Existence of Random Attractors for a 𝑝-Laplacian-Type Equation with Additive Noise

Abstract

We first establish the existence and uniqueness of a solution for a stochastic 𝑝-Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.

1. Introduction

Let 𝐷⊂ℝ^𝑛, 𝑛∈ℕ, be a bounded open set with regular boundary 𝜕𝐷. In this paper, we investigate the existence of a solution and a random attractor for the following quasilinear differential equation influenced by additive white noise𝑑𝑢+ΔΦ_𝑝(Δ𝑢)+𝑔(𝑥,𝑢)𝑑𝑡=𝑓(𝑥)𝑑𝑡+^𝑚_𝑗=1𝜙_𝑗𝑑𝑊_𝑗(𝑡),𝑥∈𝐷,𝑡≥0,(1.1) with the boundary conditions∇𝑢(𝑡)=𝟎,𝑢(𝑡)=0,𝑥∈𝜕𝐷,𝑡≥0,(1.2) and the initial condition𝑢(0,𝑥)=𝑢₀(𝑥),𝑥∈𝐷.(1.3)

In (1.1), Φ_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝≥2, 𝑊_𝑗(𝑡)(1≤𝑗≤𝑚) are mutually independent two-sided real-valued Wiener processes, 𝜙_𝑗=𝜙_𝑗(𝑥)(1≤𝑗≤𝑚,𝑥∈𝐷) are given real-valued functions that will be assumed to satisfy some conditions. The unknown 𝑢(𝑡) is a real-valued random process, sometimes denoted by 𝑢(𝑡,𝑥) or 𝑢(𝑡,𝑥,𝑤). The exterior forced function 𝑔(𝑥,𝑠) defined in 𝐷×ℝ is subjected to the following growth and monotonicity assumptions:𝑔(𝑥,𝑠)𝑠≥𝐶₁|𝑠|^𝑞−Λ₁(𝑥),Λ₁∈𝐿¹(𝐷),𝐶₁∈ℝ⁺,(1.4)||𝑔(𝑥,𝑠)||≤𝐶₂|𝑠|^𝑞−1+Λ₂(𝑥),Λ₂∈𝐿^{𝑞/(𝑞−1)}(𝐷),𝐶₂∈ℝ⁺,(1.5)𝑔𝑥,𝑠₁−𝑔𝑥,𝑠₂𝑠₁−𝑠₂≥𝐶₃||𝑠₁−𝑠₂||²,𝐶₃∈ℝ,(1.6) where 2≤𝑞≤𝑝<∞.

In deterministic case (without random perturbed term), if 𝑔(𝑥,𝑢)=𝑘𝑢, Temam [1] proved the existence and uniqueness of the solution, and then obtained the global attractor. Recently, Yang et al. [2, 3] obtained the global attractors for a general 𝑝-Laplacian-type equation on unbounded domain and bounded domain, respectively. Chen and Zhong [4] discussed the nonautonomous case where the uniform attractor was derived.

It is well known that the long-time behavior of random dynamical systems (RDS) is characterized by random attractors, which was first introduced by Crauel and Flandoli [5] as a generalization of the global attractors for deterministic dynamical system. The existence of random attractors for RDS has been extensively investigated by many authors, see [5–12] and references therein. However, most of these researches concentrate on the stochastic partial differential equations of semilinear type, such as reaction-diffusion equation [5–8], Ginzburg-Landau equation [9, 10], Navier-Stokes equation [5, 6], FitzHugh-Nagumo system [11] and so on. To our knowledge, recently, the Ladyzhenskaya model in [12] seems the first study on the random attractors for nonsemilinear type equations. It seems that the quasilinear type or complete nonlinear type evolution equations with additive noise take on severe difficulty when one wants to derive the random attractors.

In this paper, we consider the existence and uniqueness of the solution and random attractor for (1.1) with forced term 𝑔(𝑥,𝑢) satisfying (1.4)–(1.6). The additive white noise ∑^𝑚𝑗=1𝜙_𝑗𝑑𝑊_𝑗(𝑡) characterizes all kinds of stochastic influence in nature or man-made complex system which we must take into consideration in the concrete situation.

In order to deal with (1.1), we usually transform by employing a variable change the stochastic equation with a random term into a deterministic one containing a random parameter. Then the structure of the original equation (1.1) is changed by this transformation. As a result, some extra difficulties are developed in the process of the estimate of the solution, especially in the stronger norm space 𝑉, where 𝑉⊂𝐻⊂𝑉^ is the Gelfand triple; see Section 2. Hence, the methods (see [1–3]) used in unperturbed case are completely unavailable for obtaining the random attractors for (1.1).

Though we also follow the classic approach (based on the compact embedding) widely used in [5, 6, 9, 10, 12] and so on, some techniques have to be developed to overcome the difficulty of estimate of the solution to (1.1) in the Sobolev space 𝑉. Fortunately, by introducing a new inner product over the resolvent of Laplacian, we surmount this obstacle and obtain the estimate of the solution in the Sobolev space 𝑉₀, which is weaker than 𝑉, see Lemma 4.2 in Section 4. Here some basic results about Dirichlet forms of Laplacian are used. For details on the Dirichlet forms of a negative definite and self-adjoint operator please refer to [13]. The existence and uniqueness of solution, which ensure the existence of continuous RDS, are proved by employing the standard in [14]. If a restrictive assumption is imposed on the monotonicity coefficient in (1.6) we obtain a compact attractor consisting of a single point which attracts every deterministic bounded subset of 𝐻.

The organization of this paper is as follows. In the next section, we present some notions and results on the theory of RDS and Dirichlet forms which are necessary to our discussion. In Section 3, we prove the existence and uniqueness of the solution to the 𝑝-Laplacian-type equation with additive noise and obtain the corresponding RDS. In Section 4, we give some estimates for the solution satisfying (1.1)–(1.6) in given Hilbert space and then prove the existence of a random attractor for this RDS. In the last section, we prove the existence of the single point attractor under the given condition.

2. Preliminaries

In this section, we first recall some notions and results concerning the random attractor and the random flow, which can be found in [5, 6]. For more systematic theory of RDS we refer to [15]. We then list the Sobolev spaces, Laplacian and its semigroup and Dirichlet forms.

The basic notion in RDS is a measurable dynamical system (MSD). The form (Ω,ℱ,P,𝜃_𝑠) is called a MSD if (Ω,ℱ,P) is a complete probability space and {𝜃_𝑠∶Ω→Ω, 𝑠∈ℝ} is a family of measure-preserving transformations such that (𝑠,𝑤)↦𝜃_𝑠𝑤 is measurable, 𝜃₀=id and 𝜃_𝑡+𝑠=𝜃_𝑡𝜃_𝑠 for all 𝑠,𝑡∈ℝ.

A continuous RDS on a complete separable metric space (𝑋,𝑑) with Borel sigma-algebra ℬ(𝑋) over MSD (Ω,ℱ,P,𝜃_𝑠) is by definition a measurable map 𝜑∶ℝ⁺×Ω×𝑋⟶𝑋,(𝑡,𝑤,𝑥)⟼𝜑(𝑡,𝑤)𝑥(2.1) such that P-a.s. 𝑤∈Ω(i)𝜑(0,𝑤)=id on 𝑋,(ii)𝜑(𝑡+𝑠,𝑤)=𝜑(𝑡,𝜃_𝑠𝑤)𝜑(𝑠,𝑤) for all 𝑠,𝑡∈ℝ⁺ (cocycle property),(iii)𝜑(𝑡,𝑤)∶𝑋→𝑋 is continuous for all 𝑡∈ℝ⁺.

A continuous stochastic flow is by definition a family of measurable mapping 𝑆(𝑡,𝑠;𝑤)∶𝑋→𝑋,−∞≤𝑠≤𝑡≤∞, such that P-a.s. 𝑤∈Ω𝑆𝑆(𝑡(,𝑡𝑟,;𝑠𝑤;𝑤)𝑆)𝑥(𝑟=,𝑠𝑆;𝑤)𝑡𝑥−=𝑠,𝑆0.;(𝑡𝜃,_𝑠𝑠;𝑤)𝑥,𝑥∈𝑋,𝑤𝑥,𝑥∈𝑋,(2.2) for all 𝑠≤𝑟≤𝑡, and 𝑠↦𝑆(𝑡,𝑠;𝑤)𝑥 is continuous in 𝑋 for all 𝑠≤𝑡 and 𝑥∈𝑋.

A random compact set {𝐾(𝑤)}_𝑤∈Ω is a family of compact sets indexed by 𝑤 such that for every 𝑥∈𝑋 the mapping 𝑤↦𝑑(𝑥,𝐾(𝑤)) is measurable with respect to ℱ.

Let 𝒜(𝑤) be a random set. One says that 𝒜(𝑤) is attracting if for P-a.s. 𝑤∈Ω and every deterministic bounded subset 𝐵⊂𝑋lim_𝑡→∞dist𝜑𝑡,𝜃_−𝑡𝑤𝐵,𝒜(𝑤)=0,(2.3) where dist(⋅,⋅) is defined by dist(𝐴,𝐵)=sup_𝑥∈𝐴inf_𝑦∈𝐵𝑑(𝑥,𝑦).

We say that 𝒜(𝑤) absorbs 𝐵⊂𝑋 if P-a.s. 𝑤∈Ω, there exists 𝑡_𝐵(𝑤)>0 such that for all 𝑡≥𝑡_𝐵(𝑤), 𝜑𝑡,𝜃_−𝑡𝑤𝐵⊂𝒜(𝑤).(2.4)

Definition 2.1. Recall that a random compact set 𝑤↦𝒜(𝑤) is called to be a random attractor for the RDS 𝜑 if for P-a.s. 𝑤∈Ω(i)𝒜(𝑤) is invariant, that is, 𝜑(𝑡,𝑤)𝒜(𝑤)=𝒜(𝜃_𝑡𝑤), for all 𝑡≥0;(ii)𝒜(𝑤) is attracting.

Theorem 2.2 (see [5]). Let 𝜑(𝑡,𝑤) be a continuous RDS over a MDS (Ω,ℱ,P;𝜃_𝑡) with a separable Banach Space 𝑋. If there exists a compact random absorbing set 𝐾(𝑤) absorbing every deterministic bounded subset of 𝑋, then 𝜑 possesses a random attractor 𝒜(𝑤) defined by 𝒜(𝑤)=_𝐵∈_ℬ(𝑋)_𝑠≥0_𝑡≥𝑠𝜑𝑡,𝜃_−𝑡𝑤𝐵,(2.5) where ℬ(𝑋) denotes all the bounded subsets of 𝑋.

Let 𝐿^𝑝(𝐷) be the 𝑝-times integrable functions space on 𝐷 with norm denoted by ‖⋅‖_𝑝, 𝒱(𝐷) be the space consisting of infinitely continuously differential real-valued-functions with a compact support in 𝐷. We use 𝑉 to denote the norm closure of 𝒱(𝐷) in Sobolev space 𝑊^2,𝑝(𝐷), that is, 𝑉=𝑊^02,𝑝(𝐷). Since 𝐷 is a bounded smooth domain in ℝ^𝑛, we can endow the Sobolev space 𝑉 with equivalent norm (see [1, page 166]) ‖𝑣‖_𝑉=‖Δ𝑣‖_𝑝=_𝐷||Δ𝑣||^𝑝𝑑𝑥^1/𝑝,𝑣∈𝑉.(2.6) Define 𝑉^ = the dual of 𝑉, that is, 𝑉^=𝑊^−2,𝑝′(𝐷). Then we have 𝑇∈𝑊^−2,𝑝′(𝐷)⟺𝑇=_|𝛼|≤2𝐷^𝛼𝑓_𝛼,𝑓_𝛼∈𝐿^𝑝′(𝐷),(2.7) where 1/𝑝+1/𝑝^=1. Let 𝐻 denote the closure of 𝐿²(𝐷) in 𝒱(𝐷) with the usual scalar product and norm {(⋅,⋅),‖⋅‖₂}. Identifying 𝐻 with its dual space 𝐻^ by the Riesz isomorphism 𝑖∶𝐻→𝐻^, we have the following Gelfand triple: 𝑉⊂𝐻≡𝐻^⊂𝑉^,(2.8) or concretely 𝑊^02,𝑝(𝐷)⊂𝐿²(𝐷)⊂𝑊^02,𝑝(𝐷)^=𝑊^{−2,𝑝/(𝑝−1)}(𝐷),(2.9) where the injections are continuous and each space is dense in the following one.

We define the linear operator 𝐴 by 𝐴𝑢=Δ𝑢 for 𝑢∈𝐻¹⁰(𝐷)⋂𝐻²(𝐷). Then 𝐴 is negative definite and self-adjoint. It is well-known that 𝐴 (with domain 𝑊^02,𝑝(𝐷)) generates a strongly continuous semigroup 𝑀(𝑡) on 𝐿^𝑝(𝐷) which is contractive and positive. Here “contractive” means ‖𝑀(𝑡)‖_𝑝≤1 and “positive” means 𝑀(𝑡)𝑢≥0 for every 0≤𝑢∈𝐿^𝑝(𝐷). The resolvent of generator 𝐴 denoted by 𝑅(𝜆,𝐴), 𝜆∈𝜌(𝐴), where 𝜌(𝐴) is the resolvent set of 𝐴. By Lumer-Phillips Theorem in [16], it follows that (0,∞)⊂𝜌(𝐴) and for 𝑢∈𝐿^𝑝(𝐷)𝑅(𝜆,𝐴)𝑢=(𝜆−𝐴)⁻¹𝑢=^∞0𝑒^−𝜆𝑡𝑀(𝑡)𝑢𝑑𝑡,𝜆>0,(2.10)𝑀(𝑡)𝑢=lim_𝑛→∞𝑡𝑅(𝑛𝑛/𝑡,𝐴)^𝑛𝑢,𝑡>0.(2.11) Furthermore, by (2.10) for every 𝑢∈𝐿^𝑝(𝐷) we have 𝜆‖𝑅𝜆𝑅(𝜆(𝜆,𝐴,𝐴))𝑢𝑢⟶‖_𝑝≤𝑢‖,𝑢‖_𝑝,𝜆>0,as𝜆→∞,(2.12) where the convergence is in the 𝐿^𝑝-norms. Moreover, for 𝑢∈𝐷(𝐴), it follows that 𝑅(𝜆,𝐴)𝑢∈𝐷(𝐴), and 𝐴𝑅(𝜆,𝐴)𝑢=𝑅(𝜆,𝐴)𝐴𝑢.

Since 𝐴 is negative definite and self-adjoint operator on 𝐻¹⁰(𝐷)⋂𝐻²(𝐷), we associate 𝐴 with the Dirichlet forms [13] 𝜀 by 𝜀(𝑢,𝑣)=√−𝐴𝑢,√−𝐴𝑣,𝑢,𝑣∈𝐻¹⁰(𝐷).(2.13)𝜀 is unique determined by 𝐴. For 𝑢,𝑣∈𝐻¹⁰(𝐷), we define a new inner product by𝜀^(𝜆)(𝑢,𝑣)=𝜆(𝑢−𝜆𝑅(𝜆,𝐴)𝑢,𝑣),𝜆>0,(2.14) where 𝑅(𝜆,𝐴) is the resolvent of 𝐴. Then, we have the basic fact (see [13]) that 𝜀^(𝜆)(𝑢,𝑣)↑ as 𝜆→∞, andlim_𝜆→∞𝜀^(𝜆)(𝑢,𝑣)=𝜀(𝑢,𝑣),(2.15) for 𝑢,𝑣∈𝐻¹⁰(𝐷).

3. Existence and Uniqueness of RDS

We introduce an auxiliary Wiener process, which enables us to change (1.1) to a deterministic evolution equation depending on a random parameter. Here, we assume that 𝑊(𝑡) is a two-sided Wiener process on a complete probability space (Ω,ℱ,P), where Ω={𝑤∈𝐶(ℝ,ℝ^𝑚)∶𝑤(0)=0}, ℱ is the Borel sigma-algebra induced by the compact-open topology of Ω and P is the corresponding Wiener measure on (Ω,ℱ). Then we have 𝑤(𝑡)=𝑊(𝑡)=𝑊₁(𝑡),𝑊₂(𝑡),…,𝑊_𝑚(𝑡),𝑡∈ℝ.(3.1) Define the time shift by 𝜃_𝑡𝑤(𝑠)=𝑤(𝑠+𝑡)−𝑤(𝑡),𝑤∈Ω,𝑡,𝑠∈ℝ.(3.2) Then (Ω,ℱ,𝑃,𝜃_𝑡) is a ergodic measurable dynamical system.

In order to obtain the random attractor, in our following discussion, we always assume that 𝜙_𝑗(1≤𝑗≤𝑚) belong to 𝑊^04,𝑝(𝐷) and ∇𝜙_𝑗(1≤𝑗≤𝑚)=𝟎.

We now employ the approach similar to [5] to translate (1.1) by one classical change of variables𝑣(𝑡)=𝑢(𝑡)−𝑧(𝑡),(3.3) where, for short, 𝑧(𝑡)=𝑧(𝑡,𝑤)=∑^𝑚𝑗=1𝜙_𝑗𝑊_𝑗(𝑡). Then, formally, 𝑣(𝑡) satisfies the following equation parameterized by 𝑤∈Ω:𝑑𝑣𝑑𝑡+Δ(Φ(Δ𝑣+Δ𝑧))+𝑔(𝑥,𝑣+𝑧)=𝑓(𝑥),𝑥∈𝐷,𝑡≥𝑠,(3.4)𝑣(𝑠)=𝑢(𝑠)−𝑧(𝑠),𝑥∈𝐷,𝑠∈ℝ,(3.5)∇𝑣=𝟎,𝑣=0,𝑥∈𝜕𝐷,𝑡≥𝑠,(3.6) where 𝑔(𝑥,𝑢) satisfies (1.4)–(1.6) and 𝑓 is given in 𝑉^, 2≤𝑞≤𝑝<∞.

We define a nonlinear operator Ψ on 𝑉Ψ(𝑣)=ΔΦ_𝑝(Δ(𝑣+𝑧))+𝑔(𝑥,𝑣+𝑧)−𝑓(𝑥),(3.7) for 𝑣∈𝑉, 𝑥∈𝐷. Then we haveΨ(𝑣)=Ψ(𝑢),(3.8) where we define Ψ(𝑢)=Δ(Φ_𝑝(Δ𝑢))+𝑔(𝑥,𝑢)−𝑓(𝑥) with 𝑢=𝑣+𝑧 as in (3.3). So we can deduce problem (1.1) to the problem𝑑𝑣𝑑𝑡+Ψ(𝑣)=0,𝑡≥𝑠,(3.9) with initial condition 𝑣(𝑠)=𝑢(𝑠)−𝑧(𝑠) for 𝑠∈ℝ. Moreover, by (3.9), it follows that the solutions 𝑤-wise satisfy the following: 𝑣(𝑡)=𝑣(𝑠)−^𝑡𝑠Ψ(𝑣(𝜏))𝑑𝜏,(3.10) with 𝑣(𝑠)=𝑢(𝑠)−𝑧(𝑠) and 𝑡≥𝑠.

Since 𝑝≥𝑞, by our assumption (1.4)–(1.6) and 𝑓∈𝑉^, it is easy to check that the operator Ψ∶𝑣↦Ψ(𝑣) mapping 𝑊^2,𝑝(𝐷) into 𝑊^−2,𝑝′(𝐷) is well-defined, where 𝑝^=𝑝/(𝑝−1). We now prove the existence and uniqueness of solution to (3.4).

Theorem 3.1. Assume that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in 𝑉^, 2≤𝑞≤𝑝<∞. Then for all 𝑠∈ℝ and 𝑣₀∈𝐻 with 𝑣₀=𝑣(𝑠), (3.4) has a unique solution 𝑣𝑡,𝑤;𝑠,𝑣₀∈𝐿^𝑝loc([𝑠,∞),𝑉)𝐶([𝑠,∞),𝐻)(3.11) for 𝑡≥𝑠 and P-a.s. 𝑤∈Ω. Furthermore, the mapping 𝑣₀↦𝑣(𝑡,𝑤;𝑠,𝑣₀) from 𝐻 into 𝐻 is continuous for all 𝑡≥𝑠.

Proof. We will show that Ψ(𝑣) possesses hemicontinuity, monotonicity, coercivity, and boundedness properties. Then for every 𝑣₀∈𝐻 with 𝑣₀=𝑣(𝑠), the existence and uniqueness of solution 𝑣(𝑡)=𝑣(𝑡,𝑤;𝑠,𝑣₀)∈𝐿^𝑝loc([𝑠,∞),𝑉) follow from [13, Theorem 4.2.4]. If the solution 𝑣∈𝐿^𝑝([𝑠,𝑇],𝑉), 𝑇>0, then it is elementary to check that Ψ(𝑣) belongs to 𝐿^𝑝′([𝑠,𝑇],𝑉^) by our assumption 𝑝≥𝑞 and 𝑓∈𝑉^. Thus, from (3.9), we get that 𝑣_𝑡∈𝐿^𝑝′([𝑠,𝑇],𝑉^). Now by the general fact (see [1, page 164, line 1–3]) it follows that 𝑣 is almost everywhere equal to a function belonging to 𝐶([𝑠,𝑇],𝐻). The continuity of the mapping 𝑣₀↦𝑣(𝑡,𝑤;𝑠,𝑣₀) from 𝐻 into 𝐻 is easily proved by using the monotonicity of Ψ.
By [13, Theorem 4.2.4], it remains to show that Ψ(𝑣) possesses hemicontinuity, monotonicity, coercivity, and boundedness properties. For convenience of our discussion in the following, we decompose Ψ(𝑣)=Ψ₁(𝑣)+Ψ₂(𝑣), where Ψ₁(𝑣)=Δ(Φ_𝑝(Δ𝑢)) and Ψ₂(𝑣)=Ψ(𝑣)−Ψ₁(𝑣), where Ψ is as in (3.7).
We first prove the hemicontinuity, that is, for every 𝑣₁,𝑣₂,𝑣₃∈𝑉, the function 𝜆→(Ψ(𝑣₁+𝜆𝑣₂),𝑣₃) is continuous from ℝ→ℝ. But it suffices to prove the continuity at 𝜆=0. So we assume that |𝜆|<1. For 𝑣₁,𝑣₂,𝑣₃∈𝑉, by integration by parts, we see thatΨ₁𝑣₁+𝜆𝑣₂,𝑣₃=_𝐷||Δ𝑣₁+𝜆𝑣₂+𝑧||^𝑝−2Δ𝑣₁+𝜆𝑣₂+𝑧Δ𝑣₃𝑑𝑥.(3.12) By Hölder's inequality and Young's inequality, it yields that |||||Δ𝑣₁+𝜆𝑣₂+𝑧||^𝑝−2Δ𝑣₁+𝜆𝑣₂+𝑧Δ𝑣₃≤||Δ(𝑣₁|||≤2_𝑝−2+𝜆𝑣₂+𝑧)||^𝑝−1||Δ𝑣₃||||Δ𝑣¹+𝑧||^𝑝−1+||Δ𝑣₂||^𝑝−1||Δ𝑣³≤2_𝑝−2||||Δ𝑣₁+𝑧||^𝑝+||Δ𝑣₂||^𝑝+2||Δ𝑣₃||^𝑝,(3.13) which the right-hand side is in 𝐿¹(𝐷) for 𝑣₁,𝑣₂,𝑣₃∈𝑉. Hence the expression of the right-hand side of inequality (3.13) is the control function for the integrant in (3.12). Then the Lebesgue's dominated convergence theorem can be apply to (3.12) when we take the limit 𝜆→0. This proves the hemicontinuity of Ψ₁(𝑣). As for the hemicontinuity of Ψ₂(𝑣), noting that by our assumption (1.5) and 𝑓∈𝑉^ we have Ψ₂𝑣₁+𝜆𝑣₂,𝑣₃=≤𝐶_𝐷2𝑔𝑥,𝑣₁+𝜆𝑣₂+𝑧𝑣₃𝑑𝑥−_𝐷𝑓(𝑥)𝑣₃𝑑𝑥_𝐷||𝑣₁+𝜆𝑣₂+𝑧||^𝑞−1||𝑣₃||𝑑𝑥+_𝐷Λ₂(𝑥)||𝑣₃||𝑑𝑥−_𝐷𝑓(𝑥)𝑣₃𝑑𝑥.(3.14) It suffices to find the control function for the first integrand above, but we can get this by noting that 𝑞≤𝑝 and using approach similar to (3.13).
Second, we prove the monotonicity of Ψ(𝑣). We first prove the monotonicity for Ψ₁. For 𝑣₁,𝑣₂∈𝑉, since 𝑣₁=𝑢₁−𝑧, 𝑣₂=𝑢₂−𝑧, we haveΨ₁=𝑣₁||Δ−𝑢₁Ψ₁𝑣₂,𝑣₁−𝑣₂||^𝑝−2Δ𝑢₁−||Δ𝑢₂||^𝑝−2Δ𝑢₂,Δ𝑢₁−Δ𝑢₂=_𝐷||Δ𝑢¹||^𝑝+||Δ𝑢₂||^𝑝−||Δ𝑢₁||^𝑝−2Δ𝑢₁Δ𝑢₂−||Δ𝑢₂||^𝑝−2Δ𝑢₂Δ𝑢₁≥𝑑𝑥_𝐷||Δ𝑢¹=||^𝑝+||Δ𝑢₂||^𝑝−||Δ𝑢₁||^𝑝−1||Δ𝑢₂||−||Δ𝑢₂||^𝑝−1||Δ𝑢₁||𝑑𝑥_𝐷||Δ𝑢¹||^𝑝−1−||Δ𝑢₂||^𝑝−1||Δ𝑢¹||−||Δ𝑢₂||𝑑𝑥≥0.(3.15) Since 𝑝≥2, the function 𝑠^𝑝−1 is increasing for 𝑠≥0, which shows that the last inequality in the above proof is correct. On the other hand, by our assumption (1.6), we have (Ψ₂(𝑣₁)−Ψ₂(𝑣₂),𝑣₁−𝑣₂)≥𝐶₃‖𝑣₁−𝑣₂‖²², and therefore it follows that for 𝑣₁,𝑣₂∈𝑉Ψ𝑣₁−Ψ𝑣₂,𝑣₁−𝑣₂≥𝐶₃‖‖𝑣₁−𝑣₂‖‖²,(3.16) where 𝐶₃ is as in (1.6). Hence, we have showed the monotonicity of Ψ(𝑣).
As for the coercivity, for 𝑣∈𝑉, by our assumptions (1.4) and (1.5), using Hölder' inequality, we have(Ψ(𝑣),𝑣)==‖_𝐷ΔΔ𝑢‖_𝑝𝑝||Δ𝑢||^𝑝−2Δ𝑢𝑣𝑑𝑥+_𝐷𝑔(𝑥,𝑣+𝑧)𝑣𝑑𝑥−_𝐷𝑓(𝑥)𝑣𝑑𝑥≥‖Δ𝑢‖_𝑝𝑝−_𝐷||Δ𝑢||^𝑝−2Δ𝑢Δ𝑧𝑑𝑥+_𝐷𝑔(𝑥,𝑢)𝑢𝑑𝑥−_𝐷𝑔(𝑥,𝑢)𝑧𝑑𝑥−_𝐷𝑓(𝑥)𝑣𝑑𝑥−‖Δ𝑢‖^𝑝𝑝−1‖Δ𝑧‖_𝑝+𝐶₁‖𝑢‖^𝑞𝑞−‖‖Λ₁‖‖₁−𝐶₂‖𝑢‖^𝑞𝑞−1‖𝑧‖_𝑞−‖‖Λ₂‖‖_𝑞′‖𝑧‖_𝑞−‖𝑓‖_𝑉^′‖𝑣‖_𝑉,(3.17) where 𝐶₁ and 𝐶₂ are defined in (1.4) and (1.5). By employing the 𝜀-Young's inequality, that is, 𝑎𝑏≤𝜀(𝑎^𝑟/𝑟)+𝜀^{−𝑟′/𝑟}(𝑏^𝑟′/𝑟^) for 𝑟>1 and 1/𝑟+1/𝑟^=1, we find that ‖Δ𝑢‖^𝑝𝑝−1‖Δ𝑧‖_𝑝≤14𝑝−𝑝1‖Δ𝑢‖^𝑝𝑝+2^2𝑝−2𝑝‖Δ𝑧‖^𝑝𝑝≤14‖Δ𝑢‖^𝑝𝑝+2^2𝑝−2‖Δ𝑧‖^𝑝𝑝.(3.18) Similarly, we have 𝐶₂‖𝑢‖^𝑞𝑞−1‖𝑧‖_𝑞≤𝐶¹2‖𝑢‖^𝑞𝑞+2^𝑞−1𝐶^11−𝑞𝐶^𝑞2‖𝑧‖^𝑞𝑞,(3.19)‖𝑓‖_𝑉^′‖𝑣‖_𝑉≤14‖Δ𝑢‖^𝑝𝑝+2^2/(𝑝−1)‖𝑓‖^{𝑉𝑝/(𝑝−1)}_′+‖𝑓‖_𝑉^′‖Δ𝑧‖_𝑝,(3.20) then, by (3.17)–(3.20), we obtain that (Ψ(𝑣),𝑣)≥12‖Δ𝑢‖^𝑝𝑝+𝐶¹2‖𝑢‖^𝑞𝑞−𝑝₁(𝑡,𝑤),(3.21) with 𝑝₁(𝑡,𝑤)=2^2𝑝−2‖Δ𝑧‖^𝑝𝑝+2^𝑞−1𝐶^11−𝑞𝐶^𝑞2‖𝑧‖^𝑞𝑞+‖‖Λ₂‖‖_𝑞′‖𝑧‖_𝑞+2^2/(𝑝−1)‖𝑓‖^{𝑉𝑝/(𝑝−1)}_′+‖𝑓‖_𝑉^′‖Δ𝑧‖_𝑝≥0,(3.22) where 𝑞^ is the dual number of 𝑞. At the same time, (3.21) is one form of coercivity which will be used in Section 4, but in order to prove the existence and uniqueness of solution to (3.4), we will give another form.
Noting that by Hölder's inequality it follows with 𝑢(𝑡)=𝑣(𝑡)+𝑧(𝑡) that‖𝑢‖^𝑞𝑞=‖𝑣(𝑡)+𝑧(𝑡)‖^𝑞𝑞≥2^1−𝑞‖𝑣‖^𝑞𝑞−‖𝑧‖^𝑞𝑞,(3.23) then by the inverse 𝜀-Young's inequality, that is, 𝑎𝑏≥𝜀(𝑎^𝑟/𝑟)+𝜀^{−𝑟′/𝑟}(𝑏^𝑟′/𝑟^) when 𝑟<1 and 1/𝑟+1/𝑟^=1, we get from (3.23) that ‖𝑢‖^𝑞𝑞≥2^1−𝑞‖𝑣‖^𝑞𝑞−‖𝑧‖^𝑞𝑞≥2^1−𝑞𝜂^𝑞0‖𝑣‖^𝑞2−‖𝑧‖^𝑞𝑞≥𝑞2‖𝑣‖²²−‖𝑧‖^𝑞𝑞−𝐶^,(3.24) where 𝐶^=((𝑞−2)/2)2^{(2−2𝑞)/(2−𝑞)}𝜂^{02𝑞/(2−𝑞)} and 𝜂₀ is the Sobolev embedding coefficient of 𝐿^𝑞(𝐷)↪𝐿²(𝐷). Hence, it follows from (3.21) that (Ψ(𝑣),𝑣)≥12‖Δ𝑢‖^𝑝𝑝+𝑞2‖𝑣‖²²−𝑝₂(𝑡,𝑤),(3.25) with 𝑝₂(𝑡,𝑤)=𝑝₁(𝑡,𝑤)+‖𝑧‖^𝑞𝑞+𝐶^≥0, where 𝑝₁(𝑡,𝑤) is defined as in (3.22). Note that if 𝑞=2, we omit this procedure and directly (3.21) passes to (3.25). Hence we have proved the coercivity for Ψ.
We finally prove the the boundedness for Ψ(𝑣) for fixed 𝑣∈𝑉, that is, for fixed 𝑣∈𝑉, Ψ(𝑣) is a linear bounded functional on 𝑊^02,𝑝(𝐷). Indeed, for 𝑣,ℎ∈𝑉, by applying Hölder's inequality and repeatedly using Sobolev's embedding inequality, we have(Ψ(𝑣),ℎ)≤≤‖Δ_𝐷𝑢||‖Δ𝑢||^𝑝−1_𝑝𝑝−1||Δℎ||𝑑𝑥+_𝐷||𝑔(𝑥,𝑢)||||ℎ||𝑑𝑥+_𝐷||𝑓(𝑥)||||ℎ||𝑑𝑥‖Δℎ‖_𝑝+𝐶₂‖𝑢‖^𝑞𝑞−1‖ℎ‖_𝑞+‖‖Λ₂‖‖_𝑞′‖ℎ‖_𝑞+‖𝑓‖_𝑉^′‖ℎ‖_𝑉≤‖Δ𝑢‖^𝑝𝑝−1+𝑐₁‖Δ𝑢‖^𝑝𝑞−1+𝑐₂‖‖Λ₂‖‖_𝑞′+‖𝑓‖_𝑉^′‖Δℎ‖^𝑝≤2‖Δ𝑢‖^𝑝𝑝−1+𝑐₃+𝑐₂‖‖Λ₂‖‖_𝑞′+‖𝑓‖_𝑉^′‖Δℎ‖^𝑝≤2^𝑝−1‖Δ𝑣‖^𝑝𝑝−1+𝑝₃(𝑡,𝑤)‖Δℎ‖_𝑝(3.26) with the random variable 𝑝₃(𝑡,𝑤)=2^𝑝−1‖Δ𝑧‖^𝑝𝑝−1+𝑐₂‖Λ₂‖_𝑞′+𝑐₂‖𝑓‖_𝑉^′+𝑐₃≥0 and the positive constants 𝑐_𝑖(𝑖=1,2,3) independent of 𝑣,ℎ. Therefore, from (3.26) we finally find that ‖Ψ(𝑣)‖_𝑉^′≤2^𝑝−1‖Δ𝑣‖^𝑝𝑝−1+𝑝₃(𝑡,𝑤),(3.27) is a bounded linear operator on 𝑊^02,𝑝(𝐷) for fixed 𝑣∈𝑉. From the proof we know that the assumption 𝑝≥𝑞≥2 is necessary. This completes the proof of Theorem 3.1.

We now define𝑆(𝑡,𝑠;𝑤)𝑢₀=𝑣𝑡,𝑤;𝑠,𝑢₀−𝑧(𝑠,𝑤)+𝑧(𝑡,𝑤),𝑡≥𝑠∈ℝ,(3.28) with 𝑢₀=𝑢(𝑠). Then 𝑆(𝑡,𝑠;𝑤)𝑢₀ is the solution to (1.1) in certain meaning for every 𝑢₀∈𝐻 and 𝑡≥𝑠∈ℝ. By the uniqueness part of solution in Theorem 3.1, we immediately get that 𝑆(𝑡,𝑠,𝑤) is a stochastic flow, that is, for every 𝑢₀∈𝐻 and 𝑡≥𝑟≥𝑠∈ℝ𝑆(𝑡,𝑠;𝑤)𝑢₀=𝑆(𝑡,𝑟;𝑤)𝑆(𝑟,𝑠;𝑤)𝑢₀,(3.29)𝑆(𝑡,𝑠;𝑤)𝑢₀=𝑆𝑡−𝑠,0;𝜃_𝑠𝑤𝑢₀.(3.30) Hence if we define𝜑(𝑡,𝑤)𝑢₀=𝑆(𝑡,0;𝑤)𝑢₀=𝑣𝑡,𝑤;0,𝑢₀−𝑧(0,𝑤)+𝑧(𝑡,𝑤)(3.31) with 𝑢₀=𝑢(0), then by Theorem 3.1  𝜑 is a continuous stochastic dynamical system associated with quasilinear partial differential equation (1.1), with the following fact𝜑𝑡,𝜃_−𝑡𝑤𝑢₀=𝑢0,𝑤;−𝑡,𝑢₀,∀𝑡≥0,(3.32) that is to say, 𝜑(𝑡,𝜃_−𝑡𝑤)𝑢₀ can be interpreted as the position of the trajectory at time 0, which was in 𝑢₀ at time −𝑡 (see [5]).

4. Existence of Compact Random Attractor for RDS

In this section, we will compute some estimates in space 𝐻=𝐿²(𝐷) and 𝑉₀=𝐻¹⁰(𝐷). Note that 𝑝_𝑖(𝑡,𝑤)(𝑖=1,2,3) appearing in the proofs are given in Section 3. In the following computation, 𝑤∈Ω; the results will hold for P-a.s. 𝑤∈Ω.

Lemma 4.1. Suppose that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in 𝑉^. Then there exist random radii 𝑟₁(𝑤),𝑟₂(𝑤)>0, such that for all 𝜌>0 there exists 𝑠=𝑠(𝑤,𝜌)≤−1 such that for all 𝑠≤𝑠(𝑤,𝜌) and all 𝑢₀∈𝐻 with ‖𝑢₀‖₂≤𝜌, the following inequalities hold for P-a.s. 𝑤∈Ω‖‖𝑣(𝑡,𝑤;𝑠,𝑢₀−𝑧(𝑠))‖‖²²≤𝑟²¹(𝑤),∀𝑡∈[−1,0₀₋₁],‖‖Δ𝑢𝜏,𝑤;𝑠,𝑢⁰‖‖^𝑝𝑝+‖‖𝑢𝜏,𝑤;𝑠,𝑢₀‖‖^𝑞𝑞𝑑𝜏≤𝑟²²(𝑤),(4.1) where 𝑣(𝑡,𝑤;𝑠,𝑢₀−𝑧(𝑠)) is the solution to (3.4) with 𝑣(𝑡,𝑤;𝑠,𝑢₀−𝑧(𝑠))=𝑢(𝑡,𝑤;𝑠,𝑢₀)−𝑧(𝑡,𝑤) and 𝑢₀=𝑢(𝑠).

Proof. For simplicity, we abbreviate 𝑣(𝑡)∶=𝑣(𝑡,𝑤;𝑠,𝑢₀−𝑧(𝑠)) and 𝑢(𝑡)∶=𝑢(𝑡,𝑤;𝑠,𝑢₀) for fixed 𝑢₀,𝑤∈Ω and 𝑡≥𝑠 with 𝑢₀=𝑢(𝑠). Multiplying both sides of (3.9) by 𝑣(𝑡) and then integrating over 𝐷, we obtain that 12𝑑𝑑𝑡‖𝑣‖²²+(Ψ(𝑣),𝑣)=0.(4.2) Then, by (3.25), we have 𝑑𝑑𝑡‖𝑣‖²²+‖Δ𝑢‖^𝑝𝑝+𝑞‖𝑣‖²²≤2𝑝₂(𝑡,𝑤).(4.3) Applying the Gronwall's lemma to (4.3) from 𝑠 to 𝑡, 𝑡∈[−1,0], it yields that ‖𝑣(𝑡)‖²²≤≤𝑒2𝑒^{−𝑞(𝑡−𝑠)}_𝑞‖𝑣(𝑠)‖²²+2^𝑡𝑠𝑝₂(𝜏,𝑤)𝑒^{−𝑞(𝑡−𝜏)}𝑑𝜏𝑒^𝑞𝑠‖‖𝑢₀‖‖²²+𝑒^𝑞𝑠‖𝑧(𝑠)‖²²+^0−∞𝑝₂(𝜏,𝑤)𝑒^𝑞𝜏𝑑𝜏,(4.4) where 𝑝₂(𝜏,𝑤) grows at most polynomially as 𝜏→−∞ (see [5]). Since 𝑝₂(𝜏,𝑤) is multiplied by a function which decays exponentially, the integral in (4.4) is convergent.
Given every fixed 𝜌>0, we can choose 𝑠(𝑤,𝜌), depending only on 𝑤 and 𝜌, such that 𝑒^𝑞𝑠‖𝑢₀‖²²≤1. Similarly, ‖𝑧(𝑠)‖²² grows at most polynomially as 𝑠→−∞, and ‖𝑧(𝑠)‖²² is multiplied by a function which decays exponentially. Then we have sup_𝑠≤0𝑒^𝑞𝑠‖𝑧(𝑠)‖²²<+∞.(4.5) Hence by (4.4) we can give the final estimate for ‖𝑣(𝑡)‖²²‖𝑣(𝑡)‖²²≤𝑟²¹(𝑤)∶=2𝑒^𝑞1+sup_𝑠≤0𝑒^𝑞𝑠‖𝑧(𝑠)‖²²+^0−∞𝑝₂(𝜏,𝑤)𝑒^𝑞𝜏𝑑𝜏,(4.6) for 𝑡∈[−1,0]. Following (4.2), by using (3.21), we find that 𝑑𝑑𝑡‖𝑣‖²²+‖Δ𝑢‖^𝑝𝑝+𝐶₁‖𝑢‖^𝑞𝑞≤2𝑝₁(𝑡,𝑤),(4.7) where 𝑝₁(𝑡,𝑤) is the same as in (3.22). Integrating (4.7) for 𝑡 on [−1,0], we get that ⁰⁻¹‖Δ𝑢(𝜏)‖^𝑝𝑝+𝐶₁‖𝑢(𝜏)‖^𝑞𝑞𝑑𝜏≤2⁰⁻¹𝑝₁(𝜏,𝑤)𝑑𝜏+‖𝑣(−1)‖²²,(4.8) which gives an expression for 𝑟²²(𝑤).

In the following, we give the estimate of ‖∇𝑢(𝑡)‖₂. This is the most difficult part in our discussion. Because the nonlinearity of Ψ₁ and Ψ₂ in (3.4) or (3.9), it seems impossible to derive the 𝑉-norm estimate by the way as [1, page 169]. So we relax to bound the solution in a weaker Sobolev 𝑉₀=𝐻¹⁰(𝐷) with equivalent norms denoted by ‖∇𝑢‖₂ for 𝑢∈𝑉₀. Here, just as our statement in the introduction, we make the inner product over the resolvent 𝑅(𝜆,𝐴) which is defined in Section 2, then by using the Dirichlet forms of 𝐴 we obtain technically the estimate of ‖∇𝑢(𝑡)‖₂.

Lemma 4.2. Suppose that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in 𝑉^. Then there exists a random radius 𝑟₃(𝑤)>0, such that for all 𝜌>0 there exists 𝑠=𝑠(𝑤,𝜌)≤−1 such that for all 𝑠≤𝑠(𝑤,𝜌) and all 𝑢₀∈𝐻 with ‖𝑢₀‖₂≤𝜌, the following inequality holds for P-a.s. 𝑤∈Ω‖‖∇𝑢(𝑡,𝑤;𝑠,𝑢₀)‖‖²²≤𝑟²³(𝑤),∀𝑡∈[−1,0],(4.9) where 𝑢(𝑡,𝑤;𝑠,𝑢₀) is the solution to (1.1) with 𝑢₀=𝑢(𝑠). In particular, ‖‖∇𝑢(0,𝑤;𝑠,𝑢₀)‖‖²²≤𝑟²³(𝑤).(4.10)

Proof. Taking the inner product of (3.9) with −𝜆𝐴𝑅(𝜆,𝐴)𝑣 where 𝜆>0, 𝑣∈𝑉, we get −_𝐷𝑣_𝑡𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥=_𝐷Ψ₁(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥+_𝐷Ψ₂(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥.(4.11) By the semigroup theory (see [16]) we have 𝐴𝑅(𝜆,𝐴)𝑣=𝑅(𝜆,𝐴)𝐴𝑣=𝜆𝑅(𝜆,𝐴)𝑣−𝑣,(4.12) for 𝑣∈𝐷(𝐴). We now estimate every terms on the right-hand side of (4.11). The first term on the right-hand side of (4.11) is rewritten as _𝐷Ψ₁(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥=_𝐷Ψ₁(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑢𝑑𝑥−_𝐷Ψ₁(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑧𝑑𝑥.(4.13) Employing (4.12) and by integration by parts, it yields that _𝐷Ψ₁(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑢𝑑𝑥==𝜆−𝜆_𝐷_𝐷Ψ₁(𝑢)(𝜆𝑅(𝜆,𝐴)𝑢−𝑢)𝑑𝑥=−𝜆‖Δ𝑢Δ‖_𝑝𝑝||Δ𝑢||^𝑝−2Δ𝑢𝑢𝑑𝑥+𝜆_𝐷Δ||Δ𝑢||^𝑝−2Δ𝑢𝜆𝑅(𝜆,𝐴)𝑢𝑑𝑥≤−𝜆‖Δ𝑢‖_𝑝𝑝+𝜆_𝐷||Δ𝑢||^𝑝−2Δ𝑢𝜆Δ𝑅(𝜆,𝐴)𝑢𝑑𝑥≤−𝜆‖Δ𝑢‖_𝑝𝑝+𝜆_𝐷||Δ𝑢||^𝑝−1||𝜆𝑅(𝜆,𝐴)Δ𝑢||𝑑𝑥+𝜆‖Δ𝑢‖^𝑝𝑝−1‖𝜆𝑅(𝜆,𝐴)Δ𝑢‖_𝑝≤−𝜆‖Δ𝑢‖^𝑝𝑝+𝜆‖Δ𝑢‖^𝑝𝑝=0,(4.14) where we use the contraction property of 𝜆𝑅(𝜆,𝐴) on 𝐿^𝑝(𝐷), that is, ‖𝜆𝑅(𝜆,𝐴)Δ𝑢‖_𝑝≤‖Δ𝑢‖_𝑝 for Δ𝑢∈𝐿^𝑝(𝐷) and every 𝜆>0. We now bound the second term on the right-hand side of (4.13) −_𝐷Ψ₁(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑧𝑑𝑥≤‖‖Ψ₁(𝑢)‖‖_𝑉′‖𝜆𝐴𝑅(𝜆,𝐴)𝑧‖_𝑉=‖‖Ψ₁(𝑢)‖‖_𝑉′‖𝜆𝑅(𝜆,𝐴)𝐴𝑧‖_𝑉,(4.15) where we use our assumption 𝜙_𝑗(1≤𝑗≤𝑚)∈𝑊^04,𝑝(𝐷). Since Ψ₁ maps 𝑉 into 𝑉^, then for fixed 𝑢∈𝑉 and every ℎ∈𝑉, we have Ψ₁(𝑢),ℎ≤=_𝐷Δ||Δ𝑢||^𝑝−2Δ𝑢ℎ𝑑𝑥=_𝐷||Δ𝑢||^𝑝−2Δ𝑢Δℎ𝑑𝑥_𝐷||Δ𝑢||^𝑝−1||Δℎ||𝑑𝑥≤‖Δ𝑢‖^𝑝𝑝−1‖Δℎ‖_𝑝.(4.16) So for fixed 𝑢∈𝑉, ‖Ψ₁(𝑢)‖_𝑉^′≤‖Δ𝑢‖^𝑝𝑝−1, and therefore by (4.15) we obtain that −_𝐷Ψ₁(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑧𝑑𝑥≤‖Δ𝑢‖^𝑝𝑝−1‖𝜆𝑅(𝜆,𝐴)𝐴𝑧‖_𝑉≤𝐶‖Δ𝑢‖^𝑝𝑝−1‖𝐴𝑧‖_𝑉≤‖Δ𝑢‖^𝑝𝑝+𝐶^𝑝‖𝐴𝑧‖^𝑝𝑉,(4.17) where ‖𝜆𝑅(𝜆,𝐴)𝐴𝑧‖_𝑉≤𝐶‖𝐴𝑧‖_𝑉 and 𝐶 is a constant independent of 𝜆, 𝑣(𝑡) and 𝑢(𝑡). Here we should note that 𝜆𝑅(𝜆,𝐴) is a bounded linear operator on 𝑉. Hence, by (4.14)–(4.17) the first term on the right-hand side of (4.11) is finally bounded by _𝐷Ψ₁(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥≤‖Δ𝑢‖^𝑝𝑝+𝐶^𝑝‖𝐴𝑧‖^𝑝𝑉.(4.18) By our assumption (1.5), the second term on the right-hand side of (4.11) is estimated as _𝐷Ψ=₂(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥_𝐷≤(𝑔(𝑥,𝑢)−𝑓(𝑥))𝜆𝑅(𝜆,𝐴)𝐴𝑣𝑑𝑥_𝐷≤||𝑔(𝑥,𝑢)||||𝜆𝑅(𝜆,𝐴)𝐴𝑣||𝑑𝑥+_𝐷||𝑓(𝑥)||||𝜆𝐴𝑅(𝜆,𝐴)𝑣||𝑑𝑥_𝐷≤𝐶₂𝐶₂|𝑢|^𝑞−1+Λ₂(𝑥)||𝜆𝑅(𝜆,𝐴)𝐴𝑣||𝑑𝑥+_𝐷||𝑓(𝑥)||||𝜆𝑅(𝜆,𝐴)𝐴𝑣||𝑑𝑥‖𝑢‖^𝑞𝑞−1‖𝜆𝑅(𝜆,𝐴)𝐴𝑣‖_𝑞+‖‖Λ₂‖‖_𝑞′‖𝜆𝑅(𝜆,𝐴)𝐴𝑣‖_𝑞+‖𝑓‖₂‖𝜆𝑅(𝜆,𝐴)𝐴𝑣‖₂≤𝐶₂‖𝑢‖^𝑞𝑞−1‖𝐴𝑣‖_𝑞+‖‖Λ₂‖‖_𝑞′‖𝐴𝑣‖_𝑞+‖𝑓‖₂‖𝐴𝑣‖₂≤‖𝑢‖^𝑞𝑞+𝐶^𝑞2+1‖𝐴𝑣‖^𝑞𝑞+‖‖Λ₂‖‖^𝑞′_𝑞′+‖𝑓‖²²+‖𝐴𝑣‖²²,(4.19) where we employ Young's inequality 𝑎𝑏≤𝑎^𝑟+𝑏^{𝑟/(𝑟−1)} for 𝑟>1. But, by Sobolev's inequality and Young's inequality, it yields that ‖𝐴𝑣‖^𝑞𝑞≤𝜂^𝑞1‖𝐴𝑣‖^𝑞𝑝≤𝜂^𝑝1‖𝐴𝑣‖^𝑝𝑝+1≤2^𝑝−1𝜂^𝑝1‖𝐴𝑢‖^𝑝𝑝+2^𝑝−1𝜂^𝑝1‖𝐴𝑧‖^𝑝𝑝+1;(4.20) similarly ‖𝐴𝑣‖²²≤2^𝑝−1𝜂^𝑝2‖𝐴𝑢‖^𝑝𝑝+2^𝑝−1𝜂^𝑝2‖𝐴𝑧‖^𝑝𝑝+1,(4.21) where the positive constants 𝜂₁,𝜂₂ are Sobolev's embedding constants independent of 𝜆. Then by (4.19)–(4.21), there exist positive constants 𝑐₁,𝑐₂ such that _𝐷Ψ₂(𝑢)𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥≤‖𝑢‖^𝑞𝑞+𝑐₁‖𝐴𝑢‖^𝑝𝑝+𝑐₂‖𝐴𝑧‖^𝑝𝑝+‖‖Λ₂‖‖^𝑞′_𝑞′+‖𝑓‖²²+2,(4.22) where 𝑞^=𝑞/(𝑞−1). By (4.18) and (4.22), we find that (4.11) becomes −_𝐷𝑣_𝑡𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥≤𝑐₃‖Δ𝑢‖^𝑝𝑝+‖𝑢‖^𝑞𝑞+𝑝₄(𝑡,𝑤),(4.23) where 𝑝₄(𝑡,𝑤)=𝐶^𝑝‖𝐴𝑧‖^𝑝𝑉+𝑐₂‖𝐴𝑧‖^𝑝𝑝+‖Λ₂‖^𝑞′_𝑞′+‖𝑓‖²²+2≥0 and 𝑐₃=𝑐₁+1. On the other hand, by (4.12) and the Dirichlet forms (2.14), we have −_𝐷𝑣_𝑡𝜆𝐴𝑅(𝜆,𝐴)𝑣𝑑𝑥=𝜀^(𝜆)𝑣,𝑣_𝑡.(4.24) Hence by (4.24), (4.23) is rewritten as 𝜀^(𝜆)𝑣,𝑣_𝑡≤𝑐₃‖Δ𝑢‖^𝑝𝑝+‖𝑢‖^𝑞𝑞+𝑝₄(𝑡,𝑤).(4.25) Note that the right-hand side of (4.25) is independent of 𝜆. So taking limit on both side of (4.25) for 𝜆→∞, association with (2.15), we deduce that 12𝑑𝑑𝑡‖∇𝑣‖²²≤𝑐₃‖Δ𝑢‖^𝑝𝑝+‖𝑢‖^𝑞𝑞+𝑝₄(𝑡,𝑤).(4.26) Integrating (4.26) from 𝑠 to 𝑡 (−1≤𝑠≤𝑡≤0), it yields that ‖∇𝑣(𝑡)‖²²≤2𝑐₃^𝑡𝑠‖Δ𝑢(𝜏)‖^𝑝𝑝𝑑𝜏+2^𝑡𝑠‖𝑢(𝜏)‖^𝑞𝑞𝑑𝜏+2^𝑡𝑠𝑝₄(𝜏,𝑤)𝑑𝜏+‖∇𝑣(𝑠)‖²²≤2𝑐₃⁰⁻¹‖Δ𝑢(𝜏)‖^𝑝𝑝𝑑𝜏+2⁰⁻¹‖𝑢(𝜏)‖^𝑞𝑞𝑑𝜏+2⁰⁻¹𝑝₄(𝜏,𝑤)𝑑𝜏+‖∇𝑣(𝑠)‖²².(4.27) Therefore, by Lemma 4.1, we find that ‖∇𝑣(𝑡)‖²²≤2𝑐₃+1𝑟²²(𝑤)+2⁰⁻¹𝑝₄(𝜏,𝑤)𝑑𝜏+‖∇𝑣(𝑠)‖²².(4.28) Integrating (4.28) for 𝑠 from −1 to 0, we have ‖∇𝑣(𝑡)‖²²≤2𝑐₃+1𝑟²²(𝑤)+2⁰⁻¹𝑝₄(𝜏,𝑤)𝑑𝜏+⁰⁻¹‖∇𝑣(𝑠)‖²²𝑑𝑠,(4.29) for all 𝑡∈[−1,0]. By Poincare's inequality, and Young's inequality, there exist positive constants 𝑐₄, 𝑐₅, 𝑐₆ such that ‖∇𝑣(𝑠)‖²²≤𝑐₄‖Δ𝑣(𝑠)‖²²≤2𝑐₄‖Δ𝑢(𝑠)‖²²+2𝑐₄‖Δ𝑧(𝑠)‖²²≤2𝑐₅‖Δ𝑢(𝑠)‖^𝑝𝑝+2𝑐₅‖Δ𝑧(𝑠)‖²²+𝑐₆.(4.30) Hence, by (4.30) and using Lemma 4.1 again, (4.29) follows ‖∇𝑣(𝑡)‖²²≤2𝑐₃+𝑐₅+1𝑟²²(𝑤)+2⁰⁻¹𝑝₄(𝜏,𝑤)𝑑𝜏+2𝑐₅⁰⁻¹‖Δ𝑧(𝜏)‖²²𝑑𝜏+𝑐₆,(4.31) with 𝑡∈[−1,0]. See that 𝑣(𝑡)=𝑢(𝑡)−𝑧(𝑡). Then, we have ‖∇𝑢(𝑡)‖²²≤2‖∇𝑣(𝑡)‖²²+2‖∇𝑧(𝑡)‖²²≤+22𝑐𝑐₃₅+𝑐₅+1𝑟²²(𝑤)+2⁰⁻¹𝑝₄(𝜏,𝑤)𝑑𝜏⁰⁻¹‖Δ𝑧(𝜏)‖²²𝑑𝜏+2sup_{−1≤𝑡≤0}‖∇𝑧(𝑡)‖²²+𝑐₆,(4.32) which gives an expression for 𝑟²³(𝑤). This completes the proof.

By Theorem 2.2 and Lemma 4.2, we have obtained our main result in this section.

Theorem 4.3. Assume that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in 𝑉^. Then the RDS 𝜑(𝑡,𝜔) generated by the stochastic equation (1.1) possesses a random attractor 𝒜(𝑤) defined as 𝒜(𝑤)=_𝐵∈_ℬ(𝐻)_𝑠≥0_𝑡≥𝑠𝜑(𝑡,𝜃_−𝑡𝑤)𝐵,(4.33) where ℬ(𝐻) denotes all the bounded subsets of 𝐻 and the closure is the 𝐻-norm.

Remark 4.4. As stated in Theorem 3.1, under the assumptions (1.4)–(1.6), the solutions of (1.1) are in 𝑊^02,𝑝(𝐷). So it is possible in theory to obtain a compact random attractor in 𝑊^01,𝑝(𝐷) or 𝑊^02,𝑝(𝐷). But it seems most difficulty to derive the estimate of solution in 𝑊^2,𝑝(𝐷) due to the nonlinear principle part Δ(Φ_𝑝(Δ𝑢)).

5. The Single Point Attractor

In this section, we consider the attracting by a single point. In order to derive our anticipating result, we assume that 𝐶₃>0 in (1.6). This leads to the following fact that for every fixed 𝑡∈ℝ and 𝑤∈Ω, the solution 𝑢(𝑡,𝑤;𝑠,𝑢(𝑠)) to (1.1) is a Cauchy sequence in 𝐻 for the initial time 𝑠 with initial value 𝑢(𝑠) belonging to the bounded subset of 𝐻. Then we obtain a compact attractor consisting of a single point which is the limit of 𝑢(0,𝑤;𝑠,𝑢(𝑠)) as 𝑠→−∞.

Lemma 5.1. Assume that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in 𝑉^, 𝐶₃>0. Then for 𝑠₁≤𝑠₂≤𝑡 and 𝑢₀₁,𝑢₀₂∈𝐻 with 𝑢(𝑠₁)=𝑢₀₁ and 𝑢(𝑠₂)=𝑢₀₂, there exists a positive constant 𝑘<𝐶₃ such that ‖‖𝑢(𝑡,𝑤;𝑠₁,𝑢₀₁)−𝑢(𝑡,𝑤;𝑠₂,𝑢₀₂)‖‖²²≤2𝑒^−𝐶3𝑡4𝑒^{(𝐶3−𝑘)𝑠2}‖‖𝑢⁰¹‖‖²²+‖‖𝑧𝑠₁‖‖²²+^0−∞𝑝₂(𝜏,𝑤)𝑒^𝑘𝜏𝑑𝜏+2𝑒^𝑘𝑠2‖‖𝑧𝑠₂‖‖²²+𝑒^𝑘𝑠2‖‖𝑢₀₂‖‖²².(5.1) In particular, for each fixed 𝑡∈ℝ and 𝑤∈Ω there exists a single point 𝜉_𝑡(𝑤) in 𝐻 such that lim_{𝑠→−∞}𝑆(𝑡,𝑠;𝑤)𝑢₀=𝜉_𝑡(𝑤),(5.2) where 𝑢₀=𝑢(𝑠) and 𝑆(𝑡,𝑠;𝑤) is the stochastic flow defined as in (3.28) which is a version of solution to (1.1). Furthermore, the limit in the above is independent of 𝑢₀ for all 𝑢₀ belonging to a bounded subset of 𝐻.

Proof. For 𝑠₁≤𝑠₂≤𝑡 and 𝑢₀₁,𝑢₀₂∈𝐻 with 𝑢(𝑠₁)=𝑢₀₁ and 𝑢(𝑠₂)=𝑢₀₂, we can deduce from (3.9) and (3.8) that 𝑑𝑑𝑡𝑢𝑡,𝑤;𝑠¹,𝑢₀₁−𝑢𝑡,𝑤;𝑠₂,𝑢₀₂+Ψ𝑢𝑡,𝑤;𝑠₁,𝑢₀₁−Ψ𝑢𝑡,𝑤;𝑠₂,𝑢₀₂=0,(5.3) where 𝑢(𝑡)=𝑣(𝑡)+𝑧(𝑡) is the solution to problem (1.1). On the other hand, by (3.8) and (3.16), we immediately deduce that Ψ𝑢₁−Ψ𝑢₂,𝑢₁−𝑢₂≥𝐶³‖‖𝑢₁−𝑢₂‖‖²².(5.4) Then, multiplying (5.3) by 𝑢(𝑡,𝑤;𝑠₁,𝑢₀₁)−𝑢(𝑡,𝑤;𝑠₂,𝑢₀₂), integrating over 𝐷, and using (5.4), we find that 𝑑𝑑𝑡‖‖𝑢(𝑡,𝑤;𝑠¹,𝑢₀₁)−𝑢(𝑡,𝑤;𝑠₂,𝑢₀₂)‖‖²²+𝐶₃‖‖𝑢𝑡,𝑤;𝑠₁,𝑢₀₁−𝑢(𝑡,𝑤;𝑠₂,𝑢₀₂)‖‖²²≤0.(5.5) Now, applying Gronwall's lemma to (5.5) from 𝑠₂ to 𝑡, it yields that ‖‖𝑢(𝑡,𝑤;𝑠₁,𝑢₀₁)−𝑢(𝑡,𝑤;𝑠₂,𝑢₀₂)‖‖²²≤‖‖𝑢(𝑠₂,𝑤;𝑠₁,𝑢₀₁)−𝑢₀₂‖‖²²𝑒^{−𝐶3(𝑡−𝑠2)}≤2‖‖𝑢𝑠₂,𝑤;𝑠₁,𝑢₀₁‖‖²²+‖‖𝑢₀₂‖‖²²𝑒^{−𝐶3(𝑡−𝑠2)}.(5.6) We then estimate ‖𝑢(𝑠₂,𝑤;𝑠₁,𝑢₀₁)‖²². To this end, we rewrite (3.21) as (Ψ(𝑣),𝑣)≥𝐶¹2‖𝑢‖^𝑞𝑞−𝑝₁(𝑡,𝑤).(5.7) Since 𝑞≥2, by Hölder's inequality and inverse Young's inequality we can choose constant 0<𝑘<𝐶₃ such that 𝐶₁2‖𝑢‖^𝑞𝑞≥𝑘2‖𝑣‖²²−𝑐₁‖𝑧‖^𝑞𝑞−𝑐₂.(5.8) Then, by (5.7)-(5.8) it follows from (4.2) that 𝑑𝑑𝑡‖𝑣‖²²+𝑘‖𝑣‖²²≤2𝑝₅(𝑡,𝑤),(5.9) where 𝑝₅(𝑡,𝑤)=𝑝₁(𝑡,𝑤)+𝑐₁‖𝑧‖^𝑞𝑞+𝑐₂ and 𝑣(𝑡)=𝑣(𝑡,𝑤;𝑠₁,𝑢₀₁−𝑧(𝑠₁)). Using Gronwall's lemma to (5.9) from 𝑠₁ to 𝑠₂ with 𝑠₁≤𝑠₂≤0, we get that ‖‖𝑣(𝑠₂,𝑤;𝑠₁,𝑢₀₁−𝑧(𝑠₁))‖‖²²≤≤‖‖2𝑒𝑢₀₁−_−𝑘𝑠2𝑧(𝑠₁)‖‖²²𝑒^{−𝑘(𝑠2−𝑠1)}+^𝑠2_𝑠12𝑝₅(𝜏,𝑤)𝑒^{−𝑘(𝑠2−𝜏)}𝑑𝜏‖‖𝑢⁰¹‖‖²²+‖‖𝑧(𝑠₁)‖‖²²+^0−∞𝑝₅(𝜏,𝑤)𝑒^𝑘𝜏𝑑𝜏.(5.10) Similar to the argument of (4.4), we know that the integral in the above is convergent. Therefore, we have ‖‖𝑢(𝑠₂,𝑤;𝑠₁,𝑢₀₁)‖‖²²≤2‖‖𝑣(𝑠₂,𝑤;𝑠₁,𝑢₀₁)‖‖²²+2‖‖𝑧(𝑠₂)‖‖²²≤4𝑒^−𝑘𝑠2‖‖𝑢⁰¹‖‖²²+‖‖𝑧(𝑠₁)‖‖²²+^0−∞𝑝₅(𝜏,𝑤)𝑒^𝑘𝜏𝑑𝜏+2‖‖𝑧(𝑠₂)‖‖²²,(5.11) from which and (5.6) it follows for every fixed 𝑡∈ℝ that ‖‖𝑢(𝑡,𝑤;𝑠₁,𝑢₀₁)−𝑢(𝑡,𝑤;𝑠₂,𝑢₀₂)‖‖²²≤2𝑒^−𝐶3𝑡4𝑒^{(𝐶3−𝑘)𝑠2}‖‖𝑢⁰¹‖‖²²+‖‖𝑧𝑠₁‖‖²²+^0−∞𝑝₅(𝜏,𝑤)𝑒^𝑘𝜏𝑑𝜏+2𝑒^𝑘𝑠2‖‖𝑧𝑠₂‖‖²²+𝑒^𝑘𝑠2‖‖𝑢₀₂‖‖²²⟶0,as,𝑠₁,𝑠₂⟶−∞,(5.12) where the convergence is uniform with respect to 𝑢₀₁,𝑢₀₂ belonging to every bounded subset of 𝐻. Then (5.12) implies that for fixed 𝑡∈ℝ, 𝑢(𝑡,𝑤;𝑠,𝑢(𝑠)) is a Cauchy sequence in 𝐻 for 𝑠∈ℝ. Thus, by the completeness of 𝐻, for every fixed 𝑡∈ℝ and 𝑤∈Ω, 𝑢(𝑡,𝑤;𝑠,𝑢(𝑠)) has a limit in 𝐻 denoted by 𝜉_𝑡(𝑤), that is, lim_{𝑠→−∞}𝑢(𝑡,𝑤;𝑠,𝑢(𝑠))=𝜉_𝑡(𝑤).(5.13)

Theorem 5.2. Assume that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in 𝑉^, 𝐶₃>0. Then the RDS 𝜑(𝑡,𝑤) generated by the solution to (1.1) possesses a single point attractor 𝒜(𝑤), that is, there exists a single point 𝜉₀(𝑤) in 𝐻 such that 𝒜(𝑤)=𝜉₀(𝑤).(5.14)

Proof. By Lemma 5.1 we define 𝜉₀(𝑤)=lim_{𝑠→−∞}𝑆(0,𝑠;𝑤)𝑢₀,(5.15) where 𝑆(0,𝑠;𝑤)=𝑢(0,𝑤;𝑠,𝑢(𝑠)) by (3.28). Then we need prove that 𝒜(𝑤)={𝜉₀(𝑤)} is a compact attractor. It is obvious that {𝜉₀(𝑤)} is a compact random set. Hence by Definition 2.1 it suffices to prove the invariance and attracting property for {𝜉₀(𝑤)}. Since by the continuity of 𝜑(𝑡,𝑤), and relations (3.29)–(3.32), we have 𝜑(𝑡,𝑤)𝜉₀(𝑤)=𝜑(𝑡,𝑤)lim_{𝑠→−∞}𝑆(0,𝑠;𝑤)𝑢₀=lim_{𝑠→−∞}𝜑(𝑡,𝑤)𝑆(0,𝑠;𝑤)𝑢₀=lim_{𝑠→−∞}𝑆(𝑡,0;𝑤)𝑆(0,𝑠;𝑤)𝑢₀=lim_{𝑠→−∞}𝑆(𝑡,𝑠;𝑤)𝑢₀=lim_{𝑠→−∞}𝑆𝑡−𝑠,0;𝜃_𝑠𝑤𝑢₀=lim_{𝑠→−∞}𝑆0,𝑠−𝑡;𝜃_𝑡𝑤𝑢₀=𝜉𝜃_𝑡𝑤(5.16) then it follows that 𝜑(𝑡,𝑤)𝒜(𝑤)=𝒜(𝜃_𝑡𝑤). On the other hand, by Lemma 5.1, the convergence is uniform with respect to 𝑢₀ belonging to a bounded subset. Then for every bounded subset 𝐵⊂𝐻, by relations (3.32) and (3.28), it follows that dist𝜑𝑡,𝜃_−𝑡𝑤𝐵,𝒜(𝑤)=sup_𝑢0∈𝐵‖‖𝜑(𝑡,𝜃_−𝑡𝑤)𝑢₀−𝜉₀(𝑤)‖‖₂=sup_𝑢0∈𝐵‖‖𝑆(0,−𝑡,𝑤)𝑢₀−𝜉₀(𝑤)‖‖₂=sup_𝑢0∈𝐵‖‖𝑢(0,𝑤;−𝑡,𝑢₀)−𝜉₀(𝑤)‖‖₂⟶0(5.17) as 𝑡→+∞. That is to say 𝒜(𝑤) is a attracting set which attracts every deterministic bounded set of 𝐻, and therefore we complete the proof.

Acknowledgments

The authors would like to express their sincere thanks to the anonymous referee for his/her valuable comments and suggestions to improve the paper. This work was supported by China Natural Science Fund 11071199.