/2𝐴^3/2𝑣₂^1/2≤𝜈𝛼²₁𝐴^3/2𝑣²₂+𝜈𝛼²₀4‖𝐴𝑣‖²₂+𝑐𝛼²₀‖𝑣‖²₂+𝛼²₀𝐴^1/2𝑣²₂²+𝑐𝛼²₀𝐴^1/2𝑣²₂+𝛼²₀‖𝐴𝑣‖²₂²+𝑐𝛼²₀‖𝑣‖²₂+𝛼²₀𝐴^1/2𝑣²₂𝛼²₀𝐴^1/2𝑣²₂+𝛼²₀‖𝐴𝑣‖²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀‖𝑣‖²₂+𝛼²₀𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀𝐴^1/2𝑣²₂+𝛼²₀‖𝐴𝑣‖²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗⁴. (81) The other terms are bounded by 𝜈^𝑛_𝑗=1𝐴ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝐴𝑣≤𝜈𝛼²₀4‖𝐴𝑣‖²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗².𝑃𝑓(𝑥,𝑡),𝐴𝑣≤𝑓_𝑉′‖𝐴𝑣‖₂≤𝜈𝛼²₀4‖𝐴𝑣‖²₂+𝑐𝑓²_𝑉′.𝜇^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝐴𝑣≤𝜈𝛼²₀4‖𝐴𝑣‖²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗². (82) Associating with all the above inequalities, we have 𝑑𝛼𝑑𝑡²⁰𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂≤𝑐𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂²+𝑐𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂²+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗⁴+𝑐𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂×𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝑐𝑓²_𝑉′+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂. (83) Applying Gronwall’s lemma on [𝑠,𝑡+1] where 𝑠∈(𝑡,𝑡+1) and 𝑡≥𝑇₃(𝜔), we have 𝛼²₀𝐴^1/2𝑣(𝑡+1,𝜔)²₂+𝛼²₁‖𝐴𝑣(𝑡+1,𝜔)‖²₂≤𝛼²₀𝐴^1/2𝑣(𝑠,𝜔)²₂+𝛼²₁‖𝐴𝑣(𝑠,𝜔)‖²₂𝑒^{𝐽2(𝑠,𝑡+1,𝜔)}+𝑐_𝑠^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑒^{𝐽2(𝜏,𝑡+1,𝜔)}𝑑𝜏+𝑐_𝑠^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗⁴𝑒^{𝐽2(𝜏,𝑡+1,𝜔)}𝑑𝜏+𝑐_𝑠^𝑡+1𝑓(𝜏)²_𝑉′𝑒^{𝐽2(𝜏,𝑡+1,𝜔)}𝑑𝜏+𝑐_𝑠^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²×𝛼²₀‖𝑣(𝜏,𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝜏,𝜔)²₂𝑒^{𝐽2(𝜏,𝑡+1,𝜔)}𝑑𝜏+𝑐_𝑠^𝑡+1𝛼²₀‖𝑣(𝜏,𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝜏,𝜔)²₂²𝑒^{𝐽2(𝜏,𝑡+1,𝜔)}𝑑𝜏, (84) where 𝐽₂(𝑠,𝑡,𝜔)=𝑐^𝑡_𝑠𝛼²₀‖𝑣(𝜏,𝜔)‖²₂+𝛼²₁𝐴^1/2𝑣(𝜏,𝜔)²₂𝑑𝜏+𝑐^𝑡_𝑠𝛼²₀𝐴^1/2𝑣(𝜏,𝜔)²₂+𝛼²₁‖𝐴𝑣(𝜏,𝜔)‖²₂𝑑𝜏+𝑐^𝑡_𝑠^𝑛_𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏. (85) Replacing 𝜔 by 𝜃_−𝑡−1𝜔 in (84) and (85), we have 𝐽₂𝜏,𝑡+1,𝜃_−𝑡−1𝜔≤𝑐_𝑡^𝑡+1𝛼²₀𝑣𝜏,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴^1/2𝑣𝜏,𝜃_−𝑡−1𝜔²₂𝑑𝜏+𝑐_𝑡^𝑡+1𝛼²₀𝐴^1/2𝑣𝜏,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴𝑣(𝜏,𝜃_−𝑡−1𝜔)²₂𝑑𝜏+𝑐^𝑛_𝑗=1𝑡^𝑡+1𝑦_𝑗𝜃_{𝜏−𝑡−1}𝜔_𝑗²𝑑𝜏≤𝑐𝑟₅(𝜔)+𝑐𝑟₆(𝜔)+𝑐^𝑛_𝑗=1⁰₋₁𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏_▵=𝑟₇(𝜔). (86) And then 𝛼²₀𝐴^1/2𝑣𝑡+1,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴𝑣(𝑡+1,𝜃_−𝑡−1𝜔)²₂≤𝛼²₀𝐴^1/2𝑣(𝑠,𝜃_−𝑡−1𝜔)²₂+𝛼²₁𝐴𝑣𝑠,𝜃_−𝑡−1𝜔²₂𝑒^𝑟7(𝜔)+𝑐𝑒^𝑟7(𝜔)𝑟₅(𝜔)²+𝑐𝑒^{𝑟7(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏+𝑐𝑒^{𝑟7(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗⁴𝑑𝜏+𝑐𝑒^𝑟7(𝜔)_𝑡^𝑡+1𝑓(𝜏)²_𝑉′𝑑𝜏+𝑐𝑒^𝑟7(𝜔)𝑟₅(𝜔)^0𝑛_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏. (87) Integrating (87) with respect to 𝑠 over (𝑡,𝑡+1) where 𝑡≥𝑇₃(𝜔) and employing the property of normal function in (68), we have 𝛼²₀𝐴^1/2𝑣𝑡+1,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴𝑣𝑡+1,𝜃_−𝑡−1𝜔²₂≤𝑟₈(𝜔),∀𝑡≥𝑇₃(𝜔),𝑓∈Σ𝑓₀, (88) where𝑟₈(𝜔)=𝑟₆(𝜔)𝑒^𝑟7(𝜔)+𝑐𝑒^𝑟7(𝜔)𝑟₅(𝜔)²+𝑐𝑒^{𝑟7(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏+𝑐𝑒^{𝑟7(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗⁴𝑑𝜏+𝑐𝑒^𝑟7(𝜔)𝑀_𝑓0+𝑐𝑒^𝑟7(𝜔)𝑟₅(𝜔)^0𝑛_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏. (89)
We define 𝒦₂(𝜔)=𝑣(𝜔)∈𝐷(𝐴)∣𝛼²₀𝐴^1/2𝑣(𝜔)²₂+𝛼²₁‖𝐴𝑣(𝜔)‖²₂≤𝑟₈(𝜔). (90) In conclusion, 𝒦₂(𝜔) is a uniformly random absorbing set for {𝒰(𝑡,𝜏,𝜔)} in 𝐷(𝐴), which complete the proof.

So much for that we have proved the existence of bounded uniformly (with respect to 𝑓∈Σ(𝑓₀)) random absorbing sets in 𝑉 and 𝐷(𝐴). Next we derive the existence of uniformly (with respect to 𝑓∈Σ(𝑓₀)) random attractors in 𝑉 and 𝐷(𝐴).

Theorem 17. If 𝑓₀(𝑥,𝑠) is a normal function in 𝐿²_loc(ℝ,𝑉^′), then the process {𝒰_𝑓0(𝑡,𝜏,𝜔)} corresponding to (48) possess a compact uniformly (with respect to 𝜏∈ℝ) random attractor 𝒜₁(𝜔) in 𝑉 which coincides with the uniformly (with respect to 𝑓∈Σ(𝑓₀)) random attractor 𝒜_Σ(𝑓0) of the family of processes {𝒰_𝑓(𝑡,𝜏,𝜔)}, 𝑓∈Σ(𝑓₀)∶𝒜₁(𝜔)=𝒜_Σ(𝑓0)(𝜔)=𝒲_0,Σ(𝑓0)𝒦₁=_{𝑓∈Σ(𝑓0)}𝒩_𝑓(0,𝜔), (91) where 𝒦₁(𝜔) is the uniformly (with respect to 𝑓∈Σ(𝑓₀)) random absorbing set in 𝑉 and 𝒩_𝑓(𝜔) is the kernel of the process {𝒰_𝑓(𝑡,𝜏,𝜔)}. Furthermore, the kernel 𝒩_𝑓(𝜔) is nonempty for all 𝑓∈Σ(𝑓₀).

Proof. We only have to verify condition(C). As the previous section, for fixed 𝑚, let 𝐻₁ be the subspace spanned by {𝑒_𝑗}^𝑚_𝑗=1 and 𝐻₂ the orthogonal complement of 𝐻₁ in 𝐻. We write𝑣=𝑣₁+𝑣₂,𝑣₁∈𝐻₁,𝑣₂∈𝐻₂forany𝑣∈𝐻. (92) The proof of boundary of 𝑣₁ is similar to the proof in Lemma 15. We need to estimate 𝑣₂, where 𝑣=𝑣₁+𝑣₂ is a solution of (48). Letting 𝑤=𝑣₂ in (51), we have 12𝑑𝛼𝑑𝑡²⁰𝑣₂²₂+𝛼²₁𝐴^1/2𝑣₂²₂+𝜈𝛼²₀𝐴^1/2𝑣₂²₂+𝛼²₁𝐴𝑣₂²₂=−𝜈^𝑛_𝑗=1𝐴ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝑣₂−𝐵𝑣+𝑧𝜃_𝑡𝜔,𝛼²₀𝑣+𝛼²₁+𝐴𝑣_𝑛𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝑣₂+𝑃𝑓(𝑥,𝑡),𝑣₂+𝜇^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝑣₂. (93) Employing Lemma 14, the second term on the right-hand side of (93) can be bounded by 𝐵𝑣+𝑧𝜃_𝑡𝜔,𝛼²₀𝑣+𝛼²₁+𝐴𝑣_𝑛𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝑣₂≤𝑐𝑣+𝑧𝜃_𝑡𝜔₂^1/2𝐴^1/2𝑣+𝐴^1/2𝑧𝜃_𝑡𝜔₂^1/2×𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗2𝐴𝑣₂₂+𝑐𝐴^1/2𝑣+𝐴^1/2𝑧𝜃_𝑡𝜔₂×𝛼²₀𝑣+𝛼²₁𝐴𝑣+^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗2×𝐴^1/2𝑣₂₂^1/2𝐴𝑣₂₂^1/2≤𝜈𝛼²₀8𝐴^1/2𝑣₂²₂+𝜈𝛼²₁2𝐴𝑣₂²₂+𝑐𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂×𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂+𝑐𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂²+𝑐𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂²+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑗𝜔_𝑗²+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑗𝜔_𝑗⁴+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑗𝜔_𝑗²𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑗𝜔_𝑗²𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂. (94) By Poincaré’s inequality, note that 𝜈^𝑛_𝑗=1𝐴ℎ_𝑗𝑦_𝑗𝜃_𝑗𝜔_𝑗,𝑣₂≤𝜈𝛼²₀8𝐴^1/2𝑣₂²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²,𝑃𝑓,𝑣₂≤𝑓_𝑉′𝑣₂≤𝜈𝛼²₀8𝐴^1/2𝑣₂²₂+𝑐𝑓²_𝑉′,𝜇^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝑣₂≤𝜈𝛼²₀8𝐴^1/2𝑣₂²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑗𝜔_𝑗². (95) Associating with (93)–(95) and applying Poincaré inequality, we have 𝑑𝛼𝑑𝑡²⁰𝑣₂²₂+𝛼²₁𝐴^1/2𝑣₂²₂+𝜈𝜆_𝑚+1𝛼²₀𝑣₂²₂+𝛼²₁𝐴^1/2𝑣₂²₂≤𝑐𝑓²_𝑉′+𝐺(𝑡,𝜔), (96) where 𝐺()𝛼𝑡,𝜔=𝑐²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂²+𝑐𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂²+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²+𝑐𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂×𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗⁴+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂. (97) Applying Gronwall Lemma on [𝑡,𝑡+1] where 𝑡≥𝑇₃(𝜔), we have 𝛼²₀𝑣₂(𝑡+1,𝜔)²₂+𝛼²₁𝐴^1/2𝑣₂(𝑡+1,𝜔)²₂≤𝑒^{−𝜈𝜆𝑚+1}𝛼²₀𝑣₂(𝑡,𝜔)²₂+𝛼²₁𝐴^1/2𝑣₂(𝑡,𝜔)²₂+_𝑡^𝑡+1𝐺(𝑠,𝜔)𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠+𝑐_𝑡^𝑡+1𝑓(𝑠)²_𝑉′𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠. (98)
Replacing 𝜔 by 𝜃_−𝑡−1𝜔 in (98), we have 𝛼²₀𝑣₂𝑡+1,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴^1/2𝑣₂(𝑡+1,𝜃_−𝑡−1𝜔)²₂≤𝑒^{−𝜈𝜆𝑚+1}𝛼²₀𝑣₂𝑡,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴^1/2𝑣₂(𝑡,𝜃_−𝑡−1𝜔)²₂+_𝑡^𝑡+1𝐺𝑠,𝜃_−𝑡−1𝜔𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠+𝑐_𝑡^𝑡+1𝑓(𝑠)²_𝑉′𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠. (99) Now we need to estimate the second term on the right-hand side of (99). As the previous consideration in Lemma 15, for 𝑠∈[𝑡,𝑡+1) where 𝑡≥𝑇₃(𝜔), we have 𝛼²₀𝑣(𝑠,𝜃_−𝑡−2)²₂+𝛼²₁𝐴^1/2𝑣𝑠,𝜃_−𝑡−2²₂≤𝑟₉(𝜔), (100) where 𝑟₉(𝜔)=𝑒^2𝜈𝜆1𝑟₄(𝜔). As the previous consideration in Lemma 16, we have _𝑡^𝑡+1𝛼²₀𝐴^1/2𝑣(𝑠,𝜃_−𝑡−2𝜔)²₂+𝛼²₁𝐴𝑣(𝑠,𝜃_−𝑡−2𝜔)²₂𝑑𝑠≤𝑟₁₀(𝜔), (101) where 𝑟₁₀(𝜔)=𝑐𝑟₉(𝜔)^𝑛_𝑗=1⁰₋₂𝑦_𝑗𝜃_𝑠𝜔_𝑗𝑑𝑠+𝑐𝑟₉(𝜔)^𝑛_𝑗=1⁰₋₂𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠+𝑐^𝑛_𝑗=1⁰₋₂𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠+𝑟₉(𝜔). (102) Replacing 𝜔 by 𝜃_−𝑡−2𝜔 in (84), we have 𝛼²₀𝐴^1/2𝑣(𝑡+1,𝜃_−𝑡−2𝜔)²₂+𝛼²₁𝐴𝑣(𝑡+1,𝜃_−𝑡−2𝜔)²₂≤𝛼²₀𝐴^1/2𝑣(𝑠,𝜃_−𝑡−2𝜔)²₂+𝛼²₁𝐴𝑣(𝑠,𝜃_−𝑡−2𝜔)²₂𝑒^{𝐽2(𝑠,𝑡+1,𝜃−𝑡−2𝜔)}+𝑐_𝑡^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_{𝜏−𝑡−2}𝜔_𝑗²𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−2𝜔)}𝑑𝜏+𝑐_𝑡^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_{𝜏−𝑡−2}𝜔_𝑗⁴𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−2𝜔)}𝑑𝜏+𝑐_𝑡^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_{𝜏−𝑡−2}𝜔_𝑗²×𝛼²₀𝑣(𝜏,𝜃_−𝑡−2𝜔)²₂+𝛼²₁𝐴^1/2𝑣(𝜏,𝜃_−𝑡−2𝜔)²₂𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−2𝜔)}𝑑𝜏+𝑐_𝑡^𝑡+1𝛼²₀𝑣(𝜏,𝜃_−𝑡−2𝜔)²₂+𝛼²₁𝐴^1/2𝑣(𝜏,𝜃_−𝑡−2𝜔)²₂²𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−2𝜔)}𝑑𝜏+𝑐_𝑡^𝑡+1𝑓(𝜏)²_𝑉′𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−2𝜔)}𝑑𝜏. (103) Analogously to the previous estimates, we have 𝐽₂𝜏,𝑡+1,𝜃_−𝑡−2𝜔≤𝑐_𝑡^𝑡+1𝛼²₀𝑣𝜏,𝜃_−𝑡−2𝜔²₂+𝛼²₁𝐴^1/2𝑣(𝜏,𝜃_−𝑡−2𝜔)²₂𝑑𝜏+𝑐_𝑡^𝑡+1𝛼²₀𝐴^1/2𝑣𝜏,𝜃_−𝑡−2𝜔²₂+𝛼²₁𝐴𝑣𝜏,𝜃_−𝑡−2𝜔²₂𝑑𝜏+𝑐^𝑛_𝑗=1𝑡^𝑡+1𝑦_𝑗𝜃_{𝜏−𝑡−2}𝜔_𝑗²𝑑𝜏≤𝑟₁₁(𝜔), (104) where 𝑟₁₁(𝜔)=𝑐𝑟₉(𝜔)+𝑐𝑟₁₀(𝜔)+𝑐^𝑛_𝑗=1⁰₋₂𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏. (105) Integrating (103) with respect to 𝑠∈(𝑡,𝑡+1) associating (100), (101), and (104) with (68), we have 𝛼²₀𝐴^1/2𝑣(𝑡+1,𝜃_−𝑡−2𝜔)²₂+𝛼²₁𝐴𝑣(𝑡+1,𝜃_−𝑡−2𝜔)²₂≤𝑟₁₂(𝜔), (106) where 𝑟₁₂(𝜔)=𝑐𝑟₁₀(𝜔)𝑒^𝑟11(𝜔)+𝑐𝑒^{𝑟11(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏+𝑐𝑒^{𝑟11(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗⁴𝑑𝜏+𝑐𝑟₉(𝜔)𝑒^{𝑟11(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏+𝑐𝑟₉(𝜔)²𝑒^𝑟11(𝜔)+𝑐𝑀_𝑓0𝑒^𝑟11(𝜔). (107) Replacing 𝑠, 𝑡+1, and 𝜔 by 𝑡, 𝑠, and 𝜃_−𝑡−1𝜔, respectively, in (84) where 𝑠∈(𝑡,𝑡+1) and 𝑡≥𝑇₃(𝜔), we have 𝛼²₀𝐴^1/2𝑣𝑠,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴𝑣(𝑠,𝜃_−𝑡−1𝜔)²₂≤𝛼²₀𝐴^1/2𝑣(𝑡,𝜃_−𝑡−1𝜔)²₂+𝛼²₁𝐴𝑣(𝑡,𝜃_−𝑡−1𝜔)²₂𝑒^{𝐽2(𝑡,𝑡+1,𝜃−𝑡−1𝜔)}+𝑐_𝑡^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_{𝜏−𝑡−1}𝜔_𝑗²𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−1𝜔)}𝑑𝜏+𝑐_𝑡^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_{𝜏−𝑡−1}𝜔_𝑗⁴𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−1𝜔)}𝑑𝜏+𝑐_𝑡^𝑛𝑡+1_𝑗=1𝑦_𝑗𝜃_{𝜏−𝑡−1}𝜔_𝑗²×𝛼²₀𝑣𝜏,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴^1/2𝑣(𝜏,𝜃_−𝑡−1𝜔)²₂𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−1𝜔)}𝑑𝜏+𝑐_𝑡^𝑡+1𝛼²₀𝑣(𝜏,𝜃_−𝑡−1𝜔)²₂+𝛼²₁𝐴^1/2𝑣(𝜏,𝜃_−𝑡−1𝜔)²₂²𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−1𝜔)}𝑑𝜏+𝑐_𝑡^𝑡+1𝑓(𝜏)²_𝑉′𝑒^{𝐽2(𝜏,𝑡+1,𝜃−𝑡−1𝜔)}𝑑𝜏≤𝑟₁₂(𝜔)𝑒^𝑟7(𝜔)+𝑐𝑒^{𝑟7(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏+𝑐𝑒^{𝑟7(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗⁴𝑑𝜏+𝑐𝑒^𝑟7(𝜔)𝑀_𝑓0+𝑐𝑟₅(𝜔)²𝑒^𝑟7(𝜔)+𝑐𝑟₅(𝜔)𝑒^{𝑟7(𝜔)0𝑛}_−1𝑗=1𝑦_𝑗𝜃_𝜏𝜔_𝑗²𝑑𝜏_▵=𝑟₁₃(𝜔). (108) Associating with the above inequalities, we have _𝑡^𝑡+1𝐺𝑠,𝜃_−𝑡−1𝜔𝑑𝑠≤𝑟₁₄(𝜔), (109) where 𝑟₁₄(𝜔)=𝑐𝑟₅(𝜔)²+𝑐𝑟₁₃(𝜔)²+𝑐^0𝑛_−1𝑗=1𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠+𝑐𝑟₅(𝜔)𝑟₁₃(𝜔)+𝑐^0𝑛_−1𝑗=1𝑦_𝑗𝜃_𝑠𝜔_𝑗⁴𝑑𝑠+𝑐𝑟₅(𝜔)^0𝑛_−1𝑗=1𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠+𝑐𝑟₁₃(𝜔)^0𝑛_−1𝑗=1𝑦_𝑗𝜃_𝑠𝜔_𝑗²𝑑𝑠. (110) According to the property of {𝜆_𝑗}^∞_𝑗=1 mentioned in Section 3, for ℙ a.e 𝜔∈Ω and ∀𝜀>0, there exists 𝑀 such that for 𝑚≥𝑀𝑒^{−𝜈𝜆𝑚+1}𝛼²₀𝑣₂(𝑡,𝜃_−𝑡−1𝜔)²₂+𝛼²₁𝐴^1/2𝑣₂(𝑡,𝜃_−𝑡−1𝜔)²₂≤𝑟₅(𝜔)𝑒^{𝜈𝜆𝑚+1}≤𝜀𝑟₅(𝜔),_𝑡𝑡+1𝐺𝑠,𝜃_−𝑡−1𝜔𝑒^{−𝜈𝜆𝑚+1}𝑟𝑑𝑠≤¹⁴(𝜔)𝑒^{𝜈𝜆𝑚+1}≤𝜀𝑟₁₄(𝜔),𝑐_𝑡^𝑡+1𝑓(𝑠)²_𝑉′𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠≤𝑐𝑀_𝑓0𝑒^{𝜈𝜆𝑚+1}≤𝜀. (111) Therefore, we deduce from (99) that, for ℙ a.e. 𝜔∈Ω, 𝛼²₀𝑣₂(𝑡,𝜃_−𝑡𝜔)²₂+𝛼²₁𝐴^1/2𝑣₂(𝑡,𝜃_−𝑡𝜔)²₂≤𝜀+𝜀𝑟₅(𝜔)+𝜀𝑟₁₄(𝜔),∀𝑡≥𝑇₃(𝜔),𝑓∈Σ𝑓₀, (112) which indicates {𝒰_𝑓(𝑡,𝜏,𝜔)}, 𝑓∈Σ(𝑓₀) satisfying uniform (with respect to 𝑓∈Σ(𝑓₀)) condition(C) in 𝑉. According to Theorem 12, the proof is completed.

In the following we prove the existence of uniformly random attractor for the families of processes {𝒰_𝑓(𝑡,𝜏,𝜔)}, 𝑓∈Σ(𝑓₀) corresponding to (48) in 𝐷(𝐴).

Theorem 18. If 𝑓₀(𝑥,𝑠) is a normal function in 𝐿²_loc(ℝ,𝑉^′), then the process {𝒰_𝑓0(𝑡,𝜏,𝜔)} corresponding to (48) possess a compact uniformly (with respect to 𝜏∈ℝ) random attractor 𝒜₂(𝜔) in 𝐷(𝐴) which coincides with the uniformly (with respect to 𝑓∈Σ(𝑓₀)) random attractor 𝒜_Σ(𝑓0) of the family of processes {𝒰_𝑓(𝑡,𝜏,𝜔)}, 𝑓∈Σ(𝑓₀)∶𝒜₂(𝜔)=𝒜_Σ(𝑓0)(𝜔)=𝒲_0,Σ(𝑓0)𝒦₂=_{𝑓∈Σ(𝑓0)}𝒩_𝑓(0,𝜔), (113) where 𝒦₂(𝜔) is the uniformly (with respect to 𝑓∈Σ(𝑓₀)) random absorbing set in 𝐷(𝐴) and 𝒩_𝑓(𝜔) is the kernel of the process {𝒰_𝑓(𝑡,𝜏,𝜔)}. Furthermore, the kernel 𝒩_𝑓(𝜔) is nonempty for all 𝑓∈Σ(𝑓₀).

Proof. In Lemma 16, we have proved that the semigroup of processes {𝒰_𝑓(𝑡,𝜏,𝜔)}, 𝑓∈Σ(𝑓₀), has a uniformly random absorbing set in 𝐷(𝐴). Now we testify that the semigroup of processes corresponding to (48) satisfies uniform (with respect to 𝑓∈Σ(𝑓₀)) condition (C). Analogously to the proof in Lemma 16, we easily check that 𝑣₁ is bounded in 𝐷(𝐴). Letting 𝑤=𝐴𝑣₂ in (51), we have 12𝑑𝛼𝑑𝑡²⁰𝐴^1/2𝑣₂²₂+𝛼²₁𝐴𝑣₂²₂+𝜈𝛼²₀𝐴𝑣₂²₂+𝛼²₁𝐴^3/2𝑣₂²₂=−𝜈^𝑛_𝑗=1𝐴ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝐴𝑣₂−𝐵𝑣+𝑧𝜃_𝑡𝜔,𝛼²₀𝑣+𝛼²₁+𝐴𝑣_𝑛𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝐴𝑣₂+𝑃𝑓(𝑥,𝑡),𝐴𝑣₂+𝜇^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝐴𝑣₂. (114) Applying Lemma 14, the second term on the right-hand side of (114) can be bounded by 𝐵𝑣+𝑧𝜃_𝑡𝜔,𝛼²₀𝑣+𝛼²₁+𝐴𝑣_𝑛𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝐴𝑣₂≤𝜈𝛼²₀8𝐴𝑣₂²₂+𝜈𝛼²₁2𝐴^3/2𝑣₂²₂+𝑐𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂×𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂+𝑐𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂²+𝑐𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂²+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗⁴+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀‖𝑣‖²₂+𝛼²₁𝐴^1/2𝑣²₂²+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²𝛼²₀𝐴^1/2𝑣²₂+𝛼²₁‖𝐴𝑣‖²₂². (115) Note that 𝜈^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_j,𝐴𝑣₂≤𝜈𝛼²₀8𝐴𝑣₂²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗²,𝑃𝑓(𝑥,𝑡),𝐴𝑣₂≤𝑓_𝑉′𝐴𝑣₂₂≤𝜈𝛼²₀8𝐴𝑣₂²₂+𝑐𝑓²_𝑉′,𝜇^𝑛_𝑗=1ℎ_𝑗𝑦_𝑗𝜃_𝑡𝜔_𝑗,𝐴𝑣₂≤𝜈𝛼²₀8𝐴𝑣₂²₂+𝑐^𝑛_𝑗=1𝑦_𝑗𝜃_𝑡𝜔_𝑗². (116) Associating with (114)–(116) and applying Poincaré’s inequality, we have 𝑑𝛼𝑑𝑡²⁰𝐴^1/2𝑣₂²₂+𝛼²₁𝐴𝑣₂²₂+𝜈𝜆_𝑚+1𝛼²₀𝐴^1/2𝑣₂²₂+𝛼²₁𝐴𝑣₂²₂≤𝐺(𝑡,𝜔)+𝑐𝑓²_𝑉′. (117) Applying Gronwall’s lemma over 𝑠∈[𝑡,𝑡+1] where 𝑡≥𝑇₂(𝜔), we have 𝛼²₀𝐴^1/2𝑣₂(𝑡+1,𝜔)²₂+𝛼²₁𝐴𝑣₂(𝑡+1,𝜔)²₂≤𝛼²₀𝐴^1/2𝑣₂(𝑡,𝜔)²₂+𝛼²₁𝐴𝑣₂(𝑡,𝜔)²₂𝑒^{−𝜈𝜆𝑚+1}+_𝑡^𝑡+1𝐺(𝑠,𝜔)𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠+𝑐_𝑡^𝑡+1𝑓²_𝑉′𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠. (118) Replacing 𝜔 by 𝜃_−𝑡−1𝜔 in (118), we have 𝛼²₀𝐴^1/2𝑣₂𝑡+1,𝜃_−t−1𝜔²₂+𝛼²₁𝐴𝑣₂(𝑡+1,𝜃_−𝑡−1𝜔)²₂≤𝛼²₀𝐴^1/2𝑣₂(𝑡,𝜃_−𝑡−1𝜔)²₂+𝛼²₁𝐴𝑣₂(𝑡,𝜃_−𝑡−1𝜔)²₂𝑒^{−𝜈𝜆𝑚+1}+_𝑡^𝑡+1𝐺𝑠,𝜃_−𝑡−1𝜔𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠+𝑐_𝑡^𝑡+1𝑓²_𝑉′𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠. (119) Analogously to the consideration in Theorem 17, for ℙ a.e. 𝜔∈Ω and ∀𝜀>0, there exists 𝑀 such that, for 𝑚>𝑀, 𝑒^{−𝜈𝜆𝑚+1}𝛼²₀𝐴^1/2𝑣₂(𝑡,𝜃_−𝑡−1𝜔)²₂+𝛼²₁𝐴𝑣₂(𝑡,𝜃_−𝑡−1𝜔)²₂≤𝑟₁₂(𝜔)𝑒^{𝜈𝜆𝑚+1}≤𝜀𝑟₁₂(𝜔),_𝑡𝑡+1𝐺𝑠,𝜃_−𝑡−1𝜔𝑒^{−𝜈𝜆𝑚+1}𝑟𝑑𝑠≤¹⁴(𝜔)𝑒^{𝜈𝜆𝑚+1}≤𝜀𝑟₁₄(𝜔),𝑐_𝑡^𝑡+1𝑓(𝑠)²_𝑉′𝑒^{−𝜈𝜆𝑚+1}𝑑𝑠≤𝑐𝑀_𝑓0𝑒^{𝜈𝜆𝑚+1}≤𝜀. (120) Therefore, we deduce from (119) that, for ℙ a.e 𝜔∈Ω, 𝛼²₀𝐴^1/2𝑣₂𝑡+1,𝜃_−𝑡−1𝜔²₂+𝛼²₁𝐴𝑣₂𝑡+1,𝜃_−𝑡−1𝜔²₂≤𝜀+𝜀𝑟₁₂(𝜔)+𝜀𝑟₁₄(𝜔),∀𝑡≥𝑇₃(𝜔),𝑓∈Σ𝑓₀. (121) which indicates {𝒰_𝑓(𝑡,𝜏,𝜔)}, 𝑓∈Σ(𝑓₀) satisfying uniform (with respect to 𝑓∈Σ(𝑓₀)) condition (C) in 𝐷(𝐴). According to Theorem 12, the proof is completed.

Now we introduce a homeomorphism Φ(𝜃_𝑡𝜔)𝑢=𝑢+𝑧(𝜃_𝑡𝜔),𝑢∈𝐸, whose inverse homeomorphism Φ⁻¹(𝜃_𝑡𝜔)𝑢=𝑢−𝑧(𝜃_𝑡𝜔). Then the transformation 𝒱_𝜎(𝑡,𝜏,𝜔)=Φ𝜃_𝑡𝜔∘𝒰_𝜎(𝑡,𝜏,𝜔)∘Φ⁻¹𝜃_𝜏𝜔 (122) generates a process corresponding to (8). Note that the two processes are equivalent by (122). It is easy to check that 𝒱_𝜎(𝑡,𝜏,𝜔) has a uniformly (with respect to 𝜎∈Σ) random attractor provided 𝒰_𝜎(𝑡,𝜏,𝜔) possesses a uniformly (with respect to 𝜎∈Σ) random attractor. As a result Theorems 17 and 18 imply that (8) has a uniformly (with respect to 𝑓∈Σ(𝑓₀)) random attractor in 𝑉 and 𝐷(𝐴).

Acknowledgments

The authors would like to thank the reviewers for the valuable suggestions and comments. The work is supported by the National Natural Science Foundation (no. 11171115).