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 π·βŠ‚β„π‘›, π‘›βˆˆβ„•, be a bounded open set with regular boundary πœ•π·. In this paper, we investigate the existence of a solution and a random attractor for the following quasilinear differential equation influenced by additive white noiseΔΦ𝑑𝑒+𝑝(Δ𝑒)+𝑔(π‘₯,𝑒)𝑑𝑑=𝑓(π‘₯)𝑑𝑑+π‘šξ“π‘—=1πœ™π‘—π‘‘π‘Šπ‘—(𝑑),π‘₯∈𝐷,𝑑β‰₯0,(1.1) with the boundary conditionsβˆ‡π‘’(𝑑)=𝟎,𝑒(𝑑)=0,π‘₯βˆˆπœ•π·,𝑑β‰₯0,(1.2) and the initial condition𝑒(0,π‘₯)=𝑒0(π‘₯),π‘₯∈𝐷.(1.3)

In (1.1), Φ𝑝(𝑠)=|𝑠|π‘βˆ’2𝑠, 𝑝β‰₯2, π‘Šπ‘—(𝑑)(1β‰€π‘—β‰€π‘š) are mutually independent two-sided real-valued Wiener processes, πœ™π‘—=πœ™π‘—(π‘₯)(1β‰€π‘—β‰€π‘š,π‘₯∈𝐷) are given real-valued functions that will be assumed to satisfy some conditions. The unknown 𝑒(𝑑) is a real-valued random process, sometimes denoted by 𝑒(𝑑,π‘₯) or 𝑒(𝑑,π‘₯,𝑀). The exterior forced function 𝑔(π‘₯,𝑠) defined in 𝐷×ℝ is subjected to the following growth and monotonicity assumptions:𝑔(π‘₯,𝑠)𝑠β‰₯𝐢1|𝑠|π‘žβˆ’Ξ›1(π‘₯),Ξ›1∈𝐿1(𝐷),𝐢1βˆˆβ„+,(1.4)||||𝑔(π‘₯,𝑠)≀𝐢2|𝑠|π‘žβˆ’1+Ξ›2(π‘₯),Ξ›2βˆˆπΏπ‘ž/(π‘žβˆ’1)(𝐷),𝐢2βˆˆβ„+,(1.5)𝑔π‘₯,𝑠1ξ€Έξ€·βˆ’π‘”π‘₯,𝑠2𝑠1βˆ’π‘ 2ξ€Έβ‰₯𝐢3||𝑠1βˆ’π‘ 2||2,𝐢3βˆˆβ„,(1.6) where 2β‰€π‘žβ‰€π‘<∞.

In deterministic case (without random perturbed term), if 𝑔(π‘₯,𝑒)=π‘˜π‘’, 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(π‘₯,𝑒) satisfying (1.4)–(1.6). The additive white noise βˆ‘π‘šπ‘—=1πœ™π‘—π‘‘π‘Šπ‘—(𝑑) 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 π‘‰βŠ‚π»βŠ‚π‘‰ξ…ž 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 (Ξ©,β„±,P,πœƒπ‘ ) is called a MSD if (Ξ©,β„±,P) is a complete probability space and {πœƒπ‘ βˆΆΞ©β†’Ξ©, π‘ βˆˆβ„} is a family of measure-preserving transformations such that (𝑠,𝑀)β†¦πœƒπ‘ π‘€ is measurable, πœƒ0=id and πœƒπ‘‘+𝑠=πœƒπ‘‘πœƒπ‘  for all 𝑠,π‘‘βˆˆβ„.

A continuous RDS on a complete separable metric space (𝑋,𝑑) with Borel sigma-algebra ℬ(𝑋) over MSD (Ξ©,β„±,P,πœƒπ‘ ) is by definition a measurable map πœ‘βˆΆβ„+Γ—Ξ©Γ—π‘‹βŸΆπ‘‹,(𝑑,𝑀,π‘₯)βŸΌπœ‘(𝑑,𝑀)π‘₯(2.1) such that P-a.s. π‘€βˆˆΞ©(i)πœ‘(0,𝑀)=id on 𝑋,(ii)πœ‘(𝑑+𝑠,𝑀)=πœ‘(𝑑,πœƒπ‘ π‘€)πœ‘(𝑠,𝑀) for all 𝑠,π‘‘βˆˆβ„+ (cocycle property),(iii)πœ‘(𝑑,𝑀)βˆΆπ‘‹β†’π‘‹ is continuous for all π‘‘βˆˆβ„+.

A continuous stochastic flow is by definition a family of measurable mapping 𝑆(𝑑,𝑠;𝑀)βˆΆπ‘‹β†’π‘‹,βˆ’βˆžβ‰€π‘ β‰€π‘‘β‰€βˆž, such that P-a.s. π‘€βˆˆΞ©ξ€·π‘†(𝑑,π‘Ÿ;𝑀)𝑆(π‘Ÿ,𝑠;𝑀)π‘₯=𝑆(𝑑,𝑠;𝑀)π‘₯,π‘₯βˆˆπ‘‹,𝑆(𝑑,𝑠;𝑀)π‘₯=π‘†π‘‘βˆ’π‘ ,0.;πœƒπ‘ π‘€ξ€Έπ‘₯,π‘₯βˆˆπ‘‹,(2.2) for all π‘ β‰€π‘Ÿβ‰€π‘‘, and 𝑠↦𝑆(𝑑,𝑠;𝑀)π‘₯ is continuous in 𝑋 for all 𝑠≀𝑑 and π‘₯βˆˆπ‘‹.

A random compact set {𝐾(𝑀)}π‘€βˆˆΞ© is a family of compact sets indexed by 𝑀 such that for every π‘₯βˆˆπ‘‹ the mapping 𝑀↦𝑑(π‘₯,𝐾(𝑀)) is measurable with respect to β„±.

Let π’œ(𝑀) be a random set. One says that π’œ(𝑀) is attracting if for P-a.s. π‘€βˆˆΞ© and every deterministic bounded subset π΅βŠ‚π‘‹limπ‘‘β†’βˆžξ€·πœ‘ξ€·dist𝑑,πœƒβˆ’π‘‘π‘€ξ€Έξ€Έπ΅,π’œ(𝑀)=0,(2.3) where dist(β‹…,β‹…) is defined by dist(𝐴,𝐡)=supπ‘₯∈𝐴infπ‘¦βˆˆπ΅π‘‘(π‘₯,𝑦).

We say that π’œ(𝑀) absorbs π΅βŠ‚π‘‹ if P-a.s. π‘€βˆˆΞ©, there exists 𝑑𝐡(𝑀)>0 such that for all 𝑑β‰₯𝑑𝐡(𝑀), πœ‘ξ€·π‘‘,πœƒβˆ’π‘‘π‘€ξ€Έπ΅βŠ‚π’œ(𝑀).(2.4)

Definition 2.1. Recall that a random compact set π‘€β†¦π’œ(𝑀) is called to be a random attractor for the RDS πœ‘ if for P-a.s. π‘€βˆˆΞ©(i)π’œ(𝑀) is invariant, that is, πœ‘(𝑑,𝑀)π’œ(𝑀)=π’œ(πœƒπ‘‘π‘€), for all 𝑑β‰₯0;(ii)π’œ(𝑀) is attracting.

Theorem 2.2 (see [5]). Let πœ‘(𝑑,𝑀) be a continuous RDS over a MDS (Ξ©,β„±,P;πœƒπ‘‘) with a separable Banach Space 𝑋. If there exists a compact random absorbing set 𝐾(𝑀) absorbing every deterministic bounded subset of 𝑋, then πœ‘ possesses a random attractor π’œ(𝑀) defined by π’œ(𝑀)=ξšπ΅βˆˆβ„¬(𝑋)𝑠β‰₯0ξšπ‘‘β‰₯π‘ πœ‘ξ€·π‘‘,πœƒβˆ’π‘‘π‘€ξ€Έπ΅,(2.5) where ℬ(𝑋) denotes all the bounded subsets of 𝑋.

Let 𝐿𝑝(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 π‘Š2,𝑝(𝐷), that is, 𝑉=π‘Š02,𝑝(𝐷). Since 𝐷 is a bounded smooth domain in ℝ𝑛, we can endow the Sobolev space 𝑉 with equivalent norm (see [1, page 166]) ‖𝑣‖𝑉=‖Δ𝑣‖𝑝=ξ‚΅ξ€œπ·||||Δ𝑣𝑝𝑑π‘₯1/𝑝,π‘£βˆˆπ‘‰.(2.6) Define π‘‰ξ…ž = the dual of 𝑉, that is, π‘‰ξ…ž=π‘Šβˆ’2,𝑝′(𝐷). Then we have π‘‡βˆˆπ‘Šβˆ’2,𝑝′(𝐷)βŸΊπ‘‡=|𝛼|≀2𝐷𝛼𝑓𝛼,π‘“π›ΌβˆˆπΏπ‘β€²(𝐷),(2.7) where 1/𝑝+1/π‘ξ…ž=1. Let 𝐻 denote the closure of 𝐿2(𝐷) in 𝒱(𝐷) with the usual scalar product and norm {(β‹…,β‹…),β€–β‹…β€–2}. Identifying 𝐻 with its dual space π»ξ…ž by the Riesz isomorphism π‘–βˆΆπ»β†’π»ξ…ž, we have the following Gelfand triple: π‘‰βŠ‚π»β‰‘π»ξ…žβŠ‚π‘‰ξ…ž,(2.8) or concretely π‘Š02,𝑝(𝐷)βŠ‚πΏ2ξ‚€π‘Š(𝐷)βŠ‚02,𝑝(𝐷)ξ…ž=π‘Šβˆ’2,𝑝/(π‘βˆ’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 𝐴𝑒=Δ𝑒 for π‘’βˆˆπ»10⋂𝐻(𝐷)2(𝐷). Then 𝐴 is negative definite and self-adjoint. It is well-known that 𝐴 (with domain π‘Š02,𝑝(𝐷)) generates a strongly continuous semigroup 𝑀(𝑑) on 𝐿𝑝(𝐷) which is contractive and positive. Here β€œcontractive” means ‖𝑀(𝑑)‖𝑝≀1 and β€œpositive” means 𝑀(𝑑)𝑒β‰₯0 for every 0β‰€π‘’βˆˆπΏπ‘(𝐷). The resolvent of generator 𝐴 denoted by 𝑅(πœ†,𝐴), πœ†βˆˆπœŒ(𝐴), where 𝜌(𝐴) is the resolvent set of 𝐴. By Lumer-Phillips Theorem in [16], it follows that (0,∞)βŠ‚πœŒ(𝐴) and for π‘’βˆˆπΏπ‘(𝐷)𝑅(πœ†,𝐴)𝑒=(πœ†βˆ’π΄)βˆ’1ξ€œπ‘’=∞0π‘’βˆ’πœ†π‘‘π‘€(𝑑)𝑒𝑑𝑑,πœ†>0,(2.10)𝑀(𝑑)𝑒=limπ‘›β†’βˆžξ‚Έπ‘›ξ‚Ήπ‘‘π‘…(𝑛/𝑑,𝐴)𝑛𝑒,𝑑>0.(2.11) Furthermore, by (2.10) for every π‘’βˆˆπΏπ‘(𝐷) we have β€–πœ†π‘…(πœ†,𝐴)𝑒‖𝑝≀‖𝑒‖𝑝,πœ†>0,πœ†π‘…(πœ†,𝐴)π‘’βŸΆπ‘’,asπœ†β†’βˆž,(2.12) where the convergence is in the 𝐿𝑝-norms. Moreover, for π‘’βˆˆπ·(𝐴), it follows that 𝑅(πœ†,𝐴)π‘’βˆˆπ·(𝐴), and 𝐴𝑅(πœ†,𝐴)𝑒=𝑅(πœ†,𝐴)𝐴𝑒.

Since 𝐴 is negative definite and self-adjoint operator on 𝐻10⋂𝐻(𝐷)2(𝐷), we associate 𝐴 with the Dirichlet forms [13] πœ€ by ξ‚€βˆšπœ€(𝑒,𝑣)=βˆšβˆ’π΄π‘’,ξ‚βˆ’π΄π‘£,𝑒,π‘£βˆˆπ»10(𝐷).(2.13)πœ€ is unique determined by 𝐴. For 𝑒,π‘£βˆˆπ»10(𝐷), we define a new inner product byπœ€(πœ†)(𝑒,𝑣)=πœ†(π‘’βˆ’πœ†π‘…(πœ†,𝐴)𝑒,𝑣),πœ†>0,(2.14) where 𝑅(πœ†,𝐴) is the resolvent of 𝐴. Then, we have the basic fact (see [13]) that πœ€(πœ†)(𝑒,𝑣)↑ as πœ†β†’βˆž, andlimπœ†β†’βˆžπœ€(πœ†)(𝑒,𝑣)=πœ€(𝑒,𝑣),(2.15) for 𝑒,π‘£βˆˆπ»10(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 π‘Š(𝑑) is a two-sided Wiener process on a complete probability space (Ξ©,β„±,P), where Ξ©={π‘€βˆˆπΆ(ℝ,β„π‘š)βˆΆπ‘€(0)=0}, β„± is the Borel sigma-algebra induced by the compact-open topology of Ξ© and P is the corresponding Wiener measure on (Ξ©,β„±). Then we have π‘€ξ€·π‘Š(𝑑)=π‘Š(𝑑)=1(𝑑),π‘Š2(𝑑),…,π‘Šπ‘šξ€Έ(𝑑),π‘‘βˆˆβ„.(3.1) Define the time shift by πœƒπ‘‘π‘€(𝑠)=𝑀(𝑠+𝑑)βˆ’π‘€(𝑑),π‘€βˆˆΞ©,𝑑,π‘ βˆˆβ„.(3.2) Then (Ξ©,β„±,𝑃,πœƒπ‘‘) is a ergodic measurable dynamical system.

In order to obtain the random attractor, in our following discussion, we always assume that πœ™π‘—(1β‰€π‘—β‰€π‘š) belong to π‘Š04,𝑝(𝐷) and βˆ‡πœ™π‘—(1β‰€π‘—β‰€π‘š)=𝟎.

We now employ the approach similar to [5] to translate (1.1) by one classical change of variables𝑣(𝑑)=𝑒(𝑑)βˆ’π‘§(𝑑),(3.3) where, for short, βˆ‘π‘§(𝑑)=𝑧(𝑑,𝑀)=π‘šπ‘—=1πœ™π‘—π‘Šπ‘—(𝑑). Then, formally, 𝑣(𝑑) satisfies the following equation parameterized by π‘€βˆˆΞ©:𝑑𝑣𝑑𝑑+Ξ”(Ξ¦(Δ𝑣+Δ𝑧))+𝑔(π‘₯,𝑣+𝑧)=𝑓(π‘₯),π‘₯∈𝐷,𝑑β‰₯𝑠,(3.4)𝑣(𝑠)=𝑒(𝑠)βˆ’π‘§(𝑠),π‘₯∈𝐷,π‘ βˆˆβ„,(3.5)βˆ‡π‘£=𝟎,𝑣=0,π‘₯βˆˆπœ•π·,𝑑β‰₯𝑠,(3.6) where 𝑔(π‘₯,𝑒) satisfies (1.4)–(1.6) and 𝑓 is given in π‘‰ξ…ž, 2β‰€π‘žβ‰€π‘<∞.

We define a nonlinear operator Ξ¨ on 𝑉ΨΦ(𝑣)=Δ𝑝(Ξ”(𝑣+𝑧))+𝑔(π‘₯,𝑣+𝑧)βˆ’π‘“(π‘₯),(3.7) for π‘£βˆˆπ‘‰, π‘₯∈𝐷. Then we haveΞ¨(𝑣)=Ξ¨(𝑒),(3.8) where we define Ξ¨(𝑒)=Ξ”(Φ𝑝(Δ𝑒))+𝑔(π‘₯,𝑒)βˆ’π‘“(π‘₯) with 𝑒=𝑣+𝑧 as in (3.3). So we can deduce problem (1.1) to the problem𝑑𝑣𝑑𝑑+Ξ¨(𝑣)=0,𝑑β‰₯𝑠,(3.9) with initial condition 𝑣(𝑠)=𝑒(𝑠)βˆ’π‘§(𝑠) for π‘ βˆˆβ„. Moreover, by (3.9), it follows that the solutions 𝑀-wise satisfy the following: ξ€œπ‘£(𝑑)=𝑣(𝑠)βˆ’π‘‘π‘ Ξ¨(𝑣(𝜏))π‘‘πœ,(3.10) with 𝑣(𝑠)=𝑒(𝑠)βˆ’π‘§(𝑠) and 𝑑β‰₯𝑠.

Since 𝑝β‰₯π‘ž, by our assumption (1.4)–(1.6) and π‘“βˆˆπ‘‰ξ…ž, it is easy to check that the operator Ξ¨βˆΆπ‘£β†¦Ξ¨(𝑣) mapping π‘Š2,𝑝(𝐷) into π‘Šβˆ’2,𝑝′(𝐷) is well-defined, where π‘ξ…ž=𝑝/(π‘βˆ’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 π‘‰ξ…ž, 2β‰€π‘žβ‰€π‘<∞. Then for all π‘ βˆˆβ„ and 𝑣0∈𝐻 with 𝑣0=𝑣(𝑠), (3.4) has a unique solution 𝑣𝑑,𝑀;𝑠,𝑣0ξ€ΈβˆˆπΏπ‘loc([[𝑠,∞),𝑉)𝐢(𝑠,∞),𝐻)(3.11) for 𝑑β‰₯𝑠 and P-a.s. π‘€βˆˆΞ©. Furthermore, the mapping 𝑣0↦𝑣(𝑑,𝑀;𝑠,𝑣0) from 𝐻 into 𝐻 is continuous for all 𝑑β‰₯𝑠.

Proof. We will show that Ξ¨(𝑣) possesses hemicontinuity, monotonicity, coercivity, and boundedness properties. Then for every 𝑣0∈𝐻 with 𝑣0=𝑣(𝑠), the existence and uniqueness of solution 𝑣(𝑑)=𝑣(𝑑,𝑀;𝑠,𝑣0)βˆˆπΏπ‘loc([𝑠,∞),𝑉) follow from [13, Theorem 4.2.4]. If the solution π‘£βˆˆπΏπ‘([𝑠,𝑇],𝑉), 𝑇>0, then it is elementary to check that Ξ¨(𝑣) belongs to 𝐿𝑝′([𝑠,𝑇],π‘‰ξ…ž) by our assumption 𝑝β‰₯π‘ž and π‘“βˆˆπ‘‰ξ…ž. Thus, from (3.9), we get that π‘£π‘‘βˆˆπΏπ‘β€²([𝑠,𝑇],π‘‰ξ…ž). 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 𝐢([𝑠,𝑇],𝐻). The continuity of the mapping 𝑣0↦𝑣(𝑑,𝑀;𝑠,𝑣0) from 𝐻 into 𝐻 is easily proved by using the monotonicity of Ξ¨.
By [13, Theorem 4.2.4], it remains to show that Ξ¨(𝑣) possesses hemicontinuity, monotonicity, coercivity, and boundedness properties. For convenience of our discussion in the following, we decompose Ξ¨(𝑣)=Ξ¨1(𝑣)+Ξ¨2(𝑣), where Ξ¨1(𝑣)=Ξ”(Φ𝑝(Δ𝑒)) and Ξ¨2(𝑣)=Ξ¨(𝑣)βˆ’Ξ¨1(𝑣), where Ξ¨ is as in (3.7).
We first prove the hemicontinuity, that is, for every 𝑣1,𝑣2,𝑣3βˆˆπ‘‰, the function πœ†β†’(Ξ¨(𝑣1+πœ†π‘£2),𝑣3) is continuous from ℝ→ℝ. But it suffices to prove the continuity at πœ†=0. So we assume that |πœ†|<1. For 𝑣1,𝑣2,𝑣3βˆˆπ‘‰, by integration by parts, we see thatξ€·Ξ¨1𝑣1+πœ†π‘£2ξ€Έ,𝑣3ξ€Έ=ξ€œπ·||Δ𝑣1+πœ†π‘£2ξ€Έ||+π‘§π‘βˆ’2Δ𝑣1+πœ†π‘£2ξ€Έ+𝑧Δ𝑣3𝑑π‘₯.(3.12) By HΓΆlder's inequality and Young's inequality, it yields that |||||Δ𝑣1+πœ†π‘£2ξ€Έ||+π‘§π‘βˆ’2Δ𝑣1+πœ†π‘£2ξ€Έ+𝑧Δ𝑣3|||≀||Ξ”(𝑣1+πœ†π‘£2||+𝑧)π‘βˆ’1||Δ𝑣3||≀2π‘βˆ’2ξ‚€||Δ𝑣1ξ€Έ||+π‘§π‘βˆ’1+||Δ𝑣2||π‘βˆ’1||Δ𝑣3||≀2π‘βˆ’2ξ€·||Δ𝑣1ξ€Έ||+𝑧𝑝+||Δ𝑣2||𝑝||+2Δ𝑣3||𝑝,(3.13) which the right-hand side is in 𝐿1(𝐷) for 𝑣1,𝑣2,𝑣3βˆˆπ‘‰. 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 πœ†β†’0. This proves the hemicontinuity of Ξ¨1(𝑣). As for the hemicontinuity of Ξ¨2(𝑣), noting that by our assumption (1.5) and π‘“βˆˆπ‘‰ξ…ž we have ξ€·Ξ¨2𝑣1+πœ†π‘£2ξ€Έ,𝑣3ξ€Έ=ξ€œπ·π‘”ξ€·π‘₯,𝑣1+πœ†π‘£2𝑣+𝑧3ξ€œπ‘‘π‘₯βˆ’π·π‘“(π‘₯)𝑣3𝑑π‘₯≀𝐢2ξ€œπ·||𝑣1+πœ†π‘£2||+π‘§π‘žβˆ’1||𝑣3||ξ€œπ‘‘π‘₯+𝐷Λ2||𝑣(π‘₯)3||ξ€œπ‘‘π‘₯βˆ’π·π‘“(π‘₯)𝑣3𝑑π‘₯.(3.14) It suffices to find the control function for the first integrand above, but we can get this by noting that π‘žβ‰€π‘ and using approach similar to (3.13).
Second, we prove the monotonicity of Ξ¨(𝑣). We first prove the monotonicity for Ξ¨1. For 𝑣1,𝑣2βˆˆπ‘‰, since 𝑣1=𝑒1βˆ’π‘§, 𝑣2=𝑒2βˆ’π‘§, we haveξ€·Ξ¨1𝑣1ξ€Έβˆ’Ξ¨1𝑣2ξ€Έ,𝑣1βˆ’π‘£2ξ€Έ=ξ‚€||Δ𝑒1||π‘βˆ’2Δ𝑒1βˆ’||Δ𝑒2||π‘βˆ’2Δ𝑒2,Δ𝑒1βˆ’Ξ”π‘’2=ξ€œπ·ξ‚€||Δ𝑒1||𝑝+||Δ𝑒2||π‘βˆ’||Δ𝑒1||π‘βˆ’2Δ𝑒1Δ𝑒2βˆ’||Δ𝑒2||π‘βˆ’2Δ𝑒2Δ𝑒1β‰₯ξ€œπ‘‘π‘₯𝐷||Δ𝑒1||𝑝+||Δ𝑒2||π‘βˆ’||Δ𝑒1||π‘βˆ’1||Δ𝑒2||βˆ’||Δ𝑒2||π‘βˆ’1||Δ𝑒1||=ξ€œπ‘‘π‘₯𝐷||Δ𝑒1||π‘βˆ’1βˆ’||Δ𝑒2||π‘βˆ’1||Δ𝑒1||βˆ’||Δ𝑒2||𝑑π‘₯β‰₯0.(3.15) Since 𝑝β‰₯2, the function π‘ π‘βˆ’1 is increasing for 𝑠β‰₯0, which shows that the last inequality in the above proof is correct. On the other hand, by our assumption (1.6), we have (Ξ¨2(𝑣1)βˆ’Ξ¨2(𝑣2),𝑣1βˆ’π‘£2)β‰₯𝐢3‖𝑣1βˆ’π‘£2β€–22, and therefore it follows that for 𝑣1,𝑣2βˆˆπ‘‰ξ€·Ξ¨ξ€·π‘£1ξ€Έξ€·π‘£βˆ’Ξ¨2ξ€Έ,𝑣1βˆ’π‘£2ξ€Έβ‰₯𝐢3‖‖𝑣1βˆ’π‘£2β€–β€–2,(3.16) where 𝐢3 is as in (1.6). Hence, we have showed the monotonicity of Ξ¨(𝑣).
As for the coercivity, for π‘£βˆˆπ‘‰, by our assumptions (1.4) and (1.5), using HΓΆlder' inequality, we have(ξ€œΞ¨(𝑣),𝑣)=𝐷Δ||||Ξ”π‘’π‘βˆ’2ξ‚ξ€œΞ”π‘’π‘£π‘‘π‘₯+π·ξ€œπ‘”(π‘₯,𝑣+𝑧)𝑣𝑑π‘₯βˆ’π·=𝑓(π‘₯)𝑣𝑑π‘₯β€–Ξ”π‘’β€–π‘π‘βˆ’ξ€œπ·ξ‚€||||Ξ”π‘’π‘βˆ’2ξ‚ξ€œΞ”π‘’Ξ”π‘§π‘‘π‘₯+π·ξ€œπ‘”(π‘₯,𝑒)𝑒𝑑π‘₯βˆ’π·ξ€œπ‘”(π‘₯,𝑒)𝑧𝑑π‘₯βˆ’π·π‘“(π‘₯)𝑣𝑑π‘₯β‰₯β€–Ξ”π‘’β€–π‘π‘βˆ’β€–Ξ”π‘’β€–π‘π‘βˆ’1‖Δ𝑧‖𝑝+𝐢1β€–π‘’β€–π‘žπ‘žβˆ’β€–β€–Ξ›1β€–β€–1βˆ’πΆ2β€–π‘’β€–π‘žπ‘žβˆ’1β€–π‘§β€–π‘žβˆ’β€–β€–Ξ›2β€–β€–π‘žβ€²β€–π‘§β€–π‘žβˆ’β€–π‘“β€–π‘‰β€²β€–π‘£β€–π‘‰,(3.17) where 𝐢1 and 𝐢2 are defined in (1.4) and (1.5). By employing the πœ€-Young's inequality, that is, π‘Žπ‘β‰€πœ€(π‘Žπ‘Ÿ/π‘Ÿ)+πœ€βˆ’π‘Ÿβ€²/π‘Ÿ(π‘π‘Ÿβ€²/π‘Ÿξ…ž) for π‘Ÿ>1 and 1/π‘Ÿ+1/π‘Ÿξ…ž=1, we find that β€–Ξ”π‘’β€–π‘π‘βˆ’1‖Δ𝑧‖𝑝≀14π‘βˆ’1𝑝‖Δ𝑒‖𝑝𝑝+22π‘βˆ’2𝑝‖Δ𝑧‖𝑝𝑝≀14‖Δ𝑒‖𝑝𝑝+22π‘βˆ’2‖Δ𝑧‖𝑝𝑝.(3.18) Similarly, we have 𝐢2β€–π‘’β€–π‘žπ‘žβˆ’1β€–π‘§β€–π‘žβ‰€πΆ12β€–π‘’β€–π‘žπ‘ž+2π‘žβˆ’1𝐢11βˆ’π‘žπΆπ‘ž2β€–π‘§β€–π‘žπ‘ž,(3.19)‖𝑓‖𝑉′‖𝑣‖𝑉≀14‖Δ𝑒‖𝑝𝑝+22/(π‘βˆ’1)‖𝑓‖𝑉𝑝/(π‘βˆ’1)β€²+‖𝑓‖𝑉′‖Δ𝑧‖𝑝,(3.20) then, by (3.17)–(3.20), we obtain that 1(Ξ¨(𝑣),𝑣)β‰₯2‖Δ𝑒‖𝑝𝑝+𝐢12β€–π‘’β€–π‘žπ‘žβˆ’π‘1(𝑑,𝑀),(3.21) with 𝑝1(𝑑,𝑀)=22π‘βˆ’2‖Δ𝑧‖𝑝𝑝+2π‘žβˆ’1𝐢11βˆ’π‘žπΆπ‘ž2β€–π‘§β€–π‘žπ‘ž+β€–β€–Ξ›2β€–β€–π‘žβ€²β€–π‘§β€–π‘ž+22/(π‘βˆ’1)‖𝑓‖𝑉𝑝/(π‘βˆ’1)β€²+‖𝑓‖𝑉′‖Δ𝑧‖𝑝β‰₯0,(3.22) where π‘žξ…ž is the dual number of π‘ž. 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 HΓΆlder's inequality it follows with 𝑒(𝑑)=𝑣(𝑑)+𝑧(𝑑) thatβ€–π‘’β€–π‘žπ‘ž=‖𝑣(𝑑)+𝑧(𝑑)β€–π‘žπ‘žβ‰₯21βˆ’π‘žβ€–π‘£β€–π‘žπ‘žβˆ’β€–π‘§β€–π‘žπ‘ž,(3.23) then by the inverse πœ€-Young's inequality, that is, π‘Žπ‘β‰₯πœ€(π‘Žπ‘Ÿ/π‘Ÿ)+πœ€βˆ’π‘Ÿβ€²/π‘Ÿ(π‘π‘Ÿβ€²/π‘Ÿξ…ž) when π‘Ÿ<1 and 1/π‘Ÿ+1/π‘Ÿξ…ž=1, we get from (3.23) that β€–π‘’β€–π‘žπ‘žβ‰₯21βˆ’π‘žβ€–π‘£β€–π‘žπ‘žβˆ’β€–π‘§β€–π‘žπ‘žβ‰₯21βˆ’π‘žπœ‚π‘ž0β€–π‘£β€–π‘ž2βˆ’β€–π‘§β€–π‘žπ‘žβ‰₯π‘ž2‖𝑣‖22βˆ’β€–π‘§β€–π‘žπ‘žβˆ’πΆξ…ž,(3.24) where πΆξ…ž=((π‘žβˆ’2)/2)2(2βˆ’2π‘ž)/(2βˆ’π‘ž)πœ‚02π‘ž/(2βˆ’π‘ž) and πœ‚0 is the Sobolev embedding coefficient of πΏπ‘ž(𝐷)β†ͺ𝐿2(𝐷). Hence, it follows from (3.21) that 1(Ξ¨(𝑣),𝑣)β‰₯2‖Δ𝑒‖𝑝𝑝+π‘ž2‖𝑣‖22βˆ’π‘2(𝑑,𝑀),(3.25) with 𝑝2(𝑑,𝑀)=𝑝1(𝑑,𝑀)+β€–π‘§β€–π‘žπ‘ž+πΆξ…žβ‰₯0, where 𝑝1(𝑑,𝑀) is defined as in (3.22). Note that if π‘ž=2, we omit this procedure and directly (3.21) passes to (3.25). Hence we have proved the coercivity for Ξ¨.
We finally prove the the boundedness for Ξ¨(𝑣) for fixed π‘£βˆˆπ‘‰, that is, for fixed π‘£βˆˆπ‘‰, Ξ¨(𝑣) is a linear bounded functional on π‘Š02,𝑝(𝐷). Indeed, for 𝑣,β„Žβˆˆπ‘‰, by applying HΓΆlder's inequality and repeatedly using Sobolev's embedding inequality, we have(ξ€œΞ¨(𝑣),β„Ž)≀𝐷||||Ξ”π‘’π‘βˆ’1||||ξ€œΞ”β„Žπ‘‘π‘₯+𝐷||||||β„Ž||ξ€œπ‘”(π‘₯,𝑒)𝑑π‘₯+𝐷||||||β„Ž||𝑓(π‘₯)𝑑π‘₯β‰€β€–Ξ”π‘’β€–π‘π‘βˆ’1β€–Ξ”β„Žβ€–π‘+𝐢2β€–π‘’β€–π‘žπ‘žβˆ’1β€–β„Žβ€–π‘ž+β€–β€–Ξ›2β€–β€–π‘žβ€²β€–β„Žβ€–π‘ž+β€–π‘“β€–π‘‰β€²β€–β„Žβ€–π‘‰β‰€ξ‚€β€–Ξ”π‘’β€–π‘π‘βˆ’1+𝑐1β€–Ξ”π‘’β€–π‘π‘žβˆ’1+𝑐2β€–β€–Ξ›2β€–β€–π‘žβ€²+β€–π‘“β€–π‘‰β€²ξ‚β€–Ξ”β„Žβ€–π‘β‰€ξ‚€2β€–Ξ”π‘’β€–π‘π‘βˆ’1+𝑐3+𝑐2β€–β€–Ξ›2β€–β€–π‘žβ€²+β€–π‘“β€–π‘‰β€²ξ‚β€–Ξ”β„Žβ€–π‘β‰€ξ‚€2π‘βˆ’1β€–Ξ”π‘£β€–π‘π‘βˆ’1+𝑝3(𝑑,𝑀)β€–Ξ”β„Žβ€–π‘(3.26) with the random variable 𝑝3(𝑑,𝑀)=2π‘βˆ’1β€–Ξ”π‘§β€–π‘π‘βˆ’1+𝑐2β€–Ξ›2β€–π‘žβ€²+𝑐2‖𝑓‖𝑉′+𝑐3β‰₯0 and the positive constants 𝑐𝑖(𝑖=1,2,3) independent of 𝑣,β„Ž. Therefore, from (3.26) we finally find that β€–Ξ¨(𝑣)‖𝑉′≀2π‘βˆ’1β€–Ξ”π‘£β€–π‘π‘βˆ’1+𝑝3(𝑑,𝑀),(3.27) is a bounded linear operator on π‘Š02,𝑝(𝐷) for fixed π‘£βˆˆπ‘‰. From the proof we know that the assumption 𝑝β‰₯π‘žβ‰₯2 is necessary. This completes the proof of Theorem 3.1.

We now define𝑆(𝑑,𝑠;𝑀)𝑒0ξ€·=𝑣𝑑,𝑀;𝑠,𝑒0ξ€Έβˆ’π‘§(𝑠,𝑀)+𝑧(𝑑,𝑀),𝑑β‰₯π‘ βˆˆβ„,(3.28) with 𝑒0=𝑒(𝑠). Then 𝑆(𝑑,𝑠;𝑀)𝑒0 is the solution to (1.1) in certain meaning for every 𝑒0∈𝐻 and 𝑑β‰₯π‘ βˆˆβ„. By the uniqueness part of solution in Theorem 3.1, we immediately get that 𝑆(𝑑,𝑠,𝑀) is a stochastic flow, that is, for every 𝑒0∈𝐻 and 𝑑β‰₯π‘Ÿβ‰₯π‘ βˆˆβ„π‘†(𝑑,𝑠;𝑀)𝑒0=𝑆(𝑑,π‘Ÿ;𝑀)𝑆(π‘Ÿ,𝑠;𝑀)𝑒0,(3.29)𝑆(𝑑,𝑠;𝑀)𝑒0ξ€·=π‘†π‘‘βˆ’π‘ ,0;πœƒπ‘ π‘€ξ€Έπ‘’0.(3.30) Hence if we defineπœ‘(𝑑,𝑀)𝑒0=𝑆(𝑑,0;𝑀)𝑒0ξ€·=𝑣𝑑,𝑀;0,𝑒0ξ€Έβˆ’π‘§(0,𝑀)+𝑧(𝑑,𝑀)(3.31) with 𝑒0=𝑒(0), then by Theorem 3.1β€‰β€‰πœ‘ is a continuous stochastic dynamical system associated with quasilinear partial differential equation (1.1), with the following factπœ‘ξ€·π‘‘,πœƒβˆ’π‘‘π‘€ξ€Έπ‘’0ξ€·=𝑒0,𝑀;βˆ’π‘‘,𝑒0ξ€Έ,βˆ€π‘‘β‰₯0,(3.32) that is to say, πœ‘(𝑑,πœƒβˆ’π‘‘π‘€)𝑒0 can be interpreted as the position of the trajectory at time 0, which was in 𝑒0 at time βˆ’π‘‘ (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 𝑝𝑖(𝑑,𝑀)(𝑖=1,2,3) appearing in the proofs are given in Section 3. In the following computation, π‘€βˆˆΞ©; the results will hold for P-a.s. π‘€βˆˆΞ©.

Lemma 4.1. Suppose that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in π‘‰ξ…ž. Then there exist random radii π‘Ÿ1(𝑀),π‘Ÿ2(𝑀)>0, such that for all 𝜌>0 there exists 𝑠=𝑠(𝑀,𝜌)β‰€βˆ’1 such that for all 𝑠≀𝑠(𝑀,𝜌) and all 𝑒0∈𝐻 with ‖𝑒0β€–2β‰€πœŒ, the following inequalities hold for P-a.s. π‘€βˆˆΞ©β€–β€–π‘£(𝑑,𝑀;𝑠,𝑒0β€–β€–βˆ’π‘§(𝑠))22β‰€π‘Ÿ21[],ξ€œ(𝑀),βˆ€π‘‘βˆˆβˆ’1,00βˆ’1ξ‚€β€–β€–ξ€·Ξ”π‘’πœ,𝑀;𝑠,𝑒0‖‖𝑝𝑝+β€–β€–π‘’ξ€·πœ,𝑀;𝑠,𝑒0ξ€Έβ€–β€–π‘žπ‘žξ‚π‘‘πœβ‰€π‘Ÿ22(𝑀),(4.1) where 𝑣(𝑑,𝑀;𝑠,𝑒0βˆ’π‘§(𝑠)) is the solution to (3.4) with 𝑣(𝑑,𝑀;𝑠,𝑒0βˆ’π‘§(𝑠))=𝑒(𝑑,𝑀;𝑠,𝑒0)βˆ’π‘§(𝑑,𝑀) and 𝑒0=𝑒(𝑠).

Proof. For simplicity, we abbreviate 𝑣(𝑑)∢=𝑣(𝑑,𝑀;𝑠,𝑒0βˆ’π‘§(𝑠)) and 𝑒(𝑑)∢=𝑒(𝑑,𝑀;𝑠,𝑒0) for fixed 𝑒0,π‘€βˆˆΞ© and 𝑑β‰₯𝑠 with 𝑒0=𝑒(𝑠). Multiplying both sides of (3.9) by 𝑣(𝑑) and then integrating over 𝐷, we obtain that 12𝑑𝑑𝑑‖𝑣‖22+(Ξ¨(𝑣),𝑣)=0.(4.2) Then, by (3.25), we have 𝑑𝑑𝑑‖𝑣‖22+‖Δ𝑒‖𝑝𝑝+π‘žβ€–π‘£β€–22≀2𝑝2(𝑑,𝑀).(4.3) Applying the Gronwall's lemma to (4.3) from 𝑠 to 𝑑, π‘‘βˆˆ[βˆ’1,0], it yields that ‖𝑣(𝑑)β€–22β‰€π‘’βˆ’π‘ž(π‘‘βˆ’π‘ )‖𝑣(𝑠)β€–22ξ€œ+2𝑑𝑠𝑝2(𝜏,𝑀)π‘’βˆ’π‘ž(π‘‘βˆ’πœ)π‘‘πœβ‰€2π‘’π‘žξ‚΅π‘’π‘žπ‘ β€–β€–π‘’0β€–β€–22+π‘’π‘žπ‘ β€–β€–π‘§(𝑠)22+ξ€œ0βˆ’βˆžπ‘2(𝜏,𝑀)π‘’π‘žπœξ‚Ά,π‘‘πœ(4.4) where 𝑝2(𝜏,𝑀) grows at most polynomially as πœβ†’βˆ’βˆž (see [5]). Since 𝑝2(𝜏,𝑀) is multiplied by a function which decays exponentially, the integral in (4.4) is convergent.
Given every fixed 𝜌>0, we can choose 𝑠(𝑀,𝜌), depending only on 𝑀 and 𝜌, such that π‘’π‘žπ‘ β€–π‘’0β€–22≀1. Similarly, ‖𝑧(𝑠)β€–22 grows at most polynomially as π‘ β†’βˆ’βˆž, and ‖𝑧(𝑠)β€–22 is multiplied by a function which decays exponentially. Then we have sup𝑠≀0π‘’π‘žπ‘ β€–π‘§(𝑠)β€–22<+∞.(4.5) Hence by (4.4) we can give the final estimate for ‖𝑣(𝑑)β€–22‖𝑣(𝑑)β€–22β‰€π‘Ÿ21(𝑀)∢=2π‘’π‘žξ‚΅1+sup𝑠≀0π‘’π‘žπ‘ β€–β€–π‘§(𝑠)22+ξ€œ0βˆ’βˆžπ‘2(𝜏,𝑀)π‘’π‘žπœξ‚Ά,π‘‘πœ(4.6) for π‘‘βˆˆ[βˆ’1,0]. Following (4.2), by using (3.21), we find that 𝑑𝑑𝑑‖𝑣‖22+‖Δ𝑒‖𝑝𝑝+𝐢1β€–π‘’β€–π‘žπ‘žβ‰€2𝑝1(𝑑,𝑀),(4.7) where 𝑝1(𝑑,𝑀) is the same as in (3.22). Integrating (4.7) for 𝑑 on [βˆ’1,0], we get that ξ€œ0βˆ’1‖Δ𝑒(𝜏)‖𝑝𝑝+𝐢1‖𝑒(𝜏)β€–π‘žπ‘žξ€œπ‘‘πœβ‰€20βˆ’1𝑝1(𝜏,𝑀)π‘‘πœ+‖𝑣(βˆ’1)β€–22,(4.8) which gives an expression for π‘Ÿ22(𝑀).

In the following, we give the estimate of β€–βˆ‡π‘’(𝑑)β€–2. This is the most difficult part in our discussion. Because the nonlinearity of Ξ¨1 and Ξ¨2 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 β€–βˆ‡π‘’β€–2 for π‘’βˆˆπ‘‰0. Here, just as our statement in the introduction, we make the inner product over the resolvent 𝑅(πœ†,𝐴) which is defined in Section 2, then by using the Dirichlet forms of 𝐴 we obtain technically the estimate of β€–βˆ‡π‘’(𝑑)β€–2.

Lemma 4.2. Suppose that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in π‘‰ξ…ž. Then there exists a random radius π‘Ÿ3(𝑀)>0, such that for all 𝜌>0 there exists 𝑠=𝑠(𝑀,𝜌)β‰€βˆ’1 such that for all 𝑠≀𝑠(𝑀,𝜌) and all 𝑒0∈𝐻 with ‖𝑒0β€–2β‰€πœŒ, the following inequality holds for P-a.s. π‘€βˆˆΞ©β€–β€–βˆ‡π‘’(𝑑,𝑀;𝑠,𝑒0)β€–β€–22β‰€π‘Ÿ23[],(𝑀),βˆ€π‘‘βˆˆβˆ’1,0(4.9) where 𝑒(𝑑,𝑀;𝑠,𝑒0) is the solution to (1.1) with 𝑒0=𝑒(𝑠). In particular, β€–β€–βˆ‡π‘’(0,𝑀;𝑠,𝑒0)β€–β€–22β‰€π‘Ÿ23(𝑀).(4.10)

Proof. Taking the inner product of (3.9) with βˆ’πœ†π΄π‘…(πœ†,𝐴)𝑣 where πœ†>0, π‘£βˆˆπ‘‰, we get βˆ’ξ€œπ·π‘£π‘‘ξ€œπœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯=𝐷Ψ1(ξ€œπ‘’)πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯+𝐷Ψ2(𝑒)πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯.(4.11) By the semigroup theory (see [16]) we have 𝐴𝑅(πœ†,𝐴)𝑣=𝑅(πœ†,𝐴)𝐴𝑣=πœ†π‘…(πœ†,𝐴)π‘£βˆ’π‘£,(4.12) for π‘£βˆˆπ·(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 ξ€œπ·Ξ¨1(ξ€œπ‘’)πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯=𝐷Ψ1(ξ€œπ‘’)πœ†π΄π‘…(πœ†,𝐴)𝑒𝑑π‘₯βˆ’π·Ξ¨1(𝑒)πœ†π΄π‘…(πœ†,𝐴)𝑧𝑑π‘₯.(4.13) Employing (4.12) and by integration by parts, it yields that ξ€œπ·Ξ¨1(ξ€œπ‘’)πœ†π΄π‘…(πœ†,𝐴)𝑒𝑑π‘₯=πœ†π·Ξ¨1(ξ€œπ‘’)(πœ†π‘…(πœ†,𝐴)π‘’βˆ’π‘’)𝑑π‘₯=βˆ’πœ†π·Ξ”ξ‚€||||Ξ”π‘’π‘βˆ’2ξ‚ξ€œΞ”π‘’π‘’π‘‘π‘₯+πœ†π·Ξ”ξ‚€||||Ξ”π‘’π‘βˆ’2ξ‚Ξ”π‘’πœ†π‘…(πœ†,𝐴)𝑒𝑑π‘₯=βˆ’πœ†β€–Ξ”π‘’β€–π‘π‘ξ€œ+πœ†π·ξ‚€||||Ξ”π‘’π‘βˆ’2ξ‚Ξ”π‘’πœ†Ξ”π‘…(πœ†,𝐴)𝑒𝑑π‘₯β‰€βˆ’πœ†β€–Ξ”π‘’β€–π‘π‘ξ€œ+πœ†π·||||Ξ”π‘’π‘βˆ’1||||πœ†π‘…(πœ†,𝐴)Δ𝑒𝑑π‘₯β‰€βˆ’πœ†β€–Ξ”π‘’β€–π‘π‘+πœ†β€–Ξ”π‘’β€–π‘π‘βˆ’1β€–πœ†π‘…(πœ†,𝐴)Ξ”π‘’β€–π‘β‰€βˆ’πœ†β€–Ξ”π‘’β€–π‘π‘+πœ†β€–Ξ”π‘’β€–π‘π‘=0,(4.14) where we use the contraction property of πœ†π‘…(πœ†,𝐴) on 𝐿𝑝(𝐷), that is, β€–πœ†π‘…(πœ†,𝐴)Δ𝑒‖𝑝≀‖Δ𝑒‖𝑝 for Ξ”π‘’βˆˆπΏπ‘(𝐷) and every πœ†>0. We now bound the second term on the right-hand side of (4.13) βˆ’ξ€œπ·Ξ¨1(‖‖Ψ𝑒)πœ†π΄π‘…(πœ†,𝐴)𝑧𝑑π‘₯≀1β€–β€–(𝑒)π‘‰β€²β€–πœ†π΄π‘…(πœ†,𝐴)𝑧‖𝑉=β€–β€–Ξ¨1β€–β€–(𝑒)π‘‰β€²β€–πœ†π‘…(πœ†,𝐴)𝐴𝑧‖𝑉,(4.15) where we use our assumption πœ™π‘—(1β‰€π‘—β‰€π‘š)βˆˆπ‘Š04,𝑝(𝐷). Since Ξ¨1 maps 𝑉 into π‘‰ξ…ž, then for fixed π‘’βˆˆπ‘‰ and every β„Žβˆˆπ‘‰, we have ξ€·Ξ¨1(ξ€Έ=ξ€œπ‘’),β„Žπ·Ξ”ξ‚€||||Ξ”π‘’π‘βˆ’2ξ‚ξ€œΞ”π‘’β„Žπ‘‘π‘₯=𝐷||||Ξ”π‘’π‘βˆ’2ξ‚β‰€ξ€œΞ”π‘’Ξ”β„Žπ‘‘π‘₯𝐷||||Ξ”π‘’π‘βˆ’1||||Ξ”β„Žπ‘‘π‘₯β‰€β€–Ξ”π‘’β€–π‘π‘βˆ’1β€–Ξ”β„Žβ€–π‘.(4.16) So for fixed π‘’βˆˆπ‘‰, β€–Ξ¨1(𝑒)β€–π‘‰β€²β‰€β€–Ξ”π‘’β€–π‘π‘βˆ’1, and therefore by (4.15) we obtain that βˆ’ξ€œπ·Ξ¨1(𝑒)πœ†π΄π‘…(πœ†,𝐴)𝑧𝑑π‘₯β‰€β€–Ξ”π‘’β€–π‘π‘βˆ’1β€–πœ†π‘…(πœ†,𝐴)π΄π‘§β€–π‘‰β‰€πΆβ€–Ξ”π‘’β€–π‘π‘βˆ’1‖𝐴𝑧‖𝑉≀‖Δ𝑒‖𝑝𝑝+𝐢𝑝‖𝐴𝑧‖𝑝𝑉,(4.17) where β€–πœ†π‘…(πœ†,𝐴)𝐴𝑧‖𝑉≀𝐢‖𝐴𝑧‖𝑉 and 𝐢 is a constant independent of πœ†, 𝑣(𝑑) and 𝑒(𝑑). Here we should note that πœ†π‘…(πœ†,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 ξ€œπ·Ξ¨1(𝑒)πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯≀‖Δ𝑒‖𝑝𝑝+𝐢𝑝‖𝐴𝑧‖𝑝𝑉.(4.18) By our assumption (1.5), the second term on the right-hand side of (4.11) is estimated as ξ€œπ·Ξ¨2(=ξ€œπ‘’)πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯π·β‰€ξ€œ(𝑔(π‘₯,𝑒)βˆ’π‘“(π‘₯))πœ†π‘…(πœ†,𝐴)𝐴𝑣𝑑π‘₯𝐷||||||||ξ€œπ‘”(π‘₯,𝑒)πœ†π‘…(πœ†,𝐴)𝐴𝑣𝑑π‘₯+𝐷||||||||β‰€ξ€œπ‘“(π‘₯)πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯𝐷𝐢2|𝑒|π‘žβˆ’1+Ξ›2(ξ€Έ||||ξ€œπ‘₯)πœ†π‘…(πœ†,𝐴)𝐴𝑣𝑑π‘₯+𝐷||||||||𝑓(π‘₯)πœ†π‘…(πœ†,𝐴)𝐴𝑣𝑑π‘₯≀𝐢2β€–π‘’β€–π‘žπ‘žβˆ’1β€–πœ†π‘…(πœ†,𝐴)π΄π‘£β€–π‘ž+β€–β€–Ξ›2β€–β€–π‘žβ€²β€–πœ†π‘…(πœ†,𝐴)π΄π‘£β€–π‘ž+‖𝑓‖2β€–πœ†π‘…(πœ†,𝐴)𝐴𝑣‖2≀𝐢2β€–π‘’β€–π‘žπ‘žβˆ’1β€–π΄π‘£β€–π‘ž+β€–β€–Ξ›2β€–β€–π‘žβ€²β€–π΄π‘£β€–π‘ž+‖𝑓‖2‖𝐴𝑣‖2β‰€β€–π‘’β€–π‘žπ‘ž+ξ€·πΆπ‘ž2ξ€Έ+1β€–π΄π‘£β€–π‘žπ‘ž+β€–β€–Ξ›2β€–β€–π‘žβ€²π‘žβ€²+‖𝑓‖22+‖𝐴𝑣‖22,(4.19) where we employ Young's inequality π‘Žπ‘β‰€π‘Žπ‘Ÿ+π‘π‘Ÿ/(π‘Ÿβˆ’1) for π‘Ÿ>1. But, by Sobolev's inequality and Young's inequality, it yields that β€–π΄π‘£β€–π‘žπ‘žβ‰€πœ‚π‘ž1β€–π΄π‘£β€–π‘žπ‘β‰€πœ‚π‘1‖𝐴𝑣‖𝑝𝑝+1≀2π‘βˆ’1πœ‚π‘1‖𝐴𝑒‖𝑝𝑝+2π‘βˆ’1πœ‚π‘1‖𝐴𝑧‖𝑝𝑝+1;(4.20) similarly ‖𝐴𝑣‖22≀2π‘βˆ’1πœ‚π‘2‖𝐴𝑒‖𝑝𝑝+2π‘βˆ’1πœ‚π‘2‖𝐴𝑧‖𝑝𝑝+1,(4.21) where the positive constants πœ‚1,πœ‚2 are Sobolev's embedding constants independent of πœ†. Then by (4.19)–(4.21), there exist positive constants 𝑐1,𝑐2 such that ξ€œπ·Ξ¨2(𝑒)πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯β‰€β€–π‘’β€–π‘žπ‘ž+𝑐1‖𝐴𝑒‖𝑝𝑝+𝑐2‖𝐴𝑧‖𝑝𝑝+β€–β€–Ξ›2β€–β€–π‘žβ€²π‘žβ€²+‖𝑓‖22+2,(4.22) where π‘žξ…ž=π‘ž/(π‘žβˆ’1). By (4.18) and (4.22), we find that (4.11) becomes βˆ’ξ€œπ·π‘£π‘‘πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯≀𝑐3‖Δ𝑒‖𝑝𝑝+β€–π‘’β€–π‘žπ‘ž+𝑝4(𝑑,𝑀),(4.23) where 𝑝4(𝑑,𝑀)=𝐢𝑝‖𝐴𝑧‖𝑝𝑉+𝑐2‖𝐴𝑧‖𝑝𝑝+β€–Ξ›2β€–π‘žβ€²π‘žβ€²+‖𝑓‖22+2β‰₯0 and 𝑐3=𝑐1+1. On the other hand, by (4.12) and the Dirichlet forms (2.14), we have βˆ’ξ€œπ·π‘£π‘‘πœ†π΄π‘…(πœ†,𝐴)𝑣𝑑π‘₯=πœ€(πœ†)𝑣,𝑣𝑑.(4.24) Hence by (4.24), (4.23) is rewritten as πœ€(πœ†)𝑣,𝑣𝑑≀𝑐3‖Δ𝑒‖𝑝𝑝+β€–π‘’β€–π‘žπ‘ž+𝑝4(𝑑,𝑀).(4.25) Note that the right-hand side of (4.25) is independent of πœ†. So taking limit on both side of (4.25) for πœ†β†’βˆž, association with (2.15), we deduce that 12π‘‘π‘‘π‘‘β€–βˆ‡π‘£β€–22≀𝑐3‖Δ𝑒‖𝑝𝑝+β€–π‘’β€–π‘žπ‘ž+𝑝4(𝑑,𝑀).(4.26) Integrating (4.26) from 𝑠 to 𝑑 (βˆ’1≀𝑠≀𝑑≀0), it yields that β€–βˆ‡π‘£(𝑑)β€–22≀2𝑐3ξ€œπ‘‘π‘ β€–Ξ”π‘’(𝜏)β€–π‘π‘ξ€œπ‘‘πœ+2𝑑𝑠‖𝑒(𝜏)β€–π‘žπ‘žξ€œπ‘‘πœ+2𝑑𝑠𝑝4(𝜏,𝑀)π‘‘πœ+β€–βˆ‡π‘£(𝑠)β€–22≀2𝑐3ξ€œ0βˆ’1‖Δ𝑒(𝜏)β€–π‘π‘ξ€œπ‘‘πœ+20βˆ’1‖𝑒(𝜏)β€–π‘žπ‘žξ€œπ‘‘πœ+20βˆ’1𝑝4(𝜏,𝑀)π‘‘πœ+β€–βˆ‡π‘£(𝑠)β€–22.(4.27) Therefore, by Lemma 4.1, we find that β€–βˆ‡π‘£(𝑑)β€–22𝑐≀23ξ€Έπ‘Ÿ+122ξ€œ(𝑀)+20βˆ’1𝑝4(𝜏,𝑀)π‘‘πœ+β€–βˆ‡π‘£(𝑠)β€–22.(4.28) Integrating (4.28) for 𝑠 from βˆ’1 to 0, we have β€–βˆ‡π‘£(𝑑)β€–22𝑐≀23ξ€Έπ‘Ÿ+122ξ€œ(𝑀)+20βˆ’1𝑝4ξ€œ(𝜏,𝑀)π‘‘πœ+0βˆ’1β€–β€–βˆ‡π‘£(𝑠)22𝑑𝑠,(4.29) for all π‘‘βˆˆ[βˆ’1,0]. By Poincare's inequality, and Young's inequality, there exist positive constants 𝑐4, 𝑐5, 𝑐6 such that β€–βˆ‡π‘£(𝑠)β€–22≀𝑐4‖Δ𝑣(𝑠)β€–22≀2𝑐4‖Δ𝑒(𝑠)β€–22+2𝑐4‖Δ𝑧(𝑠)β€–22≀2𝑐5‖Δ𝑒(𝑠)‖𝑝𝑝+2𝑐5‖Δ𝑧(𝑠)β€–22+𝑐6.(4.30) Hence, by (4.30) and using Lemma 4.1 again, (4.29) follows β€–βˆ‡π‘£(𝑑)β€–22𝑐≀23+𝑐5ξ€Έπ‘Ÿ+122ξ€œ(𝑀)+20βˆ’1𝑝4(𝜏,𝑀)π‘‘πœ+2𝑐5ξ€œ0βˆ’1‖Δ𝑧(𝜏)β€–22π‘‘πœ+𝑐6,(4.31) with π‘‘βˆˆ[βˆ’1,0]. See that 𝑣(𝑑)=𝑒(𝑑)βˆ’π‘§(𝑑). Then, we have β€–βˆ‡π‘’(𝑑)β€–22≀2β€–βˆ‡π‘£(𝑑)β€–22+2β€–βˆ‡π‘§(𝑑)β€–22𝑐≀23+𝑐5ξ€Έπ‘Ÿ+122ξ€œ(𝑀)+20βˆ’1𝑝4(𝜏,𝑀)π‘‘πœ+2𝑐5ξ€œ0βˆ’1‖Δ𝑧(𝜏)β€–22π‘‘πœ+2supβˆ’1≀𝑑≀0β€–βˆ‡π‘§(𝑑)β€–22+𝑐6,(4.32) which gives an expression for π‘Ÿ23(𝑀). 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 π‘‰ξ…ž. Then the RDS πœ‘(𝑑,πœ”) generated by the stochastic equation (1.1) possesses a random attractor π’œ(𝑀) defined as π’œ(𝑀)=ξšπ΅βˆˆβ„¬(𝐻)𝑠β‰₯0ξšπ‘‘β‰₯π‘ πœ‘(𝑑,πœƒβˆ’π‘‘π‘€)𝐡,(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 π‘Š02,𝑝(𝐷). So it is possible in theory to obtain a compact random attractor in π‘Š01,𝑝(𝐷) or π‘Š02,𝑝(𝐷). But it seems most difficulty to derive the estimate of solution in π‘Š2,𝑝(𝐷) due to the nonlinear principle part Ξ”(Φ𝑝(Δ𝑒)).

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 𝐢3>0 in (1.6). This leads to the following fact that for every fixed π‘‘βˆˆβ„ and π‘€βˆˆΞ©, the solution 𝑒(𝑑,𝑀;𝑠,𝑒(𝑠)) to (1.1) is a Cauchy sequence in 𝐻 for the initial time 𝑠 with initial value 𝑒(𝑠) belonging to the bounded subset of 𝐻. Then we obtain a compact attractor consisting of a single point which is the limit of 𝑒(0,𝑀;𝑠,𝑒(𝑠)) as π‘ β†’βˆ’βˆž.

Lemma 5.1. Assume that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in π‘‰ξ…ž, 𝐢3>0. Then for 𝑠1≀𝑠2≀𝑑 and 𝑒01,𝑒02∈𝐻 with 𝑒(𝑠1)=𝑒01 and 𝑒(𝑠2)=𝑒02, there exists a positive constant π‘˜<𝐢3 such that ‖‖𝑒(𝑑,𝑀;𝑠1,𝑒01)βˆ’π‘’(𝑑,𝑀;𝑠2,𝑒02)β€–β€–22≀2π‘’βˆ’πΆ3𝑑4𝑒(𝐢3βˆ’π‘˜)𝑠2‖‖𝑒01β€–β€–22+‖‖𝑧𝑠1ξ€Έβ€–β€–22+ξ€œ0βˆ’βˆžπ‘2(𝜏,𝑀)π‘’π‘˜πœξ‚Άπ‘‘πœ+2π‘’π‘˜π‘ 2‖‖𝑧𝑠2ξ€Έβ€–β€–22+π‘’π‘˜π‘ 2‖‖𝑒02β€–β€–22ξ‚Ά.(5.1) In particular, for each fixed π‘‘βˆˆβ„ and π‘€βˆˆΞ© there exists a single point πœ‰π‘‘(𝑀) in 𝐻 such that limπ‘ β†’βˆ’βˆžπ‘†(𝑑,𝑠;𝑀)𝑒0=πœ‰π‘‘(𝑀),(5.2) where 𝑒0=𝑒(𝑠) and 𝑆(𝑑,𝑠;𝑀) 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 𝑒0 for all 𝑒0 belonging to a bounded subset of 𝐻.

Proof. For 𝑠1≀𝑠2≀𝑑 and 𝑒01,𝑒02∈𝐻 with 𝑒(𝑠1)=𝑒01 and 𝑒(𝑠2)=𝑒02, we can deduce from (3.9) and (3.8) that 𝑑𝑒𝑑𝑑𝑑,𝑀;𝑠1,𝑒01ξ€Έξ€·βˆ’π‘’π‘‘,𝑀;𝑠2,𝑒02+Ψ𝑒𝑑,𝑀;𝑠1,𝑒01βˆ’ξ€Έξ€ΈΞ¨ξ€·π‘’ξ€·π‘‘,𝑀;𝑠2,𝑒02ξ€Έξ€Έ=0,(5.3) where 𝑒(𝑑)=𝑣(𝑑)+𝑧(𝑑) is the solution to problem (1.1). On the other hand, by (3.8) and (3.16), we immediately deduce that Ψ𝑒1ξ€Έβˆ’Ξ¨ξ€·π‘’2ξ€Έ,𝑒1βˆ’π‘’2β‰₯𝐢3‖‖𝑒1βˆ’π‘’2β€–β€–22.(5.4) Then, multiplying (5.3) by 𝑒(𝑑,𝑀;𝑠1,𝑒01)βˆ’π‘’(𝑑,𝑀;𝑠2,𝑒02), integrating over 𝐷, and using (5.4), we find that 𝑑‖‖𝑑𝑑𝑒(𝑑,𝑀;𝑠1,𝑒01)βˆ’π‘’(𝑑,𝑀;𝑠2,𝑒02)β€–β€–22+𝐢3‖‖𝑒𝑑,𝑀;𝑠1,𝑒01ξ€Έβˆ’π‘’(𝑑,𝑀;𝑠2,𝑒02)β€–β€–22≀0.(5.5) Now, applying Gronwall's lemma to (5.5) from 𝑠2 to 𝑑, it yields that ‖‖𝑒(𝑑,𝑀;𝑠1,𝑒01)βˆ’π‘’(𝑑,𝑀;𝑠2,𝑒02)β€–β€–22≀‖‖𝑒(𝑠2,𝑀;𝑠1,𝑒01)βˆ’π‘’02β€–β€–22π‘’βˆ’πΆ3(π‘‘βˆ’π‘ 2)‖‖𝑒𝑠≀22,𝑀;𝑠1,𝑒01ξ€Έβ€–β€–22+‖‖𝑒02β€–β€–22ξ‚π‘’βˆ’πΆ3(π‘‘βˆ’π‘ 2).(5.6) We then estimate ‖𝑒(𝑠2,𝑀;𝑠1,𝑒01)β€–22. To this end, we rewrite (3.21) as 𝐢(Ξ¨(𝑣),𝑣)β‰₯12β€–π‘’β€–π‘žπ‘žβˆ’π‘1(𝑑,𝑀).(5.7) Since π‘žβ‰₯2, by HΓΆlder's inequality and inverse Young's inequality we can choose constant 0<π‘˜<𝐢3 such that 𝐢12β€–π‘’β€–π‘žπ‘žβ‰₯π‘˜2‖𝑣‖22βˆ’π‘1β€–π‘§β€–π‘žπ‘žβˆ’π‘2.(5.8) Then, by (5.7)-(5.8) it follows from (4.2) that 𝑑𝑑𝑑‖𝑣‖22+π‘˜β€–π‘£β€–22≀2𝑝5(𝑑,𝑀),(5.9) where 𝑝5(𝑑,𝑀)=𝑝1(𝑑,𝑀)+𝑐1β€–π‘§β€–π‘žπ‘ž+𝑐2 and 𝑣(𝑑)=𝑣(𝑑,𝑀;𝑠1,𝑒01βˆ’π‘§(𝑠1)). Using Gronwall's lemma to (5.9) from 𝑠1 to 𝑠2 with 𝑠1≀𝑠2≀0, we get that ‖‖𝑣(𝑠2,𝑀;𝑠1,𝑒01βˆ’π‘§(𝑠1β€–β€–))22≀‖‖𝑒01βˆ’π‘§(𝑠1)β€–β€–22π‘’βˆ’π‘˜(𝑠2βˆ’π‘ 1)+ξ€œπ‘ 2𝑠12𝑝5(𝜏,𝑀)π‘’βˆ’π‘˜(𝑠2βˆ’πœ)π‘‘πœβ‰€2π‘’βˆ’π‘˜π‘ 2‖‖𝑒01β€–β€–22+‖‖𝑧(𝑠1)β€–β€–22+ξ€œ0βˆ’βˆžπ‘5(𝜏,𝑀)π‘’π‘˜πœξ‚Ά.π‘‘πœ(5.10) Similar to the argument of (4.4), we know that the integral in the above is convergent. Therefore, we have ‖‖𝑒(𝑠2,𝑀;𝑠1,𝑒01)β€–β€–22‖‖≀2𝑣(𝑠2,𝑀;𝑠1,𝑒01)β€–β€–22β€–β€–+2𝑧(𝑠2)β€–β€–22≀4π‘’βˆ’π‘˜π‘ 2‖‖𝑒01β€–β€–22+‖‖𝑧(𝑠1)β€–β€–22+ξ€œ0βˆ’βˆžπ‘5(𝜏,𝑀)π‘’π‘˜πœξ‚Άβ€–β€–π‘‘πœ+2𝑧(𝑠2)β€–β€–22,(5.11) from which and (5.6) it follows for every fixed π‘‘βˆˆβ„ that ‖‖𝑒(𝑑,𝑀;𝑠1,𝑒01)βˆ’π‘’(𝑑,𝑀;𝑠2,𝑒02)β€–β€–22≀2π‘’βˆ’πΆ3𝑑4𝑒(𝐢3βˆ’π‘˜)𝑠2‖‖𝑒01β€–β€–22+‖‖𝑧𝑠1ξ€Έβ€–β€–22+ξ€œ0βˆ’βˆžπ‘5(𝜏,𝑀)π‘’π‘˜πœξ‚Άπ‘‘πœ+2π‘’π‘˜π‘ 2‖‖𝑧𝑠2ξ€Έβ€–β€–22+π‘’π‘˜π‘ 2‖‖𝑒02β€–β€–22ξ‚ΉβŸΆ0,as,𝑠1,𝑠2βŸΆβˆ’βˆž,(5.12) where the convergence is uniform with respect to 𝑒01,𝑒02 belonging to every bounded subset of 𝐻. Then (5.12) implies that for fixed π‘‘βˆˆβ„, 𝑒(𝑑,𝑀;𝑠,𝑒(𝑠)) is a Cauchy sequence in 𝐻 for π‘ βˆˆβ„. Thus, by the completeness of 𝐻, for every fixed π‘‘βˆˆβ„ and π‘€βˆˆΞ©, 𝑒(𝑑,𝑀;𝑠,𝑒(𝑠)) has a limit in 𝐻 denoted by πœ‰π‘‘(𝑀), that is, limπ‘ β†’βˆ’βˆžπ‘’(𝑑,𝑀;𝑠,𝑒(𝑠))=πœ‰π‘‘(𝑀).(5.13)

Theorem 5.2. Assume that 𝑔 satisfies (1.4)–(1.6) and 𝑓 is given in π‘‰ξ…ž, 𝐢3>0. Then the RDS πœ‘(𝑑,𝑀) generated by the solution to (1.1) possesses a single point attractor π’œ(𝑀), that is, there exists a single point πœ‰0(𝑀) in 𝐻 such that π’œξ€½πœ‰(𝑀)=0ξ€Ύ.(𝑀)(5.14)

Proof. By Lemma 5.1 we define πœ‰0(𝑀)=limπ‘ β†’βˆ’βˆžπ‘†(0,𝑠;𝑀)𝑒0,(5.15) where 𝑆(0,𝑠;𝑀)=𝑒(0,𝑀;𝑠,𝑒(𝑠)) by (3.28). Then we need prove that π’œ(𝑀)={πœ‰0(𝑀)} is a compact attractor. It is obvious that {πœ‰0(𝑀)} is a compact random set. Hence by Definition 2.1 it suffices to prove the invariance and attracting property for {πœ‰0(𝑀)}. Since by the continuity of πœ‘(𝑑,𝑀), and relations (3.29)–(3.32), we have πœ‘(𝑑,𝑀)πœ‰0(𝑀)=πœ‘(𝑑,𝑀)limπ‘ β†’βˆ’βˆžπ‘†(0,𝑠;𝑀)𝑒0=limπ‘ β†’βˆ’βˆžπœ‘(𝑑,𝑀)𝑆(0,𝑠;𝑀)𝑒0=limπ‘ β†’βˆ’βˆžπ‘†(𝑑,0;𝑀)𝑆(0,𝑠;𝑀)𝑒0=limπ‘ β†’βˆ’βˆžπ‘†(𝑑,𝑠;𝑀)𝑒0=limπ‘ β†’βˆ’βˆžπ‘†ξ€·π‘‘βˆ’π‘ ,0;πœƒπ‘ π‘€ξ€Έπ‘’0=limπ‘ β†’βˆ’βˆžπ‘†ξ€·0,π‘ βˆ’π‘‘;πœƒπ‘‘π‘€ξ€Έπ‘’0ξ€·πœƒ=πœ‰π‘‘π‘€ξ€Έ(5.16) then it follows that πœ‘(𝑑,𝑀)π’œ(𝑀)=π’œ(πœƒπ‘‘π‘€). On the other hand, by Lemma 5.1, the convergence is uniform with respect to 𝑒0 belonging to a bounded subset. Then for every bounded subset π΅βŠ‚π», by relations (3.32) and (3.28), it follows that ξ€·πœ‘ξ€·dist𝑑,πœƒβˆ’π‘‘π‘€ξ€Έξ€Έπ΅,π’œ(𝑀)=sup𝑒0βˆˆπ΅β€–β€–πœ‘(𝑑,πœƒβˆ’π‘‘π‘€)𝑒0βˆ’πœ‰0β€–β€–(𝑀)2=sup𝑒0βˆˆπ΅β€–β€–π‘†(0,βˆ’π‘‘,𝑀)𝑒0βˆ’πœ‰0β€–β€–(𝑀)2=sup𝑒0βˆˆπ΅β€–β€–π‘’(0,𝑀;βˆ’π‘‘,𝑒0)βˆ’πœ‰0β€–β€–(𝑀)2⟢0(5.17) as 𝑑→+∞. That is to say π’œ(𝑀) is a attracting set which attracts every deterministic bounded set of 𝐻, and therefore we complete the proof.


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.