(𝐿²,𝐻¹)-Random Attractors for Stochastic Reaction-Diffusion Equation on Unbounded Domains

Abstract

We study the random dynamical system generated by a stochastic reaction-diffusion equation with additive noise on the whole space ℝ^𝑛 and prove the existence of an (𝐿²,𝐻¹)-random attractor for such a random dynamical system. The nonlinearity 𝑓 is supposed to satisfy the growth of arbitrary order 𝑝−1 (𝑝≥2). The (𝐿²,𝐻¹)-asymptotic compactness of the random dynamical system is obtained by using an extended version of the tail estimate method introduced by Wang (1999) and the cut-off technique.

1. Introduction

In this paper, we consider the asymptotic behavior of solutions to the following stochastic reaction-diffusion equation (SRDE) with additive noise in the entire space ℝ^𝑛: 𝑑𝑢+(𝜆𝑢−Δ𝑢)𝑑𝑡=𝑓(𝑥,𝑢)+𝑔(𝑥)𝑑𝑡+^𝑚_𝑗=1ℎ_𝑗𝑑𝑤_𝑗,𝑥∈ℝ^𝑛,𝑡>0, (1) with the initial condition 𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈ℝ^𝑛, (2) where 𝜆 is a positive constant, 𝑔 is a given function in 𝐿²(ℝ^𝑛), for each 𝑗=1,…,𝑚,ℎ_𝑗∈𝐻²(ℝ^𝑛)⋂𝑊^2,𝑝(ℝ^𝑛)⋂𝐿^∞(ℝ^𝑛) for some 𝑝≥2,  {𝑤_𝑗}^𝑚_𝑗=1 are independent two-sided real-valued Wiener processes on a probability space which will be specified below, and 𝑓 is a nonlinear function satisfying the following conditions (see, e.g., [1, 2]). For all 𝑥∈ℝ^𝑛 and 𝑠∈ℝ, 𝑓(𝑥,𝑠)𝑠≤−𝛼₁|𝑠|^𝑝+𝜓₁(𝑥),𝑓(𝑥,𝑠)≤𝛼₂|𝑠|^𝑝−1+𝜓₂(𝑥),𝜕𝑓()𝜕𝑠𝑥,𝑠≤𝛽,𝜕𝑓(𝜕𝑥𝑥,𝑠)≤𝜓₃(𝑥), (3)(4)(5)(6) where 𝛼₁,𝛼₂ and 𝛽 are positive constants, 𝜓₁∈𝐿¹(ℝ^𝑛)⋂𝐿^∞(ℝ^𝑛), 𝜓₂∈𝐿²(ℝ^𝑛)⋂𝐿^𝑝(ℝ^𝑛)⋂𝐿⁴(ℝ^𝑛), and 𝜓₃∈𝐿²(ℝ^𝑛).

As we know, the asymptotic behavior of a random dynamical system (RDS) is characterized by random attractors, which were first introduced by Crauel and Flandoli [3] and Schmalfuss [4] and then developed in [1, 2, 5–12] and among others. Recently, the existence of random attractors of the RDS associated with problem (1)-(2) was studied by many authors. For example, in [1, 2] the authors proved the existence of (𝐿²(ℝ^𝑛),𝐿²(ℝ^𝑛))-random attractor and (𝐿²(ℝ^𝑛),𝐿^𝑝(ℝ^𝑛))-random attractor, respectively, in the case of additive noise. Wang and Zhou obtained (𝐿²(ℝ^𝑛),𝐿²(ℝ^𝑛))-random attractor in [12] and Li et al. proved the existence of (𝐿²(𝐷),𝐿^𝑝(𝐷))-random attractor in bounded domains in [10] in the case of multiplicative noise. A necessary and sufficient condition for the existence of random attractors for the so-called quasicontinuous RDS was established in [9], and in the most recent papers [13, 14], the author employed this result to prove the existence of random attractors for some reaction-diffusion equations with additive noise and multiplicative noise on 𝐻¹, respectively, when the domain is bounded. In this paper, we study the existence of (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-random attractor with additive noise for the same problem in the entire space ℝ^𝑛.

For our problem, there are two difficulties when we consider the existence of (𝐿²,𝐻¹)-random attractor. The first is the lack of compactness of Sobolev embeddings when the domain is unbounded. It is worth mentioning that in deterministic case differential equations of this type were extensively studied in both autonomous and nonautonomous cases and in both bounded domains and unbounded domains [15–29]. In the case of unbounded domains the difficulty of noncompact embeddings can be overcome by the energy equation approach introduced by Ball in [30, 31] and other methods. We are interested in the method used in [22] for the deterministic version of the initial problem (1)-(2) on ℝ^𝑛. In [22] the author approached ℝ^𝑛 by a bounded ball and found that the approximation error of the norm of solutions is arbitrary small uniformly for large time, and thus they proved asymptotic compactness by passing the limit of the energy equation. More recently, the idea of the tail estimate was used in [1] to prove the existence of random attractor in 𝐿²(ℝ^𝑛) for the SRDE (1)-(2). In this paper, we use an extended version of the tail estimate described in [22] to overcome the difficulty of noncompact embeddings.

Another difficulty is that one can not differentiate the stochastic equation with respect to time 𝑡 in usual sense. In the case of deterministic equation, by differentiating the reaction-diffusion equation with respect to 𝑡, one can prove the existence of (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛)) or (𝐿²(𝐷),𝐻¹₀(𝐷)) (𝐷⊂ℝ^𝑛 is bounded) attractors; see [24, 27, 29, 32] for autonomous equations and [23, 26, 33] for non-autonomous equations. But in stochastic case this idea breaks down, since, as we know, neither the Winner process nor the Ornstein-Uhlenbeck process is differentiable with respect to 𝑡 in usual sense. However, this is only a matter of method or estimate. In [25], the author used a result for compactness in 𝐻¹(ℝ^𝑛) introduced in [17] to establish the asymptotic compactness in 𝐻¹(ℝ^𝑛) without differentiating the equation. Unfortunately, the growth order 𝑝 is restricted in that case. In this paper, we overcome this drawback by using an appropriate estimate motivated by the works in [19] and the estimate is accurate enough so that we needn't differentiate the equation as usual.

This paper is organized as follows. In Section 2, we recall some basic notions of bispaces random attractors for RDS. In Section 3, we transform the problem (1)-(2) into a parameterized evolution equation and obtain the corresponding RDS. In Section 4, we give some uniform estimates of the solutions as 𝑡→∞. In Section 5, we prove the asymptotic compactness and the existence of an (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-random attractor.

Throughout this paper, we denote by ‖⋅‖_𝑋 the norm of Banach space 𝑋 and by (⋅,⋅)_𝐻 the inner product in Hilbert space 𝐻. The inner product and norm of 𝐿²(ℝ^𝑛) are written as (⋅,⋅) and ‖⋅‖, respectively. We also use ‖𝑢‖_𝑟 to denote the norm of 𝑢∈𝐿^𝑟(ℝ^𝑛) (𝑟≥1, 𝑟≠2) and |𝑢| to denote the modular of 𝑢. The letter 𝑐 denotes any positive constant which may be different from line to line or even in the same line (sometimes for special case, we also denote the different positive constants by 𝑐_𝑖(𝑖=1,2,…)).

2. Preliminaries and Abstract Results

In this section, we first recall some basic concepts related to random attractors for RDS (see [1, 3, 5–8, 34] for more detail) and then give some abstract results on the existence of (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-random attractors.

2.1. Preliminaries

Let 𝑋, 𝑌 be two Banach spaces with Borel 𝜎-algebra ℬ(𝑋) and ℬ(𝑌), respectively, and let (Ω,ℱ,ℙ) be a probability space.

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

Definition 2. An RDS on 𝑋 over an MDS (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) is a mapping 𝜙∶ℝ⁺×Ω×𝑋→𝑋, (𝑡,𝜔,𝑥)→𝜙(𝑡,𝜔,𝑥) which is (ℬ(ℝ⁺)×ℱ×ℬ(𝑋),ℬ(𝑋))-measurable and satisfies that, for ℙ-a.e. 𝜔∈Ω,(i)𝜙(0,𝜔,⋅) = id on 𝑋;(ii)𝜙(𝑡+𝑠,𝜔,⋅)=𝜙(𝑡,𝜃_𝑠𝜔,⋅)∘𝜙(𝑠,𝜔,⋅) (cocycle property) on 𝑋 for all 𝑠,𝑡∈ℝ⁺.
An RDS is said to be continuous on 𝑋 if 𝜙(𝑡,𝜔)∶𝑋→𝑋 is continuous for all 𝑡∈ℝ⁺ and ℙ-a.e. 𝜔∈Ω.

Definition 3. A random set is a set-valued map 𝐴∶Ω→2^𝑋, 𝜔↦𝐴(𝜔), which satisfies that, for each 𝑥∈𝑋, the map 𝜔↦𝑑(𝑥,𝐴(𝜔)) is measurable. A random set {𝐴(𝜔)} is called a random closed (compact) set if 𝐴(𝜔) is closed (compact) for all 𝜔∈Ω. A random set {𝐴(𝜔)} is called a random bounded set if there exist 𝑥₀∈𝑋 and a random variable 𝑟(𝜔)>0 such that, for all 𝜔∈Ω, 𝐴(𝜔)⊂𝑥∈𝑋∶𝑑𝑥,𝑥₀≤𝑟(𝜔). (7)

Definition 4. (1) A random bounded set {𝐵(𝜔)}_𝜔∈Ω of 𝑋 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, (8) where 𝑑(𝐵)=sup_𝑥∈𝐵‖𝑥‖_𝑋. (9)
(2) A random variable 𝑟(𝜔)≥0 is called tempered with respect to (𝜃_𝑡)_𝑡∈ℝ if, for ℙ-a.e. 𝜔∈Ω, lim_𝑡→∞𝑒^−𝛽𝑡𝑟𝜃_−𝑡𝜔=0,∀𝛽>0. (10)
Next, we introduce some notions about the bi-spaces random attractors which are motivated by the works in [2, 20, 25, 35]. We assume that 𝜙 is an RDS on 𝑋 and 𝑌 over an MDS (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ), respectively. Let 𝒫(𝑋) denote the family of all nonempty subsets of 𝑋 and 𝒮_𝑋 the class of all families 𝐷={𝐷(𝜔)}_𝜔∈Ω⊂𝒫(𝑋). 𝒫(𝑌), and 𝒮_𝑌 can be defined in the same way. We consider the given nonempty subclasses 𝒟_𝑋,𝒟_𝑌, where 𝒟_𝑋⊂𝒮_𝑋, 𝒟_𝑌⊂𝒮_𝑌.

Definition 5. A family 𝐵={𝐵(𝜔)}_𝜔∈Ω∈𝒟_𝑌 is said to be (𝑋,𝑌)-random absorbing for 𝜙 if, for every 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟_𝑋, there exists 𝑇𝐷(𝜔)>0 such that, for ℙ-a.e. 𝜔∈Ω, 𝜙𝑡,𝜃_−𝑡𝜔,𝐷𝜃_−𝑡𝜔(𝜔)𝐷⊂𝐵,∀𝑡≥𝑇(𝜔). (11)

Definition 6. A family 𝐶={𝐶(𝜔)}_𝜔∈Ω∈𝒟_𝑌 is said to be (𝑋,𝑌)-random attracting for 𝜙 if, for every 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟_𝑋,  we have, for ℙ-a.e. 𝜔∈Ω, lim_𝑡→∞𝑑_𝑌𝜙𝑡,𝜃_−𝑡𝜔,𝐷𝜃_−𝑡𝜔,𝐶(𝜔)=0, (12) where 𝑑_𝑌(𝐶₁,𝐶₂) denotes the Hausdorff semi-distance between 𝐶₁ and 𝐶₂ in 𝑌; that is, 𝑑_𝑌𝐶₁,𝐶₂=sup_𝑥∈𝐶1inf_𝑦∈𝐶2𝑥−𝑦_𝑌for𝐶₁,𝐶₂⊂𝑌. (13)

Definition 7. The RDS 𝜙 is said to be (𝑋,𝑌)-asymptotically compact if, for ℙ-a.e. 𝜔∈Ω, {𝜙(𝑡_𝑛,𝜃_−𝑡𝑛𝜔,𝑥_𝑛)}^∞_𝑛=1 has a convergent subsequence in 𝑌 whenever 𝑡_𝑛→∞ and 𝑥_𝑛∈𝐷(𝜃_−𝑡𝑛𝜔) with 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟_𝑋.

Definition 8. 𝐴 random set 𝐴={𝐴(𝜔)}_𝜔∈Ω∈𝒮_𝑌 is said to be an (𝑋,𝑌)-random attractor if the following conditions are satisfied for ℙ-a.e. 𝜔∈Ω, (i)𝐴(𝜔) is closed in 𝑋 and compact in 𝑌;(ii)𝐴={𝐴(𝜔)}_𝜔∈Ω is invariant; that is, 𝜙(𝑡,𝜔,𝐴(𝜔))=𝐴(𝜃_𝑡𝜔) for all 𝑡≥0;(iii)𝐴={𝐴(𝜔)}_𝜔∈Ω attracts every random set in 𝒟_𝑋 in the norm topology of 𝑌 in the sense of (12).

2.2. Abstract Results

Now, we present the main abstract results. Recall that a collection 𝒟 of random subsets is called inclusion closed if whenever {𝐸(𝜔)}_𝜔∈Ω is an arbitrary random set and {𝐹(𝜔)}_𝜔∈Ω is in 𝒟 with 𝐸(𝜔)⊂𝐹(𝜔) for all 𝜔∈Ω, then {𝐸(𝜔)}_𝜔∈Ω∈𝒟.

The following theorem is an adaptation of a result of [25] to the case of RDS. The proof is similar to that of [25], and here we omit it.

Theorem 9. Let 𝜙 be a continuous RDS on 𝑋 and an RDS on 𝑌 over (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ), respectively, and 𝒟_𝑋 and 𝒟_𝑌 are inclusion closed.(i) Case  1  (𝑋=𝑌)  (see [1]). Assume that the family 𝐵₀={𝐵₀(𝜔)}_𝜔∈Ω∈𝒟_𝑋 is a closed (𝑋,𝑋)-random absorbing set for 𝜙 and 𝜙 is (𝑋,𝑋)-asymptotically compact. Then 𝜙 has a unique (𝑋,𝑋)-random attractor 𝐴={𝐴(𝜔)}_𝜔∈Ω which is given by 𝐴(𝜔)=_{𝑠≥0𝑡≥𝑠}𝜙𝑡,𝜃_−𝑡𝜔,𝐵₀𝜃_−𝑡𝜔^𝑋, (14) where 𝐴^𝑋 denotes the closure of 𝐴 with respect to the norm topology in 𝑋.(ii)  Case  2  (𝑋≠𝑌). If the assumptions in (i) are satisfied, moreover, we assume that 𝐵₁={𝐵₁(𝜔)}_𝜔∈Ω∈𝒟_𝑌 is (𝑋,𝑌)-random absorbing and 𝜙 is (𝑋,𝑌)-asymptotically compact. Then 𝜙 has an (𝑋,𝑌)-random attractor 𝐴^′={𝐴^′(𝜔)}_𝜔∈Ω which is given by 𝐴^′(𝜔)=_{𝑠≥0𝑡≥𝑠}𝜙𝑡,𝜃_−𝑡𝜔,𝐵₀𝜃_−𝑡𝜔𝐵₁𝜃_−𝑡𝜔^𝑋=_{𝑠≥0𝑡≥𝑠}𝜙𝑡,𝜃_−𝑡𝜔,𝐵₀𝜃_−𝑡𝜔𝐵₁𝜃_−𝑡𝜔^𝑌, (15) where 𝐵₀={𝐵₀(𝜔)}_𝜔∈Ω∈𝒟_𝑋 is the (𝑋,𝑋)-random absorbing set in (i).

In the following of this paper we only consider 𝑋=𝐿²(ℝ^𝑛), 𝑌=𝐻¹(ℝ^𝑛), and 𝒟_𝑋=𝒟₂, 𝒟_𝑌=𝒟_𝑣, where 𝒟₂ and 𝒟_𝑣 denote the collections of all tempered random subsets of 𝐿²(ℝ^𝑛) and 𝐻¹(ℝ^𝑛), respectively.

Theorem 10. Assume that 𝜙 is an RDS on 𝐿²(ℝ^𝑛) and 𝐻¹(ℝ^𝑛), respectively, and then 𝜙 is (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-asymptotically compact if(i) for every 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂, ℙ-a.e. 𝜔∈Ω and every 𝜀>0, there exist 𝑘₀=𝑘₀(𝜔,𝜀)>0 and 𝑇𝐷(𝜔,𝜀)>0 such that, for all 𝐷𝑡≥𝑇(𝜔,𝜀), 𝜒_{𝑄𝑐𝑘0}⋅𝜙𝑡,𝜃_−𝑡𝜔,𝑥_{𝐻1(𝑄𝑐𝑘0)}≤𝜀,∀𝑥∈𝐷𝜃_−𝑡𝜔, (16)(ii)𝜒_𝑄𝑘⋅𝜙 is (𝐿²(ℝ^𝑛),𝐻¹(𝑄_𝑘))-asymptotically compact, ∀𝑘≥1, where 𝑄_𝑘0={𝑥∈ℝ^𝑛∶|𝑥|≤𝑘₀}, 𝑄^𝑐_𝑘0=ℝ^𝑛⧵𝑄_𝑘0, 𝑄_𝑘={𝑥∈ℝ^𝑛∶|𝑥|≤𝑘}, and 𝜒_𝐴 is the identical function on 𝐴.

Proof. It suffices to check that, for all 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and ℙ-a.e. 𝜔∈Ω, we can extract a Cauchy subsequence {𝜙(𝑡_𝑛′,𝜃_{−𝑡𝑛′}𝜔,𝑥_𝑛′)} from {𝜙(𝑡_𝑛,𝜃_−𝑡𝑛𝜔,𝑥_𝑛)}, whenever 𝑡_𝑛→∞ and 𝑥_𝑛∈𝐷(𝜃_−𝑡𝑛𝜔). We assume that there is Ω⊂Ω of full ℙ-measure such that assumption (i) holds for every 𝜔∈Ω. We now fix 𝜔∈Ω and 𝜀>0, and then by (i) there exist 𝑘₀=𝑘₀(𝜔,𝜀)>0 and 𝑇𝐷(𝜔,𝜀)>0 such that for all 𝐷𝑡≥𝑇(𝜔,𝜀), 𝜒_{𝑄𝑐𝑘0}⋅𝜙𝑡,𝜃_−𝑡𝜔,𝑥_{𝐻1(𝑄𝑐𝑘0)}≤𝜀,∀𝑥∈𝐷𝜃_−𝑡𝜔. (17) On the other hand, by (ii), 𝜒_𝑄𝑘⋅𝜙 is (𝐿²(ℝ^𝑛),𝐻¹(𝑄_𝑘))-asymptotically compact, for all 𝑘≥1. For the above 𝑘₀, there is a subsequence {𝜙(𝑡_𝑛′,𝜃_{−𝑡𝑛′}𝜔,𝑥_𝑛′)} such that {𝜒_𝑄𝑘0⋅𝜙(𝑡_𝑛′,𝜃_{−𝑡𝑛′}𝜔,𝑥_𝑛′)} is convergent in 𝐻¹(𝑄_𝑘0). Therefore, there exists an integer 𝑁(𝐷,𝜔,𝜀)>0 such that for all 𝑛^′,𝑚^′≥𝑁(𝐷,𝜔,𝜀), we have 𝜙𝑡_𝑛′,𝜃_{−𝑡𝑛′}𝜔,𝑥_𝑛′−𝜙𝑡_𝑚′,𝜃_{−𝑡𝑚′}𝜔,𝑥_𝑚′²_{𝐻1(ℝ𝑛)}=𝜒_𝑄𝑘0⋅𝜙𝑡_𝑛′,𝜃_{−𝑡𝑛′}𝜔,𝑥_𝑛′−𝜙𝑡_𝑚′,𝜃_{−𝑡𝑚′}𝜔,𝑥_𝑚′²_{𝐻1𝑄𝑘0}+𝜒_{𝑄𝑐𝑘0}⋅𝜙𝑡_𝑛′,𝜃_{−𝑡𝑛′}𝜔,𝑥_𝑛′−𝜙𝑡_𝑚′,𝜃_{−𝑡𝑚′}𝜔,𝑥_𝑚′²_{𝐻1𝑄𝑐𝑘0}≤𝜒_𝑄𝑘0⋅𝜙𝑡_𝑛′,𝜃_{−𝑡𝑛′}𝜔,𝑥_𝑛′−𝜒_𝑄𝑘0⋅𝜙𝑡_𝑚′,𝜃_{−𝑡𝑚′}𝜔,𝑥_𝑚′²_{𝐻1𝑄𝑘0}+2𝜒_{𝑄𝑐𝑘0}⋅𝜙𝑡_𝑛′,𝜃_{−𝑡𝑛′}𝜔,𝑥_𝑛′²_{𝐻1𝑄𝑐𝑘0}+2𝜒_{𝑄𝑐𝑘0}⋅𝜙𝑡_𝑚′,𝜃_{−𝑡𝑚′}𝜔,𝑥_𝑚′²_{𝐻1𝑄𝑐𝑘0}≤5𝜀. (18) The proof is complete.

Remark 11. If we replace 𝐻¹(ℝ^𝑛) by other Banach spaces in Theorem 10, such as 𝐿²(ℝ^𝑛), 𝐿^𝑝(ℝ^𝑛) and 𝐻²(ℝ^𝑛), the corresponding results also hold true. In particular, in the deterministic case, it is the exact method used in [22] when 𝐻¹(ℝ^𝑛) is replaced by 𝐿²(ℝ^𝑛).

3. The Reaction-Diffusion Equation on ℝ^𝑛 with Additive Noise

We consider the probability space (Ω,ℱ,ℙ) where Ω=𝜔=𝜔₁,𝜔₂,…,𝜔_𝑚∈𝐶ℝ,ℝ^𝑚∶𝜔(0)=0, (19)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 𝜔 with 𝑊(𝑡)≡𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑚(𝑡)=𝜔(𝑡)for𝑡∈ℝ. (20) Define the time shift by𝜃_𝑡𝜔(⋅)=𝜔(⋅+𝑡)−𝜔(𝑡),𝜔∈Ω,𝑡∈ℝ, (21) and then (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) is an MDS.

We now translate the stochastic equation (1)-(2) into a deterministic equation with a random parameter.

To this end, we consider the one-dimensional Ornstein-Uhlenbeck process given by 𝑧_𝑗(𝑡)=𝑧_𝑗𝜃_𝑡𝜔_𝑗≡−𝜆⁰_−∞𝑒^𝜆𝜏𝜃_𝑡𝜔_𝑗(𝜏)𝑑𝜏,∀𝑡∈ℝ, (22) which solves the Itô differential equation 𝑑𝑧_𝑗+𝜆𝑧_𝑗𝑑𝑡=𝑑𝜔_𝑗(𝑡). (23) Note that the random variable |𝑧_𝑗(𝜔_𝑗)| is tempered and 𝑧_𝑗(𝜃_𝑡𝜔_𝑗) is ℙ-a.e. continuous in 𝑡∈ℝ. Therefore, it follows from the Proposition 4.3.3 in [34] that there exists a tempered function 𝑟(𝜔)>0 such that 𝑝(𝜔)∶=^𝑚_𝑗=1𝑧_𝑗𝜔_𝑗²+𝑧_𝑗𝜔_𝑗⁴+𝑧_𝑗𝜔_𝑗^𝑝+𝑧_𝑗𝜔_𝑗^2𝑝+𝑧_𝑗𝜔_𝑗^2𝑝−2≤𝑟(𝜔), (24) where 𝑟(𝜔) satisfies that for ℙ-a.e. 𝜔∈Ω, 𝑟𝜃_𝑡𝜔≤𝑒^{(𝜆/2)|𝑡|}𝑟(𝜔),∀𝑡∈ℝ. (25) Therefore, for ℙ-a.e. 𝜔∈Ω, 𝑝𝜃_𝑡𝜔≤𝑒^{(𝜆/2)|𝑡|}𝑟(𝜔),∀𝑡∈ℝ. (26) Putting 𝑧𝜃_𝑡𝜔=^𝑚_𝑗=1ℎ_𝑗𝑧_𝑗𝜃_𝑡𝜔_𝑗, (27) then by (23) we have 𝑑𝑧+𝜆𝑧𝑑𝑡=^𝑚_𝑗=1ℎ_𝑗𝑑𝜔_𝑗. (28)

Remark 12. From (24) and (27), we can easily show that the sum ‖𝑧(𝜔)‖²+‖𝑧(𝜔)‖^𝑝_𝑝+‖𝑧(𝜔)‖^2𝑝_2𝑝+‖𝑧(𝜔)‖^2𝑝−2_2𝑝−2+‖𝑧(𝜔)‖⁴₄+‖∇𝑧(𝜔)‖²+‖∇𝑧(𝜔)‖²+‖Δ𝑧(𝜔)‖^𝑝_𝑝, (29) is bounded by 𝑝(𝜔) with a deterministic positive constant 𝑐₀. In the following of this paper, we use the symbols 𝑝(𝜔) and 𝑟(𝜔) to denote the random variables in (24).

In order to show that the initial problem (1)-(2) generates an RDS, we set 𝑣(𝑡)=𝑢(𝑡)−𝑧(𝜃_𝑡𝜔). Then we can consider the following evolution equation with random parameter but without white noise: 𝜕𝑣𝜃𝜕𝑡+𝜆𝑣−Δ𝑣=𝑓𝑥,𝑣+𝑧_𝑡𝜔+𝑔+Δ𝑧𝜃_𝑡𝜔, (30) with initial value condition 𝑣(𝑥,0)=𝑣₀(𝜔)=𝑢₀−𝑧(𝜔). (31)

From [1, 2], we see that for ℙ-a.e. 𝜔∈Ω and all 𝑣₀∈𝐿²(ℝ^𝑛), the parameterized evolution equation (30)-(31) with conditions (3)–(6) has a unique solution 𝑣⋅,𝜔,𝑣₀∈𝐶[0,∞);𝐿²ℝ^𝑛𝐿^𝑝(0,𝑇);𝐿^𝑝ℝ^𝑛𝐿²(0,𝑇);𝐻¹ℝ^𝑛, (32) for every 𝑇>0. Furthermore, 𝑣(𝑡,𝜔,𝑣₀) is continuous with respect to 𝑣₀ in 𝐿²(ℝ^𝑛), for all 𝑡≥0.

As 𝑢𝑡,𝜔,𝑢₀=𝑣𝑡,𝜔,𝑢₀−𝑧(𝜔)+𝑧𝜃_𝑡𝜔, (33) the process 𝑢 is the solution to the problem (1)-(2) in a certain sense. We now define a mapping 𝜙∶ℝ⁺×Ω×𝐿²(ℝ^𝑛)→𝐿²(ℝ^𝑛) by 𝜙𝑡,𝜔,𝑢₀=𝑢𝑡,𝜔,𝑢₀=𝑣𝑡,𝜔,𝑢₀−𝑧(𝜔)+𝑧𝜃_𝑡𝜔, (34) for all (𝑡,𝜔,𝑢₀)∈ℝ⁺×Ω×𝐿²(ℝ^𝑛). Then 𝜙 is a continuous RDS on 𝐿²(ℝ^𝑛) and an RDS on 𝐻¹(ℝ^𝑛) respectively associated with the initial value problem of SRDE (1)-(2) on ℝ^𝑛.

Theorem 13 (see [1, 2]). Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Then the RDS 𝜙 generated by (1)-(2) has a unique (𝐿²(ℝ^𝑛),𝐿²(ℝ^𝑛))-random attractor 𝐴₂ and has a unique (𝐿²(ℝ^𝑛),𝐿^𝑝(ℝ^𝑛))-random attractor 𝐴_𝑝; furthermore, we have 𝐴₂=𝐴_𝑝.

4. Uniform Estimates of Solutions

4.1. (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-Random Absorbing Set and Some Useful Estimates

The next lemma shows that 𝜙 has a tempered (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-random absorbing set.

Lemma 14 (see [1]). Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Let 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and 𝑢₀(𝜔)∈𝐷(𝜔). Then, for ℙ-a.e. 𝜔∈Ω, there exists 𝑇𝐷(𝜔)>0 such that, for all 𝐷𝑡≥𝑇(𝜔), 𝑢𝑡,𝜃_−𝑡𝜔,𝑢₀𝜃_−𝑡𝜔²_{𝐻1(ℝ𝑛)}≤𝑐(1+𝑟(𝜔)), (35) where 𝑐 is a constant independent of 𝑡, 𝜔, and 𝑢₀.

We now give some new estimates for the solution 𝑣(𝑡,𝜔,𝑣₀) of (30)-(31).

Lemma 15. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Let 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and 𝑢₀(𝜔)∈𝐷(𝜔). Then, for ℙ-a.e. 𝜔∈Ω, there exists 𝑇𝐷(𝜔)≥2, such that the solution 𝑣(𝑡,𝜔,𝑣₀(𝜔)) of (30)-(31) satisfies that, for all 𝐷𝑡≥𝑇(𝜔) and for all 𝑠∈[𝑡−1,𝑡+1], 𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑐(1+𝑟(𝜔)),𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^𝑝_𝑝≤𝑐(1+𝑟(𝜔)),∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑐(1+𝑟(𝜔)), (36)(37)(38) where 𝑐 is a constant independent of 𝑠, 𝑡, and 𝜔.

Proof. The first assertion was proved in [2] in the case of 𝑠∈[𝑡,𝑡+1], and the case for 𝑠∈[𝑡−1,𝑡+1] can be obtained by slightly modifying the proof of Lemma 4.4 in [2] (in fact, for 𝑠∈[𝑡−2,𝑡+1], (36) also holds true, and we will use this result in (55)), and here we omit it.
Now, we prove the second assertion. Multiplying (30) with |𝑣|^𝑝−2𝑣 and integrating over ℝ^𝑛, we get 1𝑝𝑑‖𝑣‖𝑑𝑡^𝑝𝑝+𝜆‖𝑣‖^𝑝_𝑝+−Δ𝑣,|𝑣|^𝑝−2𝑣=_ℝ𝑛𝑓𝑥,𝑣+𝑧𝜃_𝑡𝜔|𝑣|^𝑝−2+𝜃𝑣𝑑𝑥𝑔+Δ𝑧_𝑡𝜔,|𝑣|^𝑝−2𝑣. (39) Since 𝑝≥2, we have −Δ𝑣,|𝑣|^𝑝−2𝑣=∇𝑣,∇|𝑣|^𝑝−2𝑣=∇𝑣,𝑝−2|𝑣|^𝑝−4𝑣²+|𝑣|∇𝑣∇𝑣,^𝑝−2∇𝑣≥0. (40) For the nonlinearity, similar to (4.6) and (4.8) in [2], we have 𝑓𝑥,𝑣+𝑧𝜃_𝑡𝜔𝑣𝛼≤−¹2^𝑝|𝑣|^𝑝+𝑐𝑧𝜃_𝑡𝜔^𝑝+𝑧𝜃_𝑡𝜔²+𝜓₁+12𝜓²₂,_ℝ𝑛𝑓𝑥,𝑣+𝑧𝜃_𝑡𝜔|𝑣|^𝑝−2𝛼𝑣𝑑𝑥≤−¹2^𝑝+1‖𝑣‖^2𝑝−2_2𝑝−2+𝜆‖𝑣‖^𝑝_𝑝+𝑐𝑧𝜃_𝑡𝜔^2𝑝−2_2𝑝−2+𝑧𝜃_𝑡𝜔^𝑝_𝑝+𝑐𝜓₁^𝑝/2_𝑝/2+𝜓₂^𝑝_𝑝,𝑔+Δ𝑧𝜃_𝑡𝜔,|𝑣|^𝑝−2𝑣≤𝑔‖𝑣‖^𝑝−1_2𝑝−2+Δ𝑧𝜃_𝑡𝜔‖𝑣‖^𝑝−1_2𝑝−2≤𝛼₁2^𝑝+2‖𝑣‖^2𝑝−2_2𝑝−2+𝑐Δ𝑧𝜃_𝑡𝜔²+𝑐𝑔². (41)(42)(43) From (39)-(40) and (42)-(43), we get 𝑑‖𝑣‖𝑑𝑡^𝑝𝑝+𝑐‖𝑣‖^2𝑝−2_2𝑝−2≤𝑐₁𝑧𝜃_𝑡𝜔^2𝑝−2_2𝑝−2+𝑧𝜃_𝑡𝜔^𝑝_𝑝+Δ𝑧𝜃_𝑡𝜔²+𝑐₁≤𝑐₂𝑝𝜃_𝑡𝜔+𝑐₂. (44)
On the other hand, multiplying (30) by 𝑣 and integrating over ℝ^𝑛, we get the results in [1]: 𝑑‖𝑣‖𝑑𝑡²+𝜆‖𝑣‖²+‖∇𝑣‖²+𝛼₁‖𝑢‖^𝑝_𝑝≤𝑐𝑝𝜃_𝑡𝜔+𝑐,𝑣𝑡,𝜔,𝑣₀(𝜔)²≤𝑒^−𝜆𝑡𝑣₀(𝜔)²+𝑐^𝑡₀𝑒^{𝜆(𝜏−𝑡)}𝑝𝜃_𝜏𝜔𝑑𝜏+𝑐. (45)(46) By Hölder inequality, ‖𝑢‖^𝑝_𝑝≥2^1−𝑝‖𝑣‖^𝑝_𝑝−‖𝑧(𝜃_𝑡𝜔)‖^𝑝_𝑝, we can convert (45) into 𝑑‖𝑣‖𝑑𝑡²+𝜆‖𝑣‖²+‖∇𝑣‖²+𝛼₁2^1−𝑝‖𝑣‖^𝑝_𝑝≤𝑐𝑝𝜃_𝑡𝜔+𝑐. (47) For any 𝑡≥0, integrating (47) over (𝑡,𝑡+1) and using (46), we get _𝑡^𝑡+1‖𝑣(𝑠)‖^𝑝_𝑝‖𝑣(𝑡𝑑𝑠≤𝑐)‖²+_𝑡^𝑡+1𝑝𝜃_𝜏𝜔𝑒𝑑𝜏+1≤𝑐^−𝜆𝑡𝑣₀(𝜔)²+^𝑡₀𝑒^{𝜆(𝜏−𝑡)}𝑝𝜃_𝜏𝜔+𝑑𝜏_𝑡𝑡+1𝑝𝜃_𝜏𝜔𝑑𝜏+1. (48) Next, fix 𝑠∈(𝑡,𝑡+1) and integrate (44) over (𝑠,𝑡+1) to get 𝑣𝑡+1,𝜔,𝑣₀(𝜔)^𝑝_𝑝≤𝑣𝑠,𝜔,𝑣₀(𝜔)^𝑝_𝑝+𝑐_2𝑠^𝑡+1𝑝𝜃_𝜏𝜔𝑑𝜏+𝑐₂()≤𝑣𝑡+1−𝑠𝑠,𝜔,𝑣₀(𝜔)^𝑝_𝑝+𝑐_2𝑡^𝑡+1𝑝𝜃_𝜏𝜔𝑑𝜏+𝑐₂. (49) Integrating the above inequality with respect to 𝑠 over (𝑡,𝑡+1) and using (48), we obtain, for all 𝑡≥0, 𝑣𝑡+1,𝜔,𝑣₀(𝜔)^𝑝_𝑝≤_𝑡^𝑡+1𝑣𝑠,𝜔,𝑣₀(𝜔)^𝑝_𝑝𝑑𝑠+𝑐_2𝑡^𝑡+1𝑝𝜃_𝑠𝜔𝑑𝑠+𝑐₂≤𝑐𝑒^−𝜆𝑡𝑣₀(𝜔)²+^𝑡₀𝑒^{𝜆(𝜏−𝑡)}𝑝𝜃_𝜏𝜔+𝑑𝜏_𝑡𝑡+1𝑝𝜃_𝜏𝜔𝑒𝑑𝜏+1≤𝑐^{−𝜆(𝑡+1)}𝑣₀(𝜔)²+₀^𝑡+1𝑒^{𝜆(𝜏−𝑡−1)}𝑝𝜃_𝜏𝜔+𝑑𝜏_𝑡𝑡+1𝑝𝜃_𝜏𝜔𝑑𝜏+1. (50) Replacing 𝑡+1 by 𝑠 first, then substituting 𝜃_−𝑡−1𝜔 for 𝜔 in the aforementioned inequality, and noting that 𝑠∈[𝑡−1,𝑡+1](𝑡≥2), we have 𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^𝑝_𝑝≤𝑐𝑒^−𝜆𝑠𝑣₀𝜃_−𝑡−1𝜔²+^𝑠₀𝑒^{𝜆(𝜏−𝑠)}𝑝𝜃_{𝜏−𝑡−1}𝜔𝑑𝜏+^𝑠_𝑠−1𝑝𝜃_{𝜏−𝑡−1}𝜔𝑒𝑑𝜏+1≤𝑐^{𝜆(𝑡+1−𝑠)}𝑒^{−𝜆(𝑡+1)}𝑣₀𝜃_−𝑡−1𝜔²+₀^𝑡+1𝑒^{𝜆(𝑡+1−𝑠)}𝑒^{𝜆(𝜏−𝑡−1)}𝑝𝜃_{𝜏−𝑡−1}𝜔+𝑑𝜏_{𝑡+1𝑡−2}𝑝𝜃_{𝜏−𝑡−1}𝜔𝑒𝑑𝜏+1≤𝑐^{−𝜆(𝑡+1)}𝑣₀𝜃_−𝑡−1𝜔²+₀^𝑡+1𝑒^{𝜆(𝜏−𝑡−1)}𝑝𝜃_{𝜏−𝑡−1}𝜔+𝑑𝜏_{𝑡+1𝑡−2}𝑝𝜃_{𝜏−𝑡−1}𝜔𝑒𝑑𝜏+1=𝑐^{−𝜆(𝑡+1)}𝑣₀𝜃_−𝑡−1𝜔²+⁰_−𝑡−1𝑒^𝜆𝜏𝑝𝜃_𝜏𝜔+𝑑𝜏₀₋₃𝑝𝜃_𝜏𝜔𝑒𝑑𝜏+1≤𝑐^{−𝜆(𝑡+1)}𝑣₀𝜃_−𝑡−1𝜔²+𝑟(𝜔)⁰_−𝑡−1𝑒^(𝜆/2)𝜏(𝜔)𝑑𝜏+𝑟⁰₋₃𝑒^{−(𝜆/2)𝜏}𝑑𝜏+1, (51) where we have used (26) in the pervious inequality. Noting that 𝑢₀(𝜔)∈𝐷(𝜔) with 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and 𝑣₀(𝜔)=𝑢₀(𝜔)−𝑧(𝜔), we get from (51) that there exists 𝑇𝐷(𝜔)≥2 such that, for all 𝐷𝑡≥𝑇(𝜔) and for all 𝑠∈[𝑡−1,𝑡+1], 𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^𝑝_𝑝≤𝑐(1+𝑟(𝜔)). (52) That is, (37) holds true.
To prove (38), we take the inner product of (30) with −Δ𝑣 in 𝐿²(Ω), and using (4.31) in [1], we get 𝑑‖‖𝑑𝑡∇𝑣²≤𝑐‖∇𝑢‖²+‖𝑢‖^𝑝_𝑝+𝑐Δ𝑧𝜃_𝑡𝜔²+Δ𝑧𝜃_𝑡𝜔^𝑝_𝑝+1. (53) This implies that 𝑑‖‖𝑑𝑡∇𝑣²≤𝑐‖∇𝑣‖²+‖𝑣‖^𝑝_𝑝+𝑐𝑝𝜃_𝑡𝜔+1. (54) Integrating (47) over (𝑡−2,𝑡+1) and substituting 𝜃_−𝑡−1𝜔 for 𝜔, then from (36), we get ^𝑡+1_𝑡−2∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑣𝑑𝑠𝑡−2,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑐^𝑡+1_𝑡−2𝑝𝜃_{𝜏−𝑡−1}𝜔((𝜔𝐷(𝜔).𝑑𝜏+1≤𝑐1+𝑟)),∀𝑡≥𝑇 (55) Obviously, from (51) we can easily see that (37) also holds for 𝑠∈[𝑡−2,𝑡+1]; then, by (37), (47), (54)-(55), and a similar procedure as the proof of (37), one can show that, for all 𝐷𝑡≥𝑇(𝜔), and for all 𝑠∈[𝑡−1,𝑡+1], ∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑐(1+𝑟(𝜔)). (56) The proof is complete.

Lemma 16. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Let 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and 𝑢₀(𝜔)∈𝐷(𝜔). Then, for ℙ-a.e. 𝜔∈Ω, there exists 𝑇𝐷(𝜔)>0, such that the solution 𝑣(𝑡,𝜔,𝑣₀(𝜔)) of (30)-(31) satisfies that for all 𝐷𝑡≥𝑇(𝜔), _𝑡^𝑡+1𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^2𝑝−2_2𝑝−2((𝜔,𝑑𝑠≤𝑐1+𝑟))_𝑡𝑡+1𝑣_𝑠𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²((𝜔.𝑑𝑠≤𝑐1+𝑟)) (57)(58)

Proof. Integrating (44) over (𝑡,𝑡+1), we get _𝑡^𝑡+1‖𝑣(𝑠)‖^2𝑝−2_2𝑝−2𝑑𝑠≤𝑐‖𝑣(𝑡)‖^𝑝_𝑝+_𝑡^𝑡+1𝑝𝜃_𝜏𝜔𝑑𝜏+1. (59) Replacing 𝜔 by 𝜃_−𝑡−1𝜔 in the aforementioned inequality, it yields that _𝑡^𝑡+1𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^2𝑝−2_2𝑝−2𝑣𝑑𝑠≤𝑐𝑡,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^𝑝_𝑝+_𝑡^𝑡+1𝑝𝜃_{𝜏−𝑡−1}𝜔𝑣𝑑𝜏+1=𝑐𝑡,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^𝑝_𝑝+⁰₋₁𝑝𝜃_𝜏𝜔𝑣𝑑𝜏+1≤𝑐𝑡,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^𝑝_𝑝+𝑟(𝜔)⁰₋₁𝑒^{−(𝜆/2)𝜏}𝑑𝜏+1. (60) Lemma 15 and the aforementioned inequality imply that there exists 𝑇𝐷(𝜔)>0, 𝐷∀𝑡≥𝑇(𝜔), such that (57) holds.
Next, taking the inner product of (30) with 𝑣_𝑡 in 𝐿²(ℝ^𝑛), and using (4), we obtain 𝑣_𝑡²+𝜆2𝑑‖𝑣‖𝑑𝑡²+12𝑑‖‖𝑑𝑡∇𝑣²=_ℝ𝑛𝑓(𝑥,𝑢)𝑣_𝑡𝑑𝑥+_ℝ𝑛𝑔𝑣_𝑡𝑑𝑥+_ℝ𝑛Δ𝑧𝜃_𝑡𝜔𝑣_𝑡𝑑𝑥≤𝛼_2ℝ𝑛|𝑢|^𝑝−1𝑣_𝑡𝑑𝑥+_ℝ𝑛𝜓₂𝑣_𝑡𝑑𝑥+_ℝ𝑛𝑔𝑣_𝑡+𝑑𝑥_ℝ𝑛Δ𝑧𝜃_𝑡𝜔𝑣_𝑡≤1𝑑𝑥2𝑣_𝑡²+𝑐‖𝑢‖^2𝑝−2_2𝑝−2+𝜓₂²+𝑔²+Δ𝑧𝜃_𝑡𝜔²≤12𝑣_𝑡²+𝑐‖𝑣‖^2𝑝−2_2𝑝−2+𝜓₂²+𝑔²+Δ𝑧𝜃_𝑡𝜔²+𝑧𝜃_𝑡𝜔^2𝑝−2_2𝑝−2, (61) that is, 𝑣_𝑡²𝑑+𝜆‖𝑣‖𝑑𝑡²+𝑑‖‖𝑑𝑡∇𝑣²≤𝑐‖𝑣‖^2𝑝−2_2𝑝−2+𝜓₂²+𝑔²+Δ𝑧(𝜃_𝑡𝜔)²+𝑧(𝜃_𝑡𝜔)^2𝑝−2_2𝑝−2. (62) We now integrate (62) over (𝑡,𝑡+1) to obtain _𝑡^𝑡+1𝑣_𝑠𝑠,𝜔,𝑣₀(𝜔)²𝑣𝑑𝑠≤𝑐𝑡,𝜔,𝑣₀(𝜔)²+∇𝑣𝑡,𝜔,𝑣₀(𝜔)²+𝑐_𝑡^𝑡+1𝑧𝜃_𝜏𝜔^2𝑝−2_2𝑝−2+Δ𝑧𝜃_𝜏𝜔²𝑑𝜏+𝑐_𝑡^𝑡+1𝑣𝜏,𝜔,𝑣₀(𝜔)^2𝑝−2_2𝑝−2𝜓𝑑𝜏+𝑐₂²+𝑔². (63) Replacing 𝜔 by 𝜃_−𝑡−1𝜔, we get _𝑡^𝑡+1𝑣_𝑠𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝑣𝑑𝑠≤𝑐𝑡,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+∇𝑣𝑡,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑐_𝑡^𝑡+1𝑧𝜃_{𝜏−𝑡−1}𝜔^2𝑝−2_2𝑝−2+Δ𝑧𝜃_{𝜏−𝑡−1}𝜔²𝑑𝜏+𝑐_𝑡^𝑡+1𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^2𝑝−2_2𝑝−2𝜓𝑑𝑠+𝑐₂²+𝑔². (64) Equations (36), (38) and (57) together imply that (58) is also true. The proof is complete.

4.2. Tail Estimate in 𝐻¹(ℝ^𝑛)

We next estimate “the tail” of the solution to the problem (1)-(2) in 𝐻¹(ℝ^𝑛).

Lemma 17. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Let 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and 𝑢₀(𝜔)∈𝐷(𝜔). Then, for ℙ-a.e. 𝜔∈Ω and for every 𝜀>0, there exist 𝑇^∗=𝑇^∗𝐷(𝜔,𝜀)>0 and 𝑅^∗=𝑅^∗(𝜔,𝜀), such that the solution 𝑣(𝑡,𝜔,𝑣₀(𝜔)) of (30)-(31) satisfies that ∀𝑡≥𝑇^∗, _{|𝑥|≥𝑅∗}𝑣𝑡,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔²+∇𝑣𝑡,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔²𝑑𝑥≤𝜀. (65)

Proof. By Lemma 4.6 in [1], it suffices to prove that _{|𝑥|≥𝑅∗}∇𝑣𝑡,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔²𝑑𝑥≤𝜀. (66) Let 𝜑 be a smooth function defined on ℝ⁺ such that 0≤𝜑(𝑠)≤1, for all 𝑠∈ℝ⁺, and 𝜑(𝑠)=0,0≤𝑠≤1,1,𝑠≥2. (67) Then there is a positive constant 𝑐 such that |𝜑^′(𝑠)|+|𝜑^′′(𝑠)|≤𝑐 for all 𝑠∈ℝ⁺.
Multiplying (30) by −𝜑(|𝑥|²/𝑘²)Δ𝑣 and integrating with respect to 𝑥 over ℝ^𝑛, we get 12𝑑𝑑𝑡_ℝ^𝑛𝜑|𝑥|²𝑘²|∇𝑣|²𝑑𝑥+_ℝ𝑛|𝑥|∇𝜑²𝑘²⋅∇𝑣𝑣_𝑡𝑑𝑥+𝜆_ℝ𝑛𝜑|𝑥|²𝑘²|∇𝑣|²𝑑𝑥+𝜆_ℝ𝑛|𝑥|∇𝜑²𝑘²⋅∇𝑣𝑣𝑑𝑥+_ℝ𝑛𝜑|𝑥|²𝑘²|Δ𝑣|²𝑑𝑥=−_ℝ𝑛𝜑|𝑥|²𝑘²𝑓(𝑥,𝑢)Δ𝑣𝑑𝑥−_ℝ𝑛𝜑|𝑥|²𝑘²𝑔+Δ𝑧𝜃_𝑡𝜔Δ𝑣𝑑𝑥. (68) The second term of the left-hand side is bounded by _ℝ𝑛|𝑥|∇𝜑²𝑘²|∇𝑣|𝑣_𝑡=𝑑𝑥_ℝ𝑛𝜑^′|𝑥|²𝑘²2𝑥𝑘²|∇𝑣|𝑣_𝑡≤𝑐𝑑𝑥𝑘^√_{𝑘≤|𝑥|≤2𝑘}|∇𝑣|𝑣_𝑡≤𝑐𝑑𝑥𝑘𝑣_𝑡²+‖∇𝑣‖². (69) Similarly, the forth term of the left-hand side of (68) is bounded by 𝜆_ℝ𝑛|𝑥|∇𝜑²𝑘²𝑣|𝑐|∇𝑣||𝑑𝑥≤𝑘‖𝑣‖²+‖∇𝑣‖². (70) For the last term of the right-hand side of (68), we have _ℝ𝑛𝜑|𝑥|²𝑘²𝑔+Δ𝑧𝜃_𝑡𝜔≤1Δ𝑣𝑑𝑥2_ℝ^𝑛𝜑|𝑥|²𝑘²|Δ𝑣|²𝑑𝑥+_ℝ𝑛𝜑|𝑥|²𝑘²𝑔²+𝑑𝑥_ℝ𝑛𝜑|𝑥|²𝑘²Δ𝑧𝜃_𝑡𝜔²𝑑𝑥. (71) We next consider the nonlinear term in (68). Since −_ℝ𝑛𝜑|𝑥|²𝑘²𝑓(𝑥,𝑢)Δ𝑣𝑑𝑥=−_ℝ𝑛𝜑|𝑥|²𝑘²𝑓()+𝑥,𝑢Δ𝑢𝑑𝑥_ℝ𝑛𝜑|𝑥|²𝑘²𝑓(𝑥,𝑢)Δ𝑧𝜃_𝑡𝜔𝑑𝑥=_ℝ𝑛𝑓()|𝑥|𝑥,𝑢∇𝜑²𝑘²⋅∇𝑢𝑑𝑥+_ℝ𝑛𝜑|𝑥|²𝑘²𝜕𝑓()𝜕𝑥𝑥,𝑢⋅∇𝑢𝑑𝑥+_ℝ𝑛𝜑|𝑥|²𝑘²𝜕𝑓(|𝜕𝑢𝑥,𝑢)|∇𝑢²+𝑑𝑥_ℝ𝑛𝜑|𝑥|²𝑘²𝑓(𝑥,𝑢)Δ𝑧𝜃_𝑡𝜔𝑑𝑥. (72)
We now estimate each term in the right-hand side of (72). Using (4), the property of 𝜑, and Cauchy's inequality, we see that the first term of the right-hand side of (72) is bounded by _ℝ𝑛|𝑥|∇𝜑²𝑘²𝑓(𝑥,𝑢)|∇𝑢|𝑑𝑥≤𝛼_2ℝ𝑛|𝑥|∇𝜑²𝑘²|𝑢|^𝑝−1||+∇𝑢𝑑𝑥_ℝ𝑛|𝑥|∇𝜑²𝑘²𝜓₂|∇𝑢|𝑑𝑥≤𝑐_ℝ𝑛|𝑥|∇𝜑²𝑘²|𝑢|^2𝑝−2𝑑𝑥+_ℝ𝑛|𝑥|∇𝜑²𝑘²|∇𝑢|²+𝑐𝑑𝑥𝑘^√_{𝑘≤|𝑥|≤2𝑘}𝜓₂|≤𝑐|∇𝑢𝑑𝑥𝑘‖𝑢‖^2𝑝−2_2𝑝−2+‖∇𝑢‖²+1. (73) By (6), we can estimate the second term of the right-hand side of (72) as follows: _ℝ𝑛𝜑|𝑥|²𝑘²𝜕𝑓(|≤𝜕𝑥𝑥,𝑢)|∇𝑢𝑑𝑥_ℝ𝑛𝜑|𝑥|²𝑘²𝜓₃|≤𝜆|∇𝑢𝑑𝑥4_ℝ^𝑛𝜑|𝑥|²𝑘²|∇𝑢|²𝑐𝑑𝑥+𝜆_ℝ^𝑛𝜑|𝑥|²𝑘²𝜓₃²≤𝜆𝑑𝑥2_ℝ^𝑛𝜑|𝑥|²𝑘²|∇𝑣|²𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²∇𝑧𝜃_𝑡𝜔²𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²𝜓₃²𝑑𝑥. (74) For the third term of the right-hand side of (72), by using (5), we have _ℝ𝑛𝜑|𝑥|²𝑘²𝜕𝑓(|𝜕𝑢𝑥,𝑢)|∇𝑢²𝑑𝑥≤𝛽_ℝ𝑛𝜑|𝑥|²𝑘²|∇𝑢|²𝑑𝑥≤2𝛽_ℝ𝑛𝜑|𝑥|²𝑘²|∇𝑣|²𝑑𝑥+2𝛽_ℝ𝑛𝜑|𝑥|²𝑘²∇𝑧𝜃_𝑡𝜔²𝑑𝑥. (75) For the last term of the right-hand side of (72), by using (4) and Young's inequality, we find _ℝ𝑛𝜑|𝑥|²𝑘²𝑓(𝑥,𝑢)Δ𝑧𝜃_𝑡𝜔𝑑𝑥≤𝛼_2ℝ𝑛𝜑|𝑥|²𝑘²|𝑢|^𝑝−1Δ𝑧𝜃_𝑡𝜔+𝑑𝑥_ℝ𝑛𝜑|𝑥|²𝑘²𝜓₂Δ𝑧𝜃_𝑡𝜔𝑑𝑥≤𝑐_ℝ𝑛𝜑|𝑥|²𝑘²|𝑢|^𝑝𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²Δ𝑧𝜃_𝑡𝜔^𝑝+1𝑑𝑥2_ℝ^𝑛𝜑|𝑥|²𝑘²𝜓₂²+1𝑑𝑥2_ℝ^𝑛𝜑|𝑥|²𝑘²Δ𝑧𝜃_𝑡𝜔²𝑑𝑥. (76) Putting (73)–(76) together into (72), it yields that −_ℝ𝑛𝜑|𝑥|²𝑘²𝑓()≤𝑥,𝑢Δ𝑣𝑑𝑥𝜆+4𝛽2_ℝ^𝑛𝜑|𝑥|²𝑘²|∇𝑣|²𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²|𝑢|^𝑝+𝑐𝑑𝑥𝑘‖𝑢‖^2𝑝−2_2𝑝−2+‖∇𝑢‖²+1+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²𝜓₂²+𝜓₃²𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²×∇𝑧𝜃_𝑡𝜔²+Δ𝑧𝜃_𝑡𝜔²+Δ𝑧𝜃_𝑡𝜔^𝑝𝑑𝑥. (77) Then by (68)–(71) and (77), we get 𝑑𝑑𝑡_ℝ^𝑛𝜑|𝑥|²𝑘²|∇𝑣|²𝑑𝑥+𝜆_ℝ𝑛𝜑|𝑥|²𝑘²|∇𝑣|²𝑑𝑥+_ℝ𝑛𝜑|𝑥|²𝑘²|Δ𝑣|²≤𝑐𝑑𝑥𝑘𝑣_𝑡²+‖∇𝑣‖²+‖𝑣‖²+‖𝑢‖^2𝑝−2_2𝑝−2+‖∇𝑢‖²+1+4𝛽_ℝ𝑛𝜑|𝑥|²𝑘²|∇𝑣|²𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²|𝑢|^𝑝𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²∇𝑧𝜃_𝑡𝜔²+Δ𝑧𝜃_𝑡𝜔²+Δ𝑧𝜃_𝑡𝜔^𝑝𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²𝑔²+𝜓₂²+𝜓₃²𝑑𝑥. (78) In particular, 𝑑𝑑𝑡_ℝ^𝑛𝜑|𝑥|²𝑘²|∇𝑣(𝑡)|²𝑑𝑥≤4𝛽_ℝ𝑛𝜑|𝑥|²𝑘²|∇𝑣(𝑡)|²𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²|𝑢(𝑡)|^𝑝+𝑐𝑑𝑥𝑘𝑣_𝑡(𝑡)²+‖∇𝑣(𝑡)‖²+‖𝑣(𝑡)‖²+‖𝑣(𝑡)‖^2𝑝−2_2𝑝−2+𝑧𝜃_𝑡𝜔^2𝑝−2_2𝑝−2+∇𝑧𝜃_𝑡𝜔²+1+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²∇𝑧𝜃_𝑡𝜔²+Δ𝑧𝜃_𝑡𝜔²+Δ𝑧𝜃_𝑡𝜔^𝑝𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²𝑔²+𝜓₂²+𝜓₃²𝑑𝑥. (79) Let 𝑠∈(𝑡,𝑡+1) and integrate the aforementioned inequality from 𝑠 to 𝑡+1: _ℝ𝑛𝜑|𝑥|²𝑘²∇𝑣𝑡+1,𝜔,𝑣₀(𝜔)²≤𝑑𝑥_ℝ𝑛𝜑|𝑥|²𝑘²∇𝑣𝑠,𝜔,𝑣₀(𝜔)²𝑑𝑥+4𝛽_𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²∇𝑣𝜏,𝜔,𝑣₀(𝜔)²𝑑𝑥𝑑𝜏+𝑐_𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²𝑢𝜏,𝜔,𝑢₀(𝜔)^𝑝+𝑐𝑑𝑥𝑑𝜏𝑘_𝑡^𝑡+1𝑣_𝜏𝜏,𝜔,𝑣₀(𝜔)²+𝑑𝜏_𝑡𝑡+1∇𝑣𝜏,𝜔,𝑣₀(𝜔)²+𝑑𝜏_𝑡𝑡+1𝑣𝜏,𝜔,𝑣₀(𝜔)²+𝑑𝜏_𝑡𝑡+1𝑣𝜏,𝜔,𝑣₀(𝜔)^2𝑝−2_2𝑝−2+𝑑𝜏_𝑡𝑡+1𝑧𝜃_𝜏𝜔^2𝑝−2_2𝑝−2+𝑑𝜏_𝑡𝑡+1∇𝑧𝜃_𝜏𝜔²𝑑𝜏+1+𝑐_𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²×∇𝑧𝜃_𝜏𝜔²+Δ𝑧𝜃_𝜏𝜔²+Δ𝑧𝜃_𝜏𝜔^𝑝𝑑𝑥𝑑𝜏+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²𝑔²+𝜓₂²+𝜓₃²𝑑𝑥. (80) Integrating the aforementioned inequality with respect to 𝑠 over (𝑡,𝑡+1), and replacing 𝜔 by 𝜃_−𝑡−1𝜔, we obtain _ℝ𝑛𝜑|𝑥|²𝑘²∇𝑣𝑡+1,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝑑𝑥≤𝑐_3𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²×∇𝑣𝜏,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝑑𝑥𝑑𝜏+𝑐_3𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²×𝑢𝜏,𝜃_−𝑡−1𝜔,𝑢₀𝜃_−𝑡−1𝜔^𝑝+𝑐𝑑𝑥𝑑𝜏₃𝑘_𝑡^𝑡+1𝑣_𝜏𝜏,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑑𝜏_𝑡𝑡+1∇𝑣𝜏,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑑𝜏_𝑡𝑡+1𝑣𝜏,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑑𝜏_𝑡𝑡+1𝑣𝜏,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^2𝑝−2_2𝑝−2+𝑑𝜏_𝑡𝑡+1𝑧𝜃_{𝜏−𝑡−1}𝜔^2𝑝−2_2𝑝−2+𝑑𝜏_𝑡𝑡+1∇𝑧𝜃_{𝜏−𝑡−1}𝜔²𝑑𝜏+1+𝑐_3𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²×∇𝑧𝜃_{𝜏−𝑡−1}𝜔²+Δ𝑧𝜃_{𝜏−𝑡−1}𝜔²+Δ𝑧𝜃_{𝜏−𝑡−1}𝜔^𝑝𝑑𝑥𝑑𝜏+𝑐_3ℝ𝑛𝜑|𝑥|²𝑘²𝑔²+𝜓₂²+𝜓₃²𝑑𝑥. (81)
In the sequel, we estimate each term in the right-hand side of (81) to show that they are arbitrary small when 𝑘 and 𝑡 are large enough.
To estimate the first term in the right-hand side of (81), we cite the result (4.46) in [1]; that is, 12𝑑𝑑𝑡_ℝ^𝑛𝜑|𝑥|²𝑘²|𝑣|²+1𝑑𝑥2𝛼_1ℝ𝑛𝜑|𝑥|²𝑘²|𝑢|^𝑝𝑑𝑥+_ℝ𝑛𝜑|𝑥|²𝑘²|∇𝑣|²≤𝑐𝑑𝑥𝑘‖∇𝑣‖²+‖𝑣‖²+_ℝ𝑛𝜑|𝑥|²𝑘²1𝜆𝑔²+𝜓₁+12𝜓₂²𝑑𝑥+𝑐_ℝ𝑛𝜑|𝑥|²𝑘²Δ𝑧𝜃_𝑡𝜔²+𝑧𝜃_𝑡𝜔²+𝑧𝜃_𝑡𝜔^𝑝𝑑𝑥. (82) Integrating the the aforementioned inequality over (𝑡,𝑡+1) and replacing 𝜔 by 𝜃_−𝑡−1𝜔 we get _𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤1𝑑𝑥𝑑𝑠2_ℝ^𝑛𝜑|𝑥|²𝑘²𝑣𝑡,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑐𝑑𝑥𝑘_𝑡^𝑡+1∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑑𝑠_𝑡𝑡+1𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝑑𝑠+_ℝ𝑛𝜑|𝑥|²𝑘²1𝜆𝑔²+𝜓₁+12𝜓₂²𝑑𝑥+𝑐_𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²×Δ𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔^𝑝𝑑𝑥𝑑𝑠. (83) To estimate the first term of the right-hand side of (83), we need a result in [1]. Substituting 𝜃_−𝑡−1𝜔 for 𝜔 of (4.49) in [1], we get _ℝ𝑛𝜑|𝑥|²𝑘²𝑣𝑡,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝑑𝑥≤𝑒^{𝜆(𝑇−𝑡)}_ℝ𝑛𝜑|𝑥|²𝑘²𝑣𝑇,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑐𝑑𝑥𝑘^𝑡_𝑇𝑒^{𝜆(𝑠−𝑡)}𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑑𝑠_𝑡𝑇𝑒^{𝜆(𝑠−𝑡)}_ℝ𝑛𝜑|𝑥|²𝑘²×2𝜆𝑔²+2𝜓₁+𝜓₂²𝑑𝑥𝑑𝑠+𝑐^𝑡_𝑇𝑒^{𝜆(𝑠−𝑡)}_ℝ𝑛𝜑|𝑥|²𝑘²×Δ𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔^𝑝𝑑𝑥𝑑𝑠≤𝑐𝑒^{𝜆(𝑇−𝑡−1)}_ℝ𝑛𝜑|𝑥|²𝑘²×𝑣𝑇,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑐𝑑𝑥𝑘_𝑇^𝑡+1𝑒^{𝜆(𝑠−𝑡−1)}𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝑑𝑠+𝑐_𝑇^𝑡+1𝑒^{𝜆(𝑠−𝑡−1)}_ℝ𝑛𝜑|𝑥|²𝑘²×2𝜆𝑔²+2𝜓₁+𝜓₂²𝑑𝑥𝑑𝑠+𝑐_𝑇^𝑡+1𝑒^{𝜆(𝑠−𝑡−1)}_ℝ𝑛𝜑|𝑥|²𝑘²×Δ𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔^𝑝𝑑𝑥𝑑𝑠, (84) where 𝐷𝑇=𝑇(𝜔) is an absorbing time in Lemma 14 and 𝑡≥𝑇. It is a direct result of [1] that there exist 𝑇₁=𝑇¹𝐷(𝜔,𝜀)≥𝑇, 𝑅₁=𝑅₁(𝜔,𝜀) such that, for all 𝑡≥𝑇₁, 𝑘≥𝑅₁ the right-hand side of the aforementioned inequality is less than or equal to 𝜀/20𝑐₃, so we get 12_ℝ^𝑛𝜑|𝑥|²𝑘²𝑣𝑡,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝜀𝑑𝑥≤40𝑐₃. (85) By Lemma 15 we find that there exists 𝑇₂=𝑇²𝐷(𝜔)>0, for all 𝑡≥𝑇₂, 𝑐𝑘_𝑡^𝑡+1∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑑𝑠_𝑡𝑡+1𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑐𝑑𝑠𝑘(1+𝑟(𝜔)). (86) Choosing 𝑅₂=𝑅₂(𝜔,𝜀)≥0 such that (𝑐/𝑘)(1+𝑟(𝜔))≤𝜀/40𝑐₃ for 𝑘≥𝑅₂, we deduce that 𝑐𝑘_𝑡^𝑡+1∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²+𝑑𝑠_𝑡𝑡+1𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝜀𝑑𝑠40𝑐₃,∀𝑡≥𝑇₂,∀𝑘≥𝑅₂. (87) For the third term of the right-hand side of (83), we have _ℝ𝑛𝜑|𝑥|²𝑘²1𝜆𝑔²+𝜓₁+12𝜓₂²≤𝑑𝑥_{|𝑥|≥𝑘}1𝜆𝑔²+𝜓₁+12𝜓₂²𝑑𝑥, (88) since 𝜓₁∈𝐿¹(ℝ^𝑛), 𝜓₂∈𝐿²(ℝ^𝑛), and 𝑔∈𝐿²(ℝ^𝑛) there is a constant 𝑅₃=𝑅₃(𝜀) such that, for all 𝑘≥𝑅₃, _ℝ𝑛𝜑|𝑥|²𝑘²1𝜆𝑔²+𝜓₁+12𝜓₂²𝜀𝑑𝑥≤40𝑐₃. (89) Note that 𝑧(𝜃_𝑡𝜔)=∑^𝑚_𝑗=1ℎ_𝑗𝑧_𝑗(𝜃_𝑡𝜔_𝑗) and ℎ_𝑗∈𝐻²(ℝ^𝑛)⋂𝑊^2,𝑝(ℝ^𝑛)⋂𝐿^∞(ℝ^𝑛) for 𝑗=1,2,…,𝑚. Hence there exists 𝑅₄=𝑅₄(𝜔,𝜀) such that for all 𝑘≥𝑅₄ and 𝑗=1,2,…,𝑚, _{|𝑥|≥𝑘}ℎ_𝑗(𝑥)²+ℎ_𝑗(𝑥)^𝑝+Δℎ_𝑗(𝑥)²≤𝑑𝑥𝜆𝑒^−(𝜆/2)𝜀160𝑐𝑐₃𝑚^𝑝𝑟(𝜔), (90) where 𝑐 is the constant in (83), and thus we have the following estimate: 𝑐_𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²×Δ𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔^𝑝𝑑𝑥𝑑𝑠≤𝑐_𝑡^𝑡+1_{|𝑥|≥𝑘}×Δ𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔²+𝑧𝜃_{𝑠−𝑡−1}𝜔^𝑝𝑑𝑥𝑑𝑠≤𝑐𝑚^𝑝_𝑡^𝑚𝑡+1_{𝑗=1|𝑥|≥𝑘}×Δℎ_𝑗²𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗²+ℎ_𝑗²𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗²+ℎ_𝑗^𝑝𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗^𝑝𝑑𝑥𝑑𝑠=𝑐𝑚^𝑝_𝑡^𝑡+1𝑚_𝑗=1𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗²_{|𝑥|≥𝑘}Δℎ_𝑗²+𝑑𝑥_𝑚𝑗=1𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗²_{|𝑥|≥𝑘}ℎ_𝑗²+𝑑𝑥_𝑚𝑗=1𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗^𝑝_{|𝑥|≥𝑘}ℎ_𝑗^𝑝≤𝑑𝑥𝑑𝑠𝜆𝑒^−(𝜆/2)𝜀160𝑐𝑐₃𝑚^𝑝𝑟(𝜔)𝑐𝑚^𝑝_𝑡^𝑚𝑡+1_𝑗=12𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗²+𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗^𝑝≤𝑑𝑠𝜆𝑒^−(𝜆/2)𝜀80𝑐₃𝑟(𝜔)_𝑡^𝑚𝑡+1_𝑗=1𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗²+𝑧_𝑗𝜃_{𝑠−𝑡−1}𝜔_𝑗^𝑝≤𝑑𝑠𝜆𝑒^−(𝜆/2)𝜀80𝑐₃𝑟(𝜔)_𝑡^𝑡+1𝑝𝜃_{𝑠−𝑡−1}𝜔=𝑑𝑠𝜆𝑒^−(𝜆/2)𝜀80𝑐₃𝑟(𝜔)⁰₋₁𝑝𝜃_𝑠𝜔≤𝑑𝑠𝜆𝑒^−(𝜆/2)𝜀80𝑐₃𝑟(𝜔)𝑟(𝜔)⁰₋₁𝑒^{−(𝜆/2)𝑠}𝜀𝑑𝑠≤40𝑐₃. (91) Let 𝑇^∗₁=𝑇^∗₁𝐷,𝜔,𝜀=max𝑇₁,𝑇₂,𝑅^∗₁=𝑅^∗₁(𝜔,𝜀)=max𝑅₁,𝑅₂,𝑅₃,𝑅₄. (92)
From (83), (85), (87), (89), and (91), ∀𝑡≥𝑇^∗₁, 𝑘≥𝑅^∗₁, we can estimate the first term of the right-hand side of (81) as 𝑐_3𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²×∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝜀𝑑𝑥𝑑𝑠≤.10 (93)
Next, from (82) and a similar process as the proof of (93), one can also get the result for the second term of right-hand side of (81); that is, there exist 𝑅^∗₂=𝑅^∗₂(𝜔,𝜀)>0 and 𝑇^∗₂=𝑇^∗₂(𝐷,𝜔,𝜀)>0 such that, for all 𝑡≥𝑇^∗₂, 𝑘≥𝑅^∗₂, 𝑐_3𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²×𝑢𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^𝑝𝜀𝑑𝑥𝑑𝑠≤.10 (94)
From Lemmas 15 and 16 we see that there exist 𝑇^∗₃=𝑇^∗₃(𝐷,𝜔)>0 and 𝑅^∗₃=𝑅^∗₃(𝜔,𝜀)>0 such that ∀𝑡≥𝑇^∗₃,∀𝑘≥𝑅^∗₃, 𝑐₃𝑘_𝑡^𝑡+1∇𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑐𝑑𝑠𝑘((𝜔≤𝜀1+𝑟)),𝑐10₃𝑘_𝑡^𝑡+1𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑐𝑑𝑠𝑘((𝜔≤𝜀1+𝑟)),𝑐10₃𝑘_𝑡^𝑡+1𝑣_𝑠𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑐𝑑𝑠𝑘((𝜔≤𝜀1+𝑟)),𝑐10₃𝑘_𝑡^𝑡+1𝑣𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔^2𝑝−2_2𝑝−2≤𝑐𝑑𝑠𝑘((𝜔≤𝜀1+𝑟)).10 (95)
For the seventh and eighth terms of the right-hand side of (81), we have _𝑡^𝑡+1𝑧𝜃_{𝑠−𝑡−1}𝜔^2𝑝−2_2𝑝−2+∇𝑧𝜃_{𝑠−𝑡−1}𝜔²𝑑𝑠≤𝑐_0𝑡^𝑡+1𝑝𝜃_{𝑠−𝑡−1}𝜔𝑑𝑠=𝑐₀⁰₋₁𝑝𝜃_𝑠𝜔𝑑𝑠≤𝑐₀𝑟(𝜔)⁰₋₁𝑒^{−(𝜆/2)𝑠}𝑑𝑠≤𝑐𝑟(𝜔). (96) So there exists 𝑅^∗₄=𝑅^∗₄(𝜔,𝜀)>0 such that, for all 𝑘≥𝑅^∗₄ and all 𝑡≥0, we have 𝑐₃𝑘_𝑡^𝑡+1𝑧𝜃_{𝑠−𝑡−1}𝜔^2𝑝−2_2𝑝−2+∇𝑧𝜃_{𝑠−𝑡−1}𝜔²≤𝑐𝑑𝑠𝑘𝑟(𝜔)≤𝜀.10 (97)
Similar to the proof of (91), one can show that there exists 𝑅^∗₅=𝑅^∗₅(𝜔,𝜀)>0 such that, for all 𝑘≥𝑅^∗₅ and all 𝑡≥0, we have 𝑐_3𝑡^𝑡+1_ℝ𝑛𝜑|𝑥|²𝑘²∇𝑧𝜃_{𝑠−𝑡−1}𝜔²+Δ𝑧𝜃_{𝑠−𝑡−1}𝜔²+Δ𝑧𝜃_{𝑠−𝑡−1}𝜔^𝑝𝜀𝑑𝑥𝑑𝑠≤.10 (98)
Since 𝜓₂∈𝐿²(ℝ^𝑛), 𝜓₃∈𝐿²(ℝ^𝑛), and 𝑔∈𝐿²(ℝ^𝑛), we can easily show that there exists 𝑅^∗₆=𝑅^∗₆(𝜀)>0 such that the last term of the right-hand side of (81) is bounded by 𝑐_3ℝ𝑛𝜑|𝑥|²𝑘²𝑔²+𝜓₂²+𝜓₃²𝑑𝑥≤𝑐_{3|𝑥|≥𝑘}𝑔²+𝜓₂²+𝜓₃²𝜀𝑑𝑥≤,10 (99) when 𝑘≥𝑅^∗₆.
Finally, let 𝑅^∗=𝑅^∗(𝜔,𝜀)=max𝑅^∗₁,𝑅^∗₂,…,𝑅^∗₆,𝑇^∗=𝑇^∗𝐷(𝜔,𝜀)=max𝑇^∗₁,𝑇^∗₂,𝑇^∗₃. (100) From (81), (93)–(95), (97)–(99) we get _ℝ𝑛𝜑|𝑥|²𝑘²∇𝑣𝑡+1,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝑑𝑥≤𝜀,𝑡≥𝑇^∗,𝑘≥𝑅^∗. (101) Therefore, ∀𝑡≥𝑇^∗, _{√|𝑥|≥2𝑅∗}∇𝑣𝑡+1,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²≤𝑑𝑥_ℝ𝑛𝜑|𝑥|²𝑅^∗2∇𝑣𝑡+1,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²𝑑𝑥≤𝜀. (102) The proof is complete.

Lemma 18. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Let 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and 𝑢₀(𝜔)∈𝐷(𝜔). Then, for ℙ-a.e. 𝜔∈Ω and for every 𝜀>0, there exist 𝑇^∗=𝑇^∗𝐷(𝜔,𝜀)>0 and 𝑅^∗=𝑅^∗(𝜔,𝜀), such that the solution 𝑢(𝑡,𝜔,𝑢₀(𝜔)) of problem (1)-(2) satisfies that, ∀𝑡≥𝑇^∗𝐷(𝜔,𝜀), _{|𝑥|≥𝑅∗}𝑢𝑡,𝜃_−𝑡𝜔,𝑢₀𝜃_−𝑡𝜔²+∇𝑢𝑡,𝜃_−𝑡𝜔,𝑢₀𝜃_−𝑡𝜔²𝑑𝑥≤𝜀. (103)

Proof. Considering ∀𝜀>0, we choose 𝑅₅=𝑅₅(𝜔,𝜀) large enough such that _{|𝑥|≥𝑅5}ℎ_𝑗²+∇ℎ_𝑗²𝜀𝑑𝑥≤2𝑚²𝑟(𝜔),𝑗=1,2,…,𝑚, (104) and set 𝑅^∗′=max{𝑅^∗,𝑅₅}. Then by Lemma 17 and (34), one can easily show that _{|𝑥|≥𝑅∗′}𝑢𝑡,𝜃_−𝑡𝜔,𝑢₀𝜃_−𝑡𝜔²+∇𝑢𝑡,𝜃_−𝑡𝜔,𝑢₀𝜃_−𝑡𝜔²𝑑𝑥≤3𝜀, (105) with ∀𝑡≥𝑇^∗𝐷(𝜔,𝜀), where 𝑅^∗, 𝑇^∗𝐷(𝜔,𝜀) are the constants in Lemma 17. The proof is complete.

4.3. Asymptotic Compactness in Bounded Balls

In what follows, we prove the asymptotic compactness in any bounded ball, which together with Lemma 18 and Theorem 10 is a necessary condition for verifying the (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-asymptotic compactness. For this purpose, we set 𝜓=1−𝜑, where 𝜑 is the function described in Lemma 17.

For fixed 𝑘≥1, define 𝑣̃𝑡,𝜔,𝑣₀(𝜔)|𝑥|=𝜓²𝑘²𝑣𝑡,𝜔,𝑣₀(𝜔). (106) Then 𝑣(̃𝑡,𝜔,𝑣₀(𝜔))∈𝐻¹₀(𝑄_2𝑘) and 𝑣̃𝑡,𝜔,𝑣₀(𝜔)_{𝐻10(𝑄2𝑘)}≤𝑐𝑣𝑡,𝜔,𝑣₀(𝜔)_{𝐻1(ℝ𝑛)}, (107) where 𝑐 is a positive constant, independent of 𝑡, 𝜔, and 𝑘. Then we have 𝑣_𝑡̃(𝑡)|𝑥|=𝜓²𝑘²𝑣_𝑡(𝑡),Δ𝑣̃=Δ𝜓𝑣+2∇𝜓⋅∇𝑣+𝜓Δ𝑣. (108) Multiplying (30) by 𝜓(|𝑥|²/𝑘²), then we can easily show that 𝑣_𝑡̃̃̃()𝜃+𝜆𝑣−Δ𝑣=𝜓𝑓𝑥,𝑢+𝜓𝑔+𝜓Δ𝑧_𝑡𝜔−𝑣Δ𝜓−2∇𝜓⋅∇𝑣. (109) Consider the following eigenvalue problem: −Δ𝜈=𝜆𝜈,in𝑄_2𝜅,𝜈∣_{𝜗𝑄2𝜅}=0, (110) and then problem (110) has a family of eigenfunctions {𝑒_𝑗}^∞_𝑗=1 with corresponding eigenvalues {𝜆_𝑗}^∞_𝑗=1 such that {𝑒_𝑗}^∞_𝑗=1 forms an orthogonal basis in both 𝐿²(𝑄_2𝑘) and 𝐻¹₀(𝑄_2𝑘) and 𝜆₁≤𝜆₂≤⋯≤𝜆_𝑗⟶∞as𝑗⟶∞. (111) Given 𝑛, let 𝑋_𝑛= span{𝑒₁,…,𝑒_𝑛} and 𝑃_𝑛∶𝐿²(𝑄_2𝑘)→𝑋_𝑛 be the projection operator. For any 𝑣̃∈𝐻¹₀(𝑄_2𝑘), we write 𝑣̃=𝑃_𝑚𝑣̃+𝐼−𝑃_𝑚𝑣̃=∶𝑣₁̃+𝑣₂̃. (112) In order to prove the asymptotic compactness we need the following lemma, which can be found in [19].

Lemma 19 (see [19]). Let 𝑟>0, 𝜆>0, 𝜏∈ℝ, and for 𝑠>𝜏𝑦^′(𝑠)+𝜆𝑦(𝑠)≤ℎ(𝑠), (113) where the functions 𝑦, 𝑦^′, ℎ are assumed to be locally integrable and 𝑦, ℎ nonnegative on the interval 𝑡<𝑠<𝑡+𝑟, for some 𝑡≥𝜏. Then, for any 𝛿∈(0,𝑟), 𝑦(𝑡+𝑟)≤𝑒^{−𝜆(𝑟−𝛿)}1𝛿_𝑡^𝑡+𝑟𝑦(𝑠)𝑑𝑠+𝑒^{−𝜆(𝑡+𝑟)}_𝑡^𝑡+𝑟𝑒^𝜆𝑠ℎ(𝑠)𝑑𝑠. (114) In particular, let 𝛿=(𝑟/2), then 𝑦(𝑡+𝑟)≤𝑒^{−𝜆(𝑟/2)}2𝑟_𝑡^𝑡+𝑟𝑦(𝑠)𝑑𝑠+𝑒^{−𝜆(𝑡+𝑟)}∫_𝑡^𝑡+𝑟𝑒^𝜆𝑠ℎ(𝑠)𝑑𝑠. (115)

Lemma 20. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Let 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and 𝑢₀(𝜔)∈𝐷(𝜔). Then, for ℙ-a.e. 𝜔∈Ω and for every 𝜀>0, there exist 𝑀(𝜔,𝜀)>0, 𝜂(𝜔,𝜀)>0, and 𝑇𝐷(𝜔)>0 such that the solution 𝑣(𝑡,𝜔,𝑣₀(𝜔)) of (30)-(31) satisfies that, 𝐷∀𝑡≥𝑇(𝜔), ^𝑡_{𝑡−𝜂𝑄2𝑘(𝑣(𝑠,𝜃−𝑡𝜔,𝑣0(𝜃−𝑡𝜔))≥𝑀)}𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔^𝑝−1×𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀^𝑝−1𝑑𝑥𝑑𝑠≤𝜀,_{𝑡𝑡−𝜂𝑄2𝑘(𝑣(𝑠,𝜃−𝑡𝜔,𝑣0(𝜃−𝑡𝜔))≤−𝑀)}𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔^𝑝−1×𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔+𝑀^𝑝−1𝑑𝑥𝑑𝑠≤𝜀. (116)

Proof. If 𝑝=2, one can easily show the results by (36). So in the following we assume 𝑝>2. Let (𝑣−𝑀)₊=𝑣−𝑀,𝑥∈𝑄¹_2𝑘,0,elsewhere,(𝑣+𝑀)₋=𝑣+𝑀,𝑥∈𝑄²_2𝑘,0,elsewhere, (117) where 𝑄¹_2𝑘=𝑄_2𝑘(𝑣≥𝑀)=𝑄_2𝑘⋂{𝑥∶𝑣(𝑡,𝜔,𝑣₀(𝜔))≥𝑀} and 𝑄²_2𝑘=𝑄_2𝑘(𝑣≤−𝑀)=𝑄_2𝑘⋂{𝑥∶𝑣(𝑡,𝜔,𝑣₀(𝜔))≤−𝑀}.
Multiplying (30) with |(𝑣−𝑀)₊|^𝑝−1 and integrating over ℝ^𝑛, we get 1𝑝𝑑)𝑑𝑡(𝑣−𝑀₊^𝑝_𝑝+𝜆_𝑄12𝑘𝑣(𝑣−𝑀)^𝑝−1+𝑑𝑥𝑝−1_𝑄12𝑘|∇(𝑣−𝑀)|²(𝑣−𝑀)^𝑝−2𝑑𝑥=_𝑄12𝑘𝑓(𝑥,𝑢)(𝑣−𝑀)^𝑝−1+𝑑𝑥_𝑄12𝑘𝑔(𝑣−𝑀)^𝑝−1+𝑑𝑥_𝑄12𝑘Δ𝑧𝜃_𝑡𝜔(𝑣−𝑀)^𝑝−1𝑑𝑥. (118) From (41), when 𝑥∈𝑄¹_2𝑘, we have 𝑓𝑥,𝑣+𝑧𝜃_𝑡𝜔(𝑣−𝑀)^𝑝−1=𝑓𝑥,𝑣+𝑧𝜃_𝑡𝜔𝑣(𝑣−𝑀)^𝑝−1𝑣𝛼≤−¹2^𝑝|𝑣|^𝑝(𝑣−𝑀)^𝑝−1𝑣+𝑐𝑧𝜃_𝑡𝜔^𝑝+𝑧𝜃_𝑡𝜔²(𝑣−𝑀)^𝑝−1𝑣+𝜓₁+12𝜓²₂(𝑣−𝑀)^𝑝−1𝑣𝛼≤−¹2^𝑝|𝑣|^𝑝−1(𝑣−𝑀)^𝑝−1+𝑐𝑀𝑧𝜃_𝑡𝜔^𝑝+𝑧𝜃_𝑡𝜔²(𝑣−𝑀)^𝑝−1+1𝑀𝜓₁+12𝜓²₂(𝑣−𝑀)^𝑝−1𝛼≤−¹2^𝑝|𝑣|^𝑝−1(𝑣−𝑀)^𝑝−1+𝑐𝑧𝜃_𝑡𝜔^𝑝+𝑧𝜃_𝑡𝜔²(𝑣−𝑀)^𝑝−1+𝑐𝜓₁+12𝜓²₂(𝑣−𝑀)^𝑝−1, (119) where the constant 𝑐 in the right hand side of (119) is independent of 𝑀 when we assume, without loss of generality, that 𝑀≥1.
From (118)-(119), we get 1𝑝𝑑)𝑑𝑡(𝑣−𝑀₊^𝑝_𝑝+𝑐_4𝑄12𝑘|𝑣|^𝑝−1(𝑣−𝑀)^𝑝−1𝑑𝑥≤𝑐_5𝑄12𝑘𝜓₁+12𝜓₂²+𝑔(𝑣−𝑀)^𝑝−1𝑑𝑥+𝑐_5𝑄12𝑘𝑧𝜃_𝑡𝜔^𝑝+𝑧𝜃_𝑡𝜔²+Δ𝑧𝜃_𝑡𝜔(𝑣−𝑀)^𝑝−1𝑑𝑥. (120) Applying Cauchy's inequality, we obtain 1𝑝𝑑)𝑑𝑡(𝑣−𝑀₊^𝑝_𝑝+𝑐₄2_𝑄¹_2𝑘|𝑣|^𝑝−1(𝑣−𝑀)^𝑝−1+𝑐𝑑𝑥₄2_𝑄¹_2𝑘(𝑣−𝑀)^2𝑝−2≤𝑐𝑑𝑥₄2_𝑄¹_2𝑘(𝑣−𝑀)^2𝑝−2𝑑𝑥+𝑐_𝑄12𝑘𝜓₁²+𝜓₂⁴+𝑔²𝑑𝑥+𝑐_𝑄12𝑘𝑧𝜃_𝑡𝜔^2𝑝+𝑧𝜃_𝑡𝜔⁴+Δ𝑧𝜃_𝑡𝜔²𝑑𝑥, (121) that is, 2𝑝𝑑)𝑑𝑡(𝑣−𝑀₊^𝑝_𝑝+𝑐_4𝑄12𝑘|𝑣|^𝑝−1(𝑣−𝑀)^𝑝−1𝑑𝑥≤𝑐_𝑄12𝑘𝜓₁²+𝜓₂⁴+𝑔²𝑑𝑥+𝑐_𝑄12𝑘𝑧𝜃_𝑡𝜔^2𝑝+𝑧𝜃_𝑡𝜔⁴+Δ𝑧𝜃_𝑡𝜔²𝑑𝑥,2𝑝𝑑)𝑑𝑡(𝑣−𝑀₊^𝑝_𝑝+𝑐₄𝑀^𝑝−2_𝑄12𝑘(𝑣−𝑀)^𝑝𝑑𝑥≤𝑐_𝑄12𝑘𝜓₁²+𝜓₂⁴+𝑔²𝑑𝑥+𝑐_𝑄12𝑘𝑧𝜃_𝑡𝜔^2𝑝+𝑧𝜃_𝑡𝜔⁴+Δ𝑧𝜃_𝑡𝜔²𝑑𝑥. (122)(123) By (123) and Lemma 19 with 𝑟=1, we get (𝑣−𝑀)₊(𝑡)^𝑝_𝑝≤2𝑒^{−𝜇/2𝑡}_𝑡−1(𝑣−𝑀)₊(𝜏)^𝑝_𝑝𝑑𝜏+𝑐𝑒^{−𝜇𝑡𝑡}_𝑡−1𝑒^𝜇𝜏𝑧𝜃_𝜏𝜔^2𝑝_2𝑝+𝑧𝜃_𝜏𝜔⁴₄+Δ𝑧𝜃_𝜏𝜔²𝑑𝜏+𝑐𝑒^{−𝜇𝑡𝑡}_𝑡−1𝑒^𝜇𝜏𝜓₁²+𝜓₂⁴₄+𝑔²𝑑𝜏, (124) where 𝜇=𝑐₄𝑀^𝑝−2𝑝/2.
Set 𝑠∈[𝑡−1,𝑡]; we first substitute 𝑠 for 𝑡 in the aforementioned inequality and then we replace 𝜔 by 𝜃_−𝑡𝜔 to get 𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀₊^𝑝_𝑝≤2𝑒^{−𝜇/2𝑠}_𝑠−1𝑣𝜏,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀₊^𝑝_𝑝𝑑𝜏+𝑐𝑒^{−𝜇𝑠𝑠}_𝑠−1𝑒^𝜇𝜏𝑧𝜃_𝜏−𝑡𝜔^2𝑝_2𝑝+𝑧𝜃_𝜏−𝑡𝜔⁴₄+Δ𝑧𝜃_𝜏−𝑡𝜔²𝑑𝜏+𝑐𝑒^{−𝜇𝑠𝑠}_𝑠−1𝑒^𝜇𝜏𝑑𝜏𝜓₁²+𝜓₂⁴₄+𝑔²≤2𝑒^{−𝜇/2𝑡}_𝑡−2𝑣𝜏,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀₊^𝑝_𝑝𝑑𝜏+𝑐𝑒^{−𝜇𝑠𝑠}_𝑠−1𝑒^𝜇𝜏𝑧𝜃_𝜏−𝑡𝜔^2𝑝_2𝑝+𝑧𝜃_𝜏−𝑡𝜔⁴₄+Δ𝑧𝜃_𝜏−𝑡𝜔²𝑑𝜏+𝑐𝑒^{−𝜇𝑠𝑠}_𝑠−1𝑒^𝜇𝜏𝑑𝜏𝜓₁²+𝜓₂⁴₄+𝑔². (125) For the first term of the right-hand side of (125), we use Lemma 15: ^𝑡_𝑡−2𝑣𝜏,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀₊^𝑝_𝑝≤𝑑𝜏_𝑡𝑡−2𝑣𝜏,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔^𝑝_𝑝((𝜔𝐷(𝜔).𝑑𝜏≤𝑐1+𝑟)),∀𝑡≥𝑇 (126) Then there exists 𝑀₁=𝑀₁(𝜔,𝜀)>0 such that, for all 𝐷𝑡≥𝑇(𝜔) and all 𝑀≥𝑀₁, 2𝑒^{−𝜇/2𝑡}_𝑡−2𝑣𝜏,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀₊^𝑝_𝑝𝑑𝜏≤𝑐𝑒^−𝜇/2((𝜔≤𝜀1+𝑟))3. (127) For the second term of the right-hand side of (125), ∀𝑠∈[𝑡−1,𝑡], by (26) we have 𝑐𝑒^{−𝜇𝑠𝑠}_𝑠−1𝑒^𝜇𝜏𝑧𝜃_𝜏−𝑡𝜔^2𝑝_2𝑝+𝑧𝜃_𝜏−𝑡𝜔⁴₄+Δ𝑧𝜃_𝜏−𝑡𝜔²𝑑𝜏≤𝑐^𝑠_𝑠−1𝑒^{𝜇(𝜏−𝑠)}𝑝𝜃_𝜏−𝑡𝜔𝑑𝜏=𝑐⁰₋₁𝑒^𝜇𝜏𝑝𝜃_{𝜏−𝑡+𝑠}𝜔(𝜔)𝑑𝜏≤𝑐𝑟⁰₋₁𝑒^𝜇𝜏𝑒^{(𝜆/2)|𝜏−𝑡+𝑠|}(𝜔)𝑑𝜏≤𝑐𝑟⁰₋₁𝑒^𝜇𝜏𝑒^𝜆(𝜔)𝑑𝜏≤𝑐𝑟⁰₋₁𝑒^𝜇𝜏(𝜔)𝑑𝜏≤𝑐𝑟𝜇. (128) This implies that there exists 𝑀₂=𝑀₂(𝜔,𝜀)>0 such that ∀𝑀≥𝑀₂ we have 𝑐𝑒^{−𝜇𝑠𝑠}_𝑠−1𝑒^𝜇𝜏𝑧𝜃_𝜏−𝑡𝜔^2𝑝_2𝑝+𝑧𝜃_𝜏−𝑡𝜔⁴₄+Δ𝑧𝜃_𝜏−𝑡𝜔²𝜀𝑑𝜏≤3. (129) For the last term of the right-hand side of (125), we can easily see that there exists 𝑀₃=𝑀₃(𝜀)>0 such that, for all 𝑀≥𝑀₃, 𝑐𝑒^{−𝜇𝑠𝑠}_𝑠−1𝑒^𝜇𝜏𝑑𝜏𝜓₁²+𝜓₂⁴₄+𝑔²=𝑐⁰₋₁𝑒^𝜇𝜏𝑑𝜏𝜓₁²+𝜓₂⁴₄+𝑔²≤𝑐𝜇≤𝜀3. (130) Letting 𝑁^∗=𝑁^∗(𝜔,𝜀)=max{𝑀₁,𝑀₂,𝑀₃}, then combining with (125), (127) and (129)–(130), we can show that, for all 𝐷𝑡≥𝑇(𝜔) and all 𝑀≥𝑁^∗(𝜔,𝜀), 𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀₊^𝑝_𝑝≤𝜀,∀𝑠∈[𝑡−1,𝑡]. (131)
Next, we set 2𝜂=min𝜆ln𝜆𝜀6𝑐₈𝑟(𝜔),𝜀+13𝑐₇𝜓₁²+𝜓₂⁴₄+𝑔²,1, (132) where 𝑐₇=𝑐/𝑐₄, 𝑐₈=𝑐₀𝑐₇, and 𝑐 is the constants in (122). Integrating (122) over (𝑡−𝜂,𝑡), we get ^𝑡_{𝑡−𝜂𝑄12𝑘}|𝑣(𝑠)|^𝑝−1(𝑣(𝑠)−𝑀)^𝑝−1𝑑𝑥𝑑𝑠≤𝑐₆𝑣𝑡−𝜂−𝑀₊^𝑝_𝑝+𝑐₇^𝑡_{𝑡−𝜂𝑄12𝑘}𝑧𝜃_𝑠𝜔^2𝑝+𝑧𝜃_𝑠𝜔⁴+Δ𝑧𝜃_𝑠𝜔²𝑑𝑥𝑑𝑠+𝑐₇^𝑡_{𝑡−𝜂𝑄12𝑘}𝜓₁²+𝜓₂⁴+𝑔²𝑑𝑥𝑑𝑠. (133) Replacing 𝜔 by 𝜃_−𝑡𝜔 and setting 𝑄^∗_2𝑘=𝑄_2𝑘(𝑣(𝑠,𝜃_−𝑡𝜔,𝑣₀(𝜃_−𝑡𝜔))≥𝑀), we obtain ^𝑡_{𝑡−𝜂𝑄∗2𝑘}𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔^𝑝−1×𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀^𝑝−1𝑑𝑥𝑑𝑠≤𝑐₇^𝑡_{𝑡−𝜂𝑄∗2𝑘}𝑧𝜃_𝑠−𝑡𝜔^2𝑝+𝑧𝜃_𝑠−𝑡𝜔⁴+Δ𝑧𝜃_𝑠−𝑡𝜔²𝑑𝑥𝑑𝑠+𝑐₆𝑣𝑡−𝜂,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀₊^𝑝_𝑝+𝑐₇𝜂𝜓₁²+𝜓₂⁴₄+𝑔². (134) For fixed 𝜔∈Ω, by (131) and (132), 𝐷∀𝑡≥𝑇(𝜔), ∀𝑀≥𝑀^∗₁(𝜔,𝜀)=𝑁^∗(𝜔,𝜀/3𝑐₆), 𝑐₆𝑣𝑡−𝜂,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀₊^𝑝_𝑝≤𝜀3, (135) where 𝑁^∗(𝜔,𝜀) is the constant in (131).
Using (132), the first term of the right-hand side of (134) is bounded by 𝑐₇^𝑡_𝑡−𝜂𝑧𝜃_𝜏−𝑡𝜔^2𝑝_2𝑝+𝑧𝜃_𝜏−𝑡𝜔⁴₄+Δ𝑧𝜃_𝜏−𝑡𝜔²𝑑𝜏≤𝑐₈^𝑡_𝑡−𝜂𝑝𝜃_𝜏−𝑡𝜔𝑑𝜏=𝑐₈⁰_−𝜂𝑝𝜃_𝜏𝜔𝑑𝜏≤𝑐₈𝑟(𝜔)⁰_−𝜂𝑒^{−(𝜆/2)𝜏}𝑑𝜏=2𝑐₈𝜆𝑟(𝜔)𝑒^𝜆𝜂/2≤𝜀−13. (136) For the last term of right-hand side of (134), by using (132), we have 𝑐₇𝜂𝜓₁²+𝜓₂⁴₄+𝑔²≤𝜀3. (137) By (134)–(137), ∀𝑀≥𝑀^∗₁, 𝐷∀𝑡≥𝑇(𝜔), and ∀𝑢₀(𝜔)∈𝐷(𝜔) with 𝑣₀(𝜔)=𝑢₀(𝜔)−𝑧(𝜔), we have ^𝑡_{𝑡−𝜂𝑄∗2𝑘}𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔^𝑝−1×𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔−𝑀^𝑝−1𝑑𝑥𝑑𝑠≤𝜀. (138)
Similarly, multiplying (30) with |(𝑣+𝑀)₋|^𝑝−1, we can prove that there exists 𝑀^∗₂=𝑀^∗₂(𝜔,𝜀) such that ∀𝑀≥𝑀^∗₂𝐷,∀𝑡≥𝑇(𝜔), and ∀𝑢₀(𝜔)∈𝐷(𝜔) with 𝑣₀(𝜔)=𝑢₀(𝜔)−𝑧(𝜔), over the region 𝑄^∗∗_2𝑘=𝑄_2𝑘(𝑣(𝑠,𝜃_−𝑡𝜔,𝑣₀(𝜃_−𝑡𝜔))≤−𝑀); we have ^𝑡_{𝑡−𝜂𝑄∗∗2𝑘}𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔^𝑝−1×𝑣𝑠,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔+𝑀^𝑝−1𝑑𝑥𝑑𝑠≤𝜀. (139) Let 𝑀^∗=max{𝑀^∗₁,𝑀^∗₂}; from (138) and (139) we can obtain our results. The proof is complete.

Remark 21. The idea of the proof of the above lemma comes from [19] (this idea can be further traced back to Marion [16] and Robinson [18]). We see from (132) that the constant 𝜂 in Lemma 20 is independent of 𝑡, which is different from the 𝜂 in the Lemma 3.4 in [19]. This is crucial in the following estimates. As we know, the time 𝑡 will vary to infinite when we consider the asymptotic behavior of an RDS. It means that if 𝜂 is not a fixed constant with respect to 𝑡, the following estimate will be invalid. In other words, if the function 𝑔 in (1) is dependent on 𝑡, our method will fail.

Lemma 22. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Let 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂ and 𝑢₀(𝜔)∈𝐷(𝜔). Then, for ℙ-a.e. 𝜔∈Ω and for every 𝜀>0 and all 𝑘≥1, there exist 𝑇𝐷(𝜔)>0 and 𝑁(𝑘,𝜔,𝜀)>0 such that the solution 𝑣(𝑡,𝜔,𝑣₀(𝜔)) of (30)-(31) satisfies that, 𝐷∀𝑡≥𝑇(𝜔), ∀𝑚≥𝑁(𝑘,𝜔,𝜀), 𝑣₂̃𝑡,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔²_{𝐻10𝑄2𝑘}≤𝜀, (140) where 𝑣₂̃=(𝐼−𝑃_𝑚)𝑣̃.

Proof. Multiplying (109) with −Δ𝑣₂̃ and integrating over 𝑄_2𝑘, we get 12𝑑𝑣̃𝑑𝑡^{22𝐻10𝑄2𝑘}+𝜆𝑣̃₂²_{𝐻10𝑄2𝑘}+Δ𝑣̃₂²_{𝐿2𝑄2𝑘}=−𝜓𝑓(𝑥,𝑢),Δ𝑣̃_{2𝐿2(𝑄2𝑘)}−𝜓Δ𝑧𝜃_𝑡𝜔,Δ𝑣̃_{2𝐿2(𝑄2𝑘)}−𝜓𝑔,Δ𝑣̃_{2𝐿2(𝑄2𝑘)}+𝑣Δ𝜓,Δ𝑣̃_{2𝐿2(𝑄2𝑘)}+2∇𝜓⋅∇𝑣,Δ𝑣̃_{2𝐿2(𝑄2𝑘)}. (141)
We now estimate each term in the right-hand side of the aforementioned equality. For the first term, by using Cauchy inequality and (4), we have 𝜓𝑓(𝑥,𝑢),Δ𝑣̃_{2𝐿2(𝑄2𝑘)}≤1Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_𝑄2𝑘𝑓(𝑥,𝑢)²≤1𝑑𝑥Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_𝑄2𝑘|𝑢|^2𝑝−2𝑑𝑥+𝑐_𝑄2𝑘𝜓₂²≤1𝑑𝑥Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_𝑄2𝑘|𝑣|^2𝑝−2𝑑𝑥+𝑐_𝑄2𝑘𝑧𝜃_𝑡𝜔^2𝑝−2≤1𝑑𝑥+𝑐Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_{𝑄2𝑘(|𝑣|≤𝑀)}|𝑣|^2𝑝−2𝑑𝑥+𝑐_{𝑄2𝑘(|𝑣|≥𝑀)}|𝑣|^2𝑝−2𝑑𝑥+𝑐_𝑄2𝑘𝑧𝜃_𝑡𝜔^2𝑝−2≤1𝑑𝑥+𝑐Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_{𝑄2𝑘(|𝑣|≥𝑀)}|𝑣|^2𝑝−2𝑧𝜃𝑑𝑥+𝑐_𝑡𝜔^2𝑝−2_2𝑝−2+𝑐𝜓₂²,𝑄_2𝑘,𝑝,𝑀, (142) where |𝑄_2𝑘| denotes the Lebesgue measure of 𝑄_2𝑘.
For the second to fifth term, we can estimate them as follows: 𝜓Δ𝑧𝜃_𝑡𝜔,Δ𝑣̃_{2𝐿2(𝑄2𝑘)}≤1Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐Δ𝑧𝜃_𝑡𝜔²,𝜓𝑔,Δ𝑣̃_{2𝐿2(𝑄2𝑘)}≤1Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐𝑔²,𝑣Δ𝜓,Δ𝑣̃_{2𝐿2(𝑄2𝑘)}≤1Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_𝑄2𝑘|𝑣|²Δ𝜓²≤1𝑑𝑥Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_ℝ𝑛|𝑣|²Δ𝜓²≤1𝑑𝑥Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_ℝ𝑛|𝑣|²𝜑^′′|𝑥|²𝑘²4|𝑥|²𝑘⁴+𝜑^′|𝑥|²𝑘²2𝑛𝑘²²≤1𝑑𝑥Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐𝑘^{4√𝑘≤|𝑥|≤2𝑘}|𝑣|²≤1𝑑𝑥Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐‖𝑣‖²,2∇𝜓⋅∇𝑣,Δ𝑣̃_{2𝐿2(𝑄2𝑘)}≤1Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐_𝑄2𝑘∇𝜓²|∇𝑣|²≤1𝑑𝑥Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐𝑘²‖∇𝑣‖²≤1Δ𝑣̃10^{22𝐿2𝑄2𝑘}+𝑐‖∇𝑣‖², (143)(144)(145)(146) where the constant 𝑐 in (145) and (146) is independent of 𝑘, since 𝑘≥1.
Combining (141)–(146) we get 𝑑𝑣̃𝑑𝑡^{22𝐻10𝑄2𝑘}+2𝜆𝑣̃₂²_{𝐻10𝑄2𝑘}+Δ𝑣̃₂²_{𝐿2𝑄2𝑘}≤𝑐_{𝑄2𝑘(|𝑣|≥𝑀)}|𝑣|^2𝑝−2𝑑𝑥+𝑐‖𝑣‖²+‖∇𝑣‖²+𝑐𝑧𝜃_𝑡𝜔^2𝑝−2_2𝑝−2+Δ𝑧𝜃_𝑡𝜔²+𝑐𝜓₂²,𝑔²,𝑄_2𝑘,𝑝,𝑀, (147) and this implies that 𝑑𝑣̃𝑑𝑡^{22𝐻10𝑄2𝑘}+𝜆_𝑚+1𝑣̃₂²_{𝐻10𝑄2𝑘}≤𝑐_{𝑄2𝑘(|𝑣|≥𝑀)}|𝑣|^2𝑝−2𝑑𝑥+𝑐‖𝑣‖²+‖∇𝑣‖²+𝑐𝑧𝜃_𝑡𝜔^2𝑝−2_2𝑝−2+Δ𝑧𝜃_𝑡𝜔²+𝑐𝜓₂²,𝑔²,𝑄_2𝑘,𝑝,𝑀. (148) By using Lemma 19 with 𝑟=1, we get 𝑣₂̃(𝑡+1)²_{𝐻10𝑄2𝑘}≤2𝑒^{−𝜆𝑚+1/2}_𝑡^𝑡+1𝑣₂̃(𝑠)²_{𝐻10𝑄2𝑘}𝑑𝑠+𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}_{𝑄2𝑘(|𝑣(𝑠)|≥𝑀)}|𝑣(𝑠)|^2𝑝−2𝑑𝑥𝑑𝑠+𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}×𝑧𝜃_𝑠𝜔^2𝑝−2_2𝑝−2+Δ𝑧𝜃_𝑠𝜔²𝑑𝑠+𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}‖𝑣(𝑠)‖²+‖∇𝑣(𝑠)‖²𝜓𝑑𝑠+𝑐₂²,𝑔²,𝑄_2𝑘,𝑝,𝑀𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝑑𝑠. (149) Substituting 𝜃_−𝑡−1𝜔 for 𝜔, we get 𝑣₂̃𝑡+1,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²_{𝐻10𝑄2𝑘}≤2𝑒^{−𝜆𝑚+1/2}_𝑡^𝑡+1𝑣₂̃𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²_{𝐻10𝑄2𝑘}𝑑𝑠+𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}×|𝑣(∗)|^2𝑝−2𝑑𝑥𝑑𝑠+𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝑧𝜃_{𝑠−𝑡−1}𝜔^2𝑝−2_2𝑝−2+Δ𝑧𝜃_{𝑠−𝑡−1}𝜔²𝑑𝑠+𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}‖𝑣(∗)‖²+‖∇𝑣(∗)‖²𝜓𝑑𝑠+𝑐₂²,𝑔²,𝑄_2𝑘,𝑝,𝑀𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝑑𝑠, (150) for simplicity, hereafter, we write 𝑣(∗)=𝑣(𝑠,𝜃_−𝑡−1𝜔,𝑣₀(𝜃_−𝑡−1𝜔)).
In the sequel, we estimate each term in the right-hand side of (150). For the first term, we use Lemma 15 and (107) to obtain that, for 𝐷𝑡≥𝑇(𝜔), 2𝑒^{−𝜆𝑚+1/2}_𝑡^𝑡+1𝑣₂̃𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²_{𝐻10𝑄2𝑘}𝑑𝑠≤2𝑒^{−𝜆𝑚+1/2}_𝑡^𝑡+1𝑣̃𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²_{𝐻10𝑄2𝑘}𝑑𝑠≤𝑐𝑒^{−𝜆𝑚+1/2}_𝑡^𝑡+1‖𝑣(∗)‖²_{𝐻1(ℝ𝑛)}𝑑𝑠≤𝑐𝑒^{−𝜆𝑚+1/2}(1+𝑟(𝜔)). (151) Thus, there exists 𝑁₁=𝑁₁(𝜔,𝜀), for all 𝑚≥𝑁₁ and all 𝐷𝑡≥𝑇(𝜔), we get 2𝑒^{−𝜆𝑚+1/2}_𝑡^𝑡+1𝑣₂̃𝑠,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²_{𝐻10𝑄2𝑘}𝜀𝑑𝑠≤5. (152)
For the second term in the right-hand side of (150), we can estimate it as follows: _{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^2𝑝−2=𝑑𝑥_{𝑄2𝑘(𝑣(∗)≥𝑀)}|𝑣(∗)|^2𝑝−2+𝑑𝑥_{𝑄2𝑘(𝑣(∗)≤−𝑀)}|𝑣(∗)|^2𝑝−2𝑑𝑥≤2^𝑝−2_{𝑄2𝑘(𝑣(∗)≥𝑀)}|𝑣(∗)|^𝑝−1×|𝑣(∗)−𝑀|^𝑝−1+𝑀^𝑝−1𝑑𝑥+_{𝑄2𝑘(𝑣(∗)≤−𝑀)}|𝑣(∗)|^𝑝−1×|𝑣(∗)+𝑀|^𝑝−1+𝑀^𝑝−1𝑑𝑥≤2^𝑝−2_{𝑄2𝑘(𝑣(∗)≥𝑀)}|𝑣(∗)|^𝑝−1|𝑣(∗)−𝑀|^𝑝−1+𝑑𝑥_{𝑄2𝑘(𝑣(∗)≤−𝑀)}|𝑣(∗)|^𝑝−1|𝑣(∗)+𝑀|^𝑝−1𝑑𝑥+2^𝑝−2𝑀^𝑝−2_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^𝑝𝑑𝑥, (153)𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^2𝑝−2𝑑𝑥𝑑𝑠=𝑐𝑒^{−𝜆𝑚+1(𝑡+1)𝑡+1}_{𝑡+1−𝜂}+_𝑡^{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}×_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^2𝑝−2𝑑𝑥𝑑𝑠≤𝑐𝑒^{−𝜆𝑚+1(𝑡+1)𝑡+1}_{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}2^𝑝−2×_{𝑄2𝑘(𝑣(∗)≥𝑀)}|𝑣(∗)|^𝑝−1|𝑣(∗)−𝑀|^𝑝−1+𝑑𝑥_{𝑄2𝑘(𝑣(∗)≤−𝑀)}|𝑣(∗)|^𝑝−1×|𝑣(∗)+𝑀|^𝑝−1𝑑𝑥𝑑𝑠+𝑐2^𝑝−2𝑀^𝑝−2𝑒^{−𝜆𝑚+1(𝑡+1)}×^𝑡+1_{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^𝑝𝑑𝑥𝑑𝑠+𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}×_𝑡^{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^2𝑝−2𝑑𝑥𝑑𝑠. (154) The first term in the right-hand side of (154) is bounded by 2^𝑝−2𝑐^𝑡+1_{𝑡+1−𝜂}×_{𝑄2𝑘(𝑣(∗)≥𝑀)}|𝑣(∗)|^𝑝−1|𝑣(∗)−𝑀|^𝑝−1+𝑑𝑥_{𝑄2𝑘(𝑣(∗)≤−𝑀)}|𝑣(∗)|^𝑝−1|𝑣(∗)+𝑀|^𝑝−1≤𝜀𝑑𝑥𝑑𝑠𝐷(𝜔),15,∀𝑡≥𝑇 (155) when we choose appropriate 𝜂 and 𝑀 by Lemma 20.
For the second term of the right-hand side of (154), we use Lemma 15, 𝐷∀𝑡≥𝑇(𝜔), 𝑐2^𝑝−2𝑀^𝑝−2𝑒^{−𝜆𝑚+1(𝑡+1)}×^𝑡+1_{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^𝑝𝑑𝑥𝑑𝑠≤𝑐2^𝑝−2𝑀^𝑝−2𝑒^{−𝜆𝑚+1(𝑡+1)𝑡+1}_{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}‖𝑣(∗)‖^𝑝_𝑝𝑑𝑠≤𝑐2^𝑝−2𝑀^𝑝−2𝑒^{−𝜆𝑚+1(𝑡+1)}(1+𝑟(𝜔))^𝑡+1_{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}𝑑𝑠≤𝑐2^𝑝−2𝑀^𝑝−2(1+𝑟(𝜔))^𝑡+1_{𝑡+1−𝜂}𝑒^{𝜆𝑚+1(𝑠−𝑡−1)}𝑑𝑠=𝑐2^𝑝−2𝑀^𝑝−2(1+𝑟(𝜔))⁰_−𝜂𝑒^{𝜆𝑚+1𝑠}≤𝑑𝑠𝑐2^𝑝−2𝑀^𝑝−2(1+𝑟(𝜔))𝜆_𝑚+1, (156) where 𝑀 is determined in (155), and then there exists 𝑁¹₂(𝜔,𝜀)>0, such that ∀𝑚≥𝑁¹₂𝐷,∀𝑡≥𝑇(𝜔), we get 𝑐2^𝑝−2𝑀^𝑝−2𝑒^{−𝜆𝑚+1(𝑡+1)}×^𝑡+1_{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^𝑝≤𝜀𝑑𝑥𝑑𝑠.15 (157) By Lemma 16, the last term of the right-hand side of (154) is bounded by 𝑐𝑒^{−𝜆𝑚+1𝜂}_𝑡^{𝑡+1−𝜂}_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^2𝑝−2𝑑𝑥𝑑𝑠≤𝑐𝑒^{−𝜆𝑚+1𝜂}_𝑡^𝑡+1‖𝑣(∗)‖^2𝑝−2_2𝑝−2𝑑𝑠≤𝑐𝑒^{−𝜆𝑚+1𝜂}(1+𝑟(𝜔)), (158) for all 𝐷𝑡≥𝑇(𝜔). This implies that there exists 𝑁²₂=𝑁²₂(𝜔,𝜀)>0, such that, for all 𝑚≥𝑁²₂, 𝐷𝑡≥𝑇(𝜔), we have 𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}×_𝑡^{𝑡+1−𝜂}𝑒^{𝜆𝑚+1𝑠}_{𝑄2𝑘(|𝑣(∗)|≥𝑀)}|𝑣(∗)|^2𝑝−2≤𝜀𝑑𝑥𝑑𝑠.15 (159) Combining (154), (155), (157), and (159) and setting 𝑁₂=𝑁₂(𝜔,𝜀)=max{𝑁¹₂(𝜔,𝜀),𝑁²₂(𝜔,𝜀)}, then we have that, for all 𝑚≥𝑁₂, 𝐷𝑡≥𝑇(𝜔), 𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}×_{𝑄2𝑘𝑣(∗)(|)|≥𝑀}|𝑣(∗)|^2𝑝−2𝜀𝑑𝑥𝑑𝑠≤5. (160) The third term of the right-hand side of (150) is bounded by 𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝑝𝜃_{𝑠−𝑡−1}𝜔𝑑𝑠=𝑐_𝑡^𝑡+1𝑒^{𝜆𝑚+1(𝑠−𝑡−1)}𝑝𝜃_{𝑠−𝑡−1}𝜔𝑑𝑠=𝑐⁰₋₁𝑒^{𝜆𝑚+1𝑠}𝑝𝜃_𝑠𝜔(𝜔)𝑑𝑠≤𝑐𝑟⁰₋₁𝑒^{𝜆𝑚+1𝑠}𝑒^{−(𝜆/2)𝑠}(𝜔)𝑑𝑠≤𝑐𝑟𝜆_𝑚+1,−𝜆/2 (161) so there exists 𝑁₃=𝑁₃(𝜔,𝜀)>0, ∀𝑚≥𝑁₃,∀𝑡≥0; we have 𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}×_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝑧𝜃_{𝑠−𝑡−1}𝜔^2𝑝−2_2𝑝−2+Δ𝑧𝜃_{𝑠−𝑡−1}𝜔²≤𝜀𝑑𝑠5. (162) For the forth term of the right-hand side of (150), by using Lemma 15, we have 𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}‖𝑣(∗)‖²+‖∇𝑣(∗)‖²𝑑𝑠≤𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝑑𝑠(1+𝑟(𝜔))≤𝑐⁰₋₁𝑒^{𝜆𝑚+1𝑠}((𝜔≤𝑐𝑑𝑠1+𝑟))𝜆_𝑚+1((𝜔𝐷1+𝑟)),∀𝑡≥𝑇(𝜔), (163) and then there exists 𝑁₄=𝑁₄(𝜔,𝜀)>0, for all 𝑚≥𝑁₄ and all 𝐷𝑡≥𝑇(𝜔), 𝑐𝑒^{−𝜆𝑚+1(𝑡+1)}×_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}‖𝑣(∗)‖²+‖∇𝑣(∗)‖²≤𝜀𝑑𝑠5. (164) For the last term of the right-hand side of (150), we have 𝑐𝜓₂²,𝑔²,𝑄_2𝑘,𝑝,𝑀𝑒^{−𝜆𝑚+1(𝑡+1)}×_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝑐𝜓𝑑𝑠≤²²,𝑔²,𝑄_2𝑘,𝑝,𝑀𝜆_𝑚+1, (165) this implies that there exists 𝑁₅=𝑁₅(𝑘,𝜀) (𝑀 is determined in (155)), for all 𝑚≥𝑁₅ and all 𝑡≥0, 𝑐𝜓₂²,𝑔²,𝑄_2𝑘,𝑝,𝑀𝑒^{−𝜆𝑚+1(𝑡+1)}×_𝑡^𝑡+1𝑒^{𝜆𝑚+1𝑠}𝜀𝑑𝑠≤5. (166)
Finally, let 𝑁(𝑘,𝜔,𝜀)=max{𝑁₁,𝑁₂,𝑁₃,𝑁₄,𝑁₅}, and then from (150), (152), (160), (162), (164), and (166) we have that, for all 𝑚≥𝑁(𝑘,𝜔,𝜀), 𝐷𝑡≥𝑇(𝜔), 𝑣₂̃𝑡+1,𝜃_−𝑡−1𝜔,𝑣₀𝜃_−𝑡−1𝜔²_{𝐻10𝑄2𝑘}≤𝜀. (167) The proof is complete.

Lemma 23. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold, and 𝜙 is the RDS generated by problem (1)-(2); then, 𝜒_𝑄𝑘⋅𝜙 is (𝐿²(ℝ^𝑛),𝐻¹(𝑄_𝑘))-asymptotically compact for all 𝑘≥1.

Proof. We first prove that the sequence {𝑣(̃𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑦_𝑚)}^∞_𝑚=1 is precompact in 𝐻¹₀(𝑄_2𝑘), for any 𝑡_𝑚→∞, and any 𝑥_𝑚∈𝐷(𝜃_−𝑡𝑚𝜔) with 𝐷={𝐷(𝜔)}_𝜔∈Ω∈𝒟₂, where the relationship between 𝑥_𝑚 and 𝑦_𝑚 is determined by (31), that is, 𝑦_𝑚=𝑥_𝑚−𝑧(𝜃_−𝑡𝑚𝜔). By Lemma 15 we see that, for all 𝐷𝑡≥𝑇(𝜔), 𝑣𝑡,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔²_{𝐻1(ℝ𝑛)}≤𝑐(1+𝑟(𝜔)), (168) where 𝑣₀(𝜃_−𝑡𝜔)=𝑢₀(𝜃_−𝑡𝜔)−𝑧(𝜃_−𝑡𝜔) with 𝑢₀(𝜔)∈𝐷(𝜔). Thus there is 𝑀₁=𝑀₁(𝐷,𝜔) such that, for all 𝑚≥𝑀₁, we have 𝑡_𝑚𝐷≥𝑇(𝜔) and 𝑣𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑦_𝑚²_{𝐻1(ℝ𝑛)}≤𝑐(1+𝑟(𝜔)),∀𝑚≥𝑀₁. (169) By (107) and the aforementioned inequality we find that 𝑣̃𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑦_𝑚²_{𝐻10𝑄2𝑘}≤𝑐(1+𝑟(𝜔)),∀𝑚≥𝑀₁. (170)
Given 𝜀>0, it follows from Lemma 22 that there are 𝑇𝐷(𝜔) and 𝑁(𝑘,𝜔,𝜀) such that, for all 𝐷𝑡≥𝑇(𝜔), 𝐼−𝑃_𝑁𝑣̃𝑡,𝜃_−𝑡𝜔,𝑣₀𝜃_−𝑡𝜔²_{𝐻10𝑄2𝑘}≤𝜀. (171) Taking 𝑀₂=𝑀₂(𝐷,𝜔) large enough such that 𝑡_𝑚𝐷≥𝑇(𝜔) for 𝑚≥𝑀₂, then we get from (171) that 𝐼−𝑃_𝑁𝑣̃𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑦_𝑚²_{𝐻10𝑄2𝑘}≤𝜀,∀𝑚≥𝑀₂. (172) On the other hand, (170) shows that the sequence {𝑃_𝑁𝑣(̃𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑦_𝑚)} is bounded in the finite-dimensional space 𝑃_𝑁𝐻¹₀(𝑄_2𝑘) and hence is precompact in 𝑃_𝑁𝐻¹₀(𝑄_2𝑘), which along with (172) implies the precompactness of {𝑣(̃𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑦_𝑚)} in 𝐻¹₀(𝑄_2𝑘).
Next we prove the (𝐿²(ℝ^𝑛),𝐻¹(𝑄_𝑘))-asymptotic compactness for 𝜒_𝑄𝑘⋅𝜙. From the definition of 𝑣̃, we see that 𝑣=𝑣̃ in 𝑄_𝑘, so 𝑣(𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑦_𝑚)}^∞_𝑚=1 is precompact in 𝐻¹(𝑄_𝑘). By the relationship 𝑢𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑥_𝑚=𝑣𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑦_𝑚+𝑧(𝜔), (173) and the aforementioed assert, we obtain that {𝑢(𝑡_𝑚,𝜃_−𝑡𝑚𝜔,𝑥_𝑚)}^∞_𝑚=1 is precompact in 𝐻¹(𝑄_𝑘), for all 𝑘≥1. The proof is complete.

5. Asymptotic Compactness and Random Attractors

In this section, we prove our main result, that is, the existence of an (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-random attractor for the RDS 𝜙 associated with the initial value problem of SRDE (1)-(2). To this end, we should show the (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-asymptotic compactness of 𝜙. From Theorem 10 and Lemmas 18 and 23, we can immediately obtain the asymptotic compactness of 𝜙.

Lemma 24. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Then the RDS 𝜙 generated by (1)-(2) is (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛)) asymptotically compact.

Now, we are in a position to present our main result.

Theorem 25. Assume that 𝑔∈𝐿²(ℝ^𝑛) and (3)–(6) hold. Then the RDS 𝜙 generated by (1)-(2) has an (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-random attractor 𝐴_𝑣.

Proof. The result can be obtained by Theorems 9, and 13, Lemmas 14, and 24 immediately.

Remark 26. Our methods can be used to prove the existence of (𝐿²(ℝ^𝑛),𝐻¹(ℝ^𝑛))-pullback attractors for the following non-autonomous reaction-diffusion equation on unbounded domains: 𝜕𝑢()()𝜕𝑡+𝜆𝑢−Δ𝑢=𝑓𝑥,𝑢+𝑔𝑥,𝑡,𝑥∈ℝ^𝑛, (174) with the initial condition 𝑢(𝑥,𝜏)=𝑢_𝜏(𝑥),𝑥∈ℝ^𝑛, (175) for every 𝜏∈ℝ and 𝑡>𝜏 where the nonlinear term 𝑓 satisfies (3)–(6) in this paper. 𝑔 is a given function in 𝐿²_loc(ℝ,ℝ^𝑛) with ^𝑡_−∞𝑒^𝜆𝑠𝑔(𝑠)²𝑑𝑠<∞,∀𝑡∈ℝ, (176) and the result is new in this case.

Acknowledgments

The authors are grateful to the anonymous referees for the helpful comments and suggestions that greatly improved the presentation of this paper. This work was supported by NSF of China 10871078 and FRF for the Central Universities of China 2012QN034, and 2013ZZGH027.