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Asymptotic behavior of stochastic p-Laplacian-type equation with multiplicative noise

Wenqiang Zhao

Author Affiliations

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

Boundary Value Problems 2012, 2012:61  doi:10.1186/1687-2770-2012-61


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/61


Received:28 December 2011
Accepted:13 April 2012
Published:22 June 2012

© 2012 Zhao; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The unique existence of solutions to stochastic p-Laplacian-type equation with forced term satisfying some growth and dissipative conditions is established for the initial value in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M1">View MathML</a>. The generation of a continuous random dynamical system and the existence of a random attractor for stochastic p-Laplacian-type equation driven by multiplicative noise are obtained. Furthermore, we obtain a random attractor consisting of a single point and thus the system possesses a unique stationary solution.

MSC: 60H15, 35B40, 35B41.

Keywords:
random dynamical systems; stochastic p-Laplacian-type equation; random attractors

1 Introduction

The purpose of this paper is to investigate the long-time behavior of solutions to stochastic p-Laplacian-type equation with multiplicative noise, which reads

(1.1)

(1.2)

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M6">View MathML</a>; D is an open and bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M7">View MathML</a> with regular boundary ∂D; Δ is the Laplacian with regard to the variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M8">View MathML</a>; b is a positive constant; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M9">View MathML</a> a real-valued variable of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M11">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M12">View MathML</a> is mutually independent two-sided real-valued Wiener process defined on a complete probability space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M13">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M14">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M15">View MathML</a> is the Borel σ-algebra induced by the compact-open topology of Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a> is the corresponding Wiener measure on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M17">View MathML</a>. Then we can identify <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M18">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M19">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M20">View MathML</a>

It is known that the random attractor, which characterizes the long-time behavior of random dynamical systems (RDS) perfectly, was first introduced by [6,13] as a generalization of a global attractor for deterministic PDE. The existences of the random attractor for RDS have been richly developed by many authors for all kinds of SPDEs, see [2,5,6,9,10,15-18,21-25] and references therein.

In deterministic case, there is a large number of works about the p-Laplacian-type equation. Temam [14] obtained the global attractor for (1.1) with exterior forcing term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M21">View MathML</a>, a simple case. In recent years, Yang et al. [19,20] considered the global attractor for a general p-Laplacian-type equation defined both on unbounded domain and bounded domain, respectively. The uniform attractor was also investigated by Chen and Zhong [3] in nonautonomous case. In random case, Zhao [23] obtained random attractors for the p-Laplacian-type equation driven by additive noise.

In this paper, we consider the existence of a random attractor for (1.1)-(1.3) with exterior forcing term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22">View MathML</a> satisfying some growth conditions. The multiplicative noise <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M23">View MathML</a> characterizes, to some extent, some of the minimal fluctuations among environment or a man-made complex system, which we should take into consideration in order to model perfectly the concrete problem.

One difficulty in our discussions is to estimate the solution operator in the stronger norm space V, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M24">View MathML</a> is the Gelfand triple, see Section 2. It seems that the methods used in unperturbed case (see [14,19,20]) are completely unavailable because of the leading term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M25">View MathML</a> with high order differentials and the forcing term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M27">View MathML</a> times growth.

We need to develop some techniques to surmount the obstacle, though we also follow the classic approach (based on the compact embedding) widely used in [5,6,17,21-24] and so on. By using the properties of Dirichlet form for the Laplacian, we overcome this obstacle and obtain the estimate of the solution in the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M28">View MathML</a>, which is weaker than V. Here some basic results about the Laplacian are used. We refer to [8] to obtain the details on Dirichlet forms for a negative definite and self-adjoint operator. The existence and uniqueness of a continuous RDS are proved by employing the standard in [12].

We give the outline of this paper. In Section 2, we present some preliminaries for the theory of RDS and the results about the Laplacian which are necessary to our discussion. In Section 3, we prove the existence and uniqueness of a continuous RDS which is generated by the solution to stochastic p-Laplacian-type equation with multiplicative noise. In Section 4, we give some estimates for the solution operators in given Hilbert space and then obtain a random attractor for this RDS. In the last part, we show that the system possesses a unique stationary point under a given condition.

2 Preliminaries

In this section, we present some basic notions about RDS, which can be found in [1,4-6]. We also list the Sobolev spaces, some results about the Laplacian and its Dirichlet forms.

The basic notion in RDS is a metric dynamical system (MSD) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M29">View MathML</a>, which is a probability space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M30">View MathML</a> with a group <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M31">View MathML</a>, of measure preserving transformations of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M30">View MathML</a>. MSD θ is said to be ergodic under <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a> if for any θ-invariant set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M34">View MathML</a> we have either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M35">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M36">View MathML</a>, where the θ-invariant set is in the sense <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M37">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M34">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39">View MathML</a>.

RDS is an object consisting of a MSD and a cocycle over this MSD, where the MSD is used to model random perturbations. Let X be complete and separable metric space with metric d and Borel sigma-algebra <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M40">View MathML</a>, i.e., the smallest σ-algebra on X which contains all open subsets.

Definition 2.1 (1) A continuous RDS on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M41">View MathML</a> over a MSD θ is a family of measurable mappings

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M42">View MathML</a>

such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M44">View MathML</a>, the mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M45">View MathML</a> satisfy the cocycle property

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M46">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M47">View MathML</a>, and the mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M48">View MathML</a> are continuous in X for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M49">View MathML</a>.

(2) A continuous stochastic flow is a family of measurable mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M51">View MathML</a>, such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M44">View MathML</a>,

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M55">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M56">View MathML</a> are continuous in X for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M57">View MathML</a>.

(3) A random compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M58">View MathML</a> is a family of compact sets indexed by ω such that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M59">View MathML</a> the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M60">View MathML</a> is measurable with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M61">View MathML</a>.

(4) A random set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62">View MathML</a> is an attracting set if for every deterministic bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M44">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M66">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M67">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M68">View MathML</a>.

(5) A random set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62">View MathML</a> is an absorbing set if for every deterministic bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M44">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M73">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M74">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M75">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M76">View MathML</a>.

It is obvious that an absorbing set is an attracting set. The attraction in the definition of the attracting set is a form of pathwise convergence. In fact, the attracting set also attracts in the weaker convergence in probability, in the sense, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M77">View MathML</a> and every bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M63">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M79">View MathML</a>

Definition 2.2 A random compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M80">View MathML</a> is called to be a random attractor for the RDS φ if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62">View MathML</a> is an attracting set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M82">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M11">View MathML</a>.

Theorem 2.3 (see [4])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M85">View MathML</a>is a continuous RDS onXover MDSθ. If there exists a compact random absorbing set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M86">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M85">View MathML</a>possesses a random attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M89">View MathML</a>

(2.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M90">View MathML</a>denotes all the bounded subsets ofX.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M91">View MathML</a> be the p-times integrable functions space on D with norm denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M92">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M93">View MathML</a> with Sobolev equivalent norm (see p.166 of [14])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M94">View MathML</a>

Put the dual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95">View MathML</a> of V by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M96">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M97">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M98">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M99">View MathML</a> with the usual scalar product and norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M100">View MathML</a>. Then we have the following Gelfand triple

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M101">View MathML</a>

or concretely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M102">View MathML</a>

where the injections are continuous and each space is dense in the following one.

We know that the Laplacian Δ, which is negative definite and self-adjoint, is the generator (with domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M103">View MathML</a>) of a strongly continuous semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M104">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M91">View MathML</a> which is contractive and positive. Here “contractive” means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M106">View MathML</a> and “positive” means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M107">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M108">View MathML</a>. The resolvent of generator Δ is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M109">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M110">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M111">View MathML</a> is the resolvent set of Δ. By the Lumer-Phillips Theorem in [11], it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M112">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M113">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M114">View MathML</a>

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M115">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M117">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M118">View MathML</a> is the domain of Δ.

Since Δ is negative definite and self-adjoint, then Δ is associated with the Dirichlet forms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M119">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M120">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M119">View MathML</a> is unique determined by Δ. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M122">View MathML</a>, we define a new inner product by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M123">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M109">View MathML</a> is the resolvent of Δ. Then it follows from Ref. [8] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M125">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M126">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M127">View MathML</a>

(2.4)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M122">View MathML</a>.

3 Existence and uniqueness of RDS

In this section, we show the existence and uniqueness of a continuous RDS for the following stochastic p-Laplacian-type equation with multiplicative noise,

(3.1)

(3.2)

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M6">View MathML</a>. To study System (3.1)-(3.3), we assume that the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22">View MathML</a> defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M135">View MathML</a> satisfies the following conditions:

(3.4)

(3.5)

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M139">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>, we define a nonlinear operator A on V by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M141">View MathML</a>

(3.7)

Then (3.1) reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M142">View MathML</a>

(3.8)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M143">View MathML</a>, by our assumption (3.4)-(3.6) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M144">View MathML</a>, it is easy to check that for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M146">View MathML</a> mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M147">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M148">View MathML</a> is well defined, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M149">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M150">View MathML</a> be the probability space as in the introduction. Define the Wiener shift by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M151">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M152">View MathML</a> is an ergodic MDS.

In order to obtain the existence of a continuous RDS, it is necessary to translate (3.1)-(3.3) into a deterministic system parameterized by ω. To this end, we consider the Ornstein-Uhlenbeck process. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M153">View MathML</a>

which solves the Itô differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M154">View MathML</a>

where the Ornstein-Uhlenbeck constant equals to 1.

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M155">View MathML</a> is a Gaussian process with mathematical expectation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M156">View MathML</a> and variance <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M157">View MathML</a>, see [7], whereas <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M158">View MathML</a>. Furthermore, from [2,15,18], the random variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M159">View MathML</a> is continuous in t for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a> and grows sublinearly, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M162">View MathML</a>.

We now translate (3.1) by one classical change of variables

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M163">View MathML</a>

(3.9)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M164">View MathML</a>

Then, formally, the variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M165">View MathML</a> satisfies the following equations parameterized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a> but without white noise:

(3.10)

(3.11)

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22">View MathML</a> satisfies (3.4)-(3.6) and f is given in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M172">View MathML</a>.

For convenience, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M173">View MathML</a>

(3.13)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M174">View MathML</a>

(3.14)

Note that System (3.1)-(3.3) and System (3.10)-(3.12) are equivalent by (3.9). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M176">View MathML</a> be the solution of System (3.1)-(3.3) and System (3.10)-(3.12) respectively. It is easy to check that if System (3.1)-(3.3) possess a unique solution in V for all initial values in H then System (3.10)-(3.12) possess a unique solution in V for the same initial value in H. Moreover, if the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M177">View MathML</a> is continuous in H for the initial value in H, then the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M178">View MathML</a> is also continuous in H, vice verse.

We now show the existence and uniqueness of solution to System (3.1)-(3.6).

Theorem 3.1Assume thatgsatisfies (3.4)-(3.6) andfis given in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M180">View MathML</a>. Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M181">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M182">View MathML</a>, System (3.1)-(3.3) has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M183">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M184">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>. Furthermore, the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M187">View MathML</a>fromHintoHis continuous for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M184">View MathML</a>.

Proof We first show that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M181">View MathML</a> there exists a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M190">View MathML</a>. By Theorem 4.2.4 and Exercise 4.1.2 in [12], it suffices to show that for every fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M192">View MathML</a> possesses Hemi-continuity, Monotonicity, Coercivity, and Bounded-ness properties (for the definitions of these notions please refer to p.56 of [12]). But the proofs are an analogy of the corresponding works in [23]. So we omit them here.

We then show that the solution is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M193">View MathML</a>. By our assumptions that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M144">View MathML</a>, we can check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M196">View MathML</a> maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M197">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M198">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>. Thus if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M200">View MathML</a>, then (3.14) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M201">View MathML</a>. Now by the general fact (see p.164 of [14]) it follows that v is almost everywhere equal to a function belonging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M202">View MathML</a>. Hence by the transformation (3.9) and the continuous property of Ornstein-Uhlenbeck process, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M203">View MathML</a> is almost everywhere equal to a function belonging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M193">View MathML</a>.

We finally prove the continuity of the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M205">View MathML</a> from H into H. It suffices to prove that the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M206">View MathML</a> is continuous from H into H.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M208">View MathML</a> be two different initial values at initial value time s, and corresponding solutions be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M209">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M210">View MathML</a> respectively. Then it follows from (3.14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M211">View MathML</a>

(3.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M212">View MathML</a> is defined in (3.13). Note that

Because the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M214">View MathML</a> is increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M215">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M6">View MathML</a>, the last inequality in the above proof is correct. Then by a simple computation we find that for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M218">View MathML</a>

(3.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M219">View MathML</a> is in (3.6). Hence, multiplying (3.15) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M220">View MathML</a>, integrating over D, and using (3.16), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M221">View MathML</a>

(3.17)

Using Gronwall’s lemma to (3.17) from s to t, it yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M222">View MathML</a>

(3.18)

Then, the continuity of the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M206">View MathML</a> from H into H is followed from the contraction property (3.18). This finishes the total proofs of Theorem 3.1. □

We now define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M224">View MathML</a>

(3.19)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M182">View MathML</a>. By the uniqueness part of the solution in Theorem 3.1, we immediately get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M226">View MathML</a> is a stochastic flow; that is, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M228">View MathML</a>

(3.20)

(3.21)

Hence if we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M231">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M232">View MathML</a>, then by Theorem 3.1 ψ is a continuous RDS associated with System (3.1)-(3.3).

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M233">View MathML</a>

Then φ is a continuous RDS associated with System (3.10)-(3.12), with the following fact

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M234">View MathML</a>

(3.22)

That is to say, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M235">View MathML</a> can be interpreted as the position of the trajectory at time 0, which was in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M236">View MathML</a> at time −t (see [5]).

It is easy to check that ψ possesses a random attractor provided that φ possesses a random attractor. Hence in the following we only concentrate on the RDS φ.

4 Existence of compact random attractor for RDS

In this section, we will compute some estimates in space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M238">View MathML</a>. Note that in the following <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>; the results will hold for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a> and the generic constants c or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M243">View MathML</a> are independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M244">View MathML</a> in the context, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M110">View MathML</a>.

Lemma 4.1Suppose thatgsatisfies (3.4)-(3.6) andfis given in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95">View MathML</a>. Then there exist random radii<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M247">View MathML</a>, such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M248">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M249">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M250">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M251">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M252">View MathML</a>, the following inequalities hold for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>,

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M176">View MathML</a>is the solution to Equation (3.10) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M257">View MathML</a>.

Proof For simplicity, we abbreviate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M258">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M184">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M257">View MathML</a>. Multiplying both sides of (3.10) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M165">View MathML</a> and then integrating over D, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M262">View MathML</a>

(4.1)

where

(4.2)

(4.3)

(4.4)

Then by (4.1)-(4.4), we have

(4.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M267">View MathML</a>, then by using Sobolev’s embedding inequality and inverse Young’s inequality we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M268">View MathML</a>

(4.6)

Then it follows from (4.5) and (4.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M269">View MathML</a>

(4.7)

By employing Gronwall’s lemma over interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M270">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M271">View MathML</a>, we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M272">View MathML</a>

(4.8)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M273">View MathML</a>. By the properties of the Ornstein-Uhlenbeck process, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M274">View MathML</a>

(4.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M275">View MathML</a>

(4.10)

Hence, given every fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M273">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M278">View MathML</a>, depending only on ω and ϱ, such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M271">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M281">View MathML</a>

(4.11)

which gives an expression for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M282">View MathML</a>. Replacing t by τ in (4.5) and integrating for τ over intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M283">View MathML</a>, then using (4.11) it yields that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M279">View MathML</a>,

(4.12)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M286">View MathML</a>

(4.13)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M287">View MathML</a>

Thus the right-hand side of (4.13) gives an expression for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M288">View MathML</a>. □

In the following, we shall obtain the regularity of the solution to stochastic p-Laplacian-type equation. This is the most challenging part in our discussion. Because of the nonlinearity of driven <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M289">View MathML</a> and function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M22">View MathML</a> in Equation (3.10), it seems difficult to derive the V-norm estimate as in [14], where the author only deals with a linear case, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M21">View MathML</a>. So we relax to estimate the solution in a weaker Sobolev <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M238">View MathML</a> with equivalent norms denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M293">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M294">View MathML</a>. Here, just as stated in the introduction, we use the properties of Dirichlet forms for the Laplacian Δ.

Lemma 4.2Suppose thatgsatisfies (3.4)-(3.6) andfis given in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95">View MathML</a>. Then there exists a random radius<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M296">View MathML</a>, such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M248">View MathML</a>there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M298">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M250">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M251">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M252">View MathML</a>, the following inequality holds for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M16">View MathML</a>-a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M304">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M176">View MathML</a>is the solution to (3.10) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M257">View MathML</a>.

Proof Taking the inner product of (3.10) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M307">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M309">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M310">View MathML</a>

(4.14)

By the semigroup theory (see [14]) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M311">View MathML</a>

(4.15)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M312">View MathML</a>, the domain of Laplacian Δ. We now estimate all terms on the right-hand side of (4.14). Employing (4.15) and integrating by parts, it yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M313">View MathML</a>

(4.16)

where we use the contraction property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M314">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M91">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M316">View MathML</a>

(4.17)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M317">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M244">View MathML</a>. By our assumption (3.5), along with (4.17) for q, the second term on the right-hand side of (4.14) is estimated as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M319">View MathML</a>

(4.18)

where we employ Young’s inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M320">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M321">View MathML</a> twice. But, by Sobolev’s inequality and Young’s inequality, it yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M322">View MathML</a>

(4.19)

and by (4.19) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M323">View MathML</a>

(4.20)

Then by (4.18)-(4.20), there exist positive constants c such that

(4.21)

For the third term on the right-hand side of (4.14), by (4.17) we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M325">View MathML</a>

(4.22)

On the other hand, by (4.15) and the Dirichlet forms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M119">View MathML</a> (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M327">View MathML</a>

(4.23)

Then it follows from (4.14), (4.16) and (4.21)-(4.23) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M328">View MathML</a>

(4.24)

where

and c is a positive constant independent of λ. So taking limit on both sides of (4.24) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M330">View MathML</a> and associating with (2.2) and (2.4), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M331">View MathML</a>

(4.25)

Replacing t by τ in (4.25) and integrating τ from s to t (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M332">View MathML</a>), it yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M333">View MathML</a>

(4.26)

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M334">View MathML</a>

Then by Lemma 4.1, (4.26) reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M335">View MathML</a>

(4.27)

Integrating (4.27) for s over intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M283">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M337">View MathML</a>

(4.28)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M271">View MathML</a>. By Poincare’s inequality and Young’s inequality, there exists positive constant c such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M339">View MathML</a>

(4.29)

Hence by using Lemma 4.1 again, along with (4.29), it follows from (4.28) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M340">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M341">View MathML</a>, which gives an expression for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M342">View MathML</a>. This completes the proof. □

By Theorem 2.3 and Lemma 4.2, we have obtained our main result in this section.

Theorem 4.3Assume thatgsatisfies (3.4)-(3.6) andfis given in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95">View MathML</a>. Then the RDS<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M344">View MathML</a>generated by System (3.10)-(3.12) possesses a random attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M345">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M346">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M347">View MathML</a>denotes all the bounded subsets ofHand the closure is theH-norm.

5 The single point attractor

In this section, we consider a special case, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M348">View MathML</a> in (3.6), in which case we find that the random attractor is just composed of a single point. This shows that System (3.10)-(3.12) possesses an unique stationary solution for every given initial value in the space H. We begin with a lemma.

Lemma 5.1Assume thatgsatisfies (3.4)-(3.6) andfis given in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M348">View MathML</a>. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M351">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M352">View MathML</a>, there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M353">View MathML</a>such that

In particular, for each fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>there exists a single point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M357">View MathML</a>inHsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M358">View MathML</a>

(5.1)

for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M359">View MathML</a>belonging to the bounded subsetBofH. Furthermore, the convergence in (5.1) is uniform with respect to all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M360">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M361">View MathML</a> be the solutions to (3.10) with initial values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M363">View MathML</a>. Then we can deduce from (3.14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M364">View MathML</a>

(5.2)

Multiplying (5.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M365">View MathML</a>, integrating over D and using (3.16), we find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M366">View MathML</a>

(5.3)

Now, applying Gronwall’s lemma to (5.3) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M367">View MathML</a> to t, it yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M368">View MathML</a>

(5.4)

We then estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M369">View MathML</a>. By (4.5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M370">View MathML</a>

(5.5)

By the Sobolev’s embedding inequality and the inverse Young’s inequality, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M371">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M372">View MathML</a>

(5.6)

So by (5.5) and (5.6) we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M373">View MathML</a>

(5.7)

Using Gronwall’s lemma to (5.7) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M374">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M367">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M376">View MathML</a>, we get that

(5.8)

Similar to the argument of (4.10), we know that the integral in the last term on the right-hand side of (5.8) is convergent. Hence, it follows from (5.8) and (5.4) that for every fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39">View MathML</a>,

(5.9)

So for every bounded subset B of H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M380">View MathML</a>, it follows from (5.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M381">View MathML</a>

(5.10)

since by the properties of the Ornstein-Uhlenbeck process, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M382">View MathML</a>

(5.11)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M383">View MathML</a>

(5.12)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M384">View MathML</a>

(5.13)

Moreover, the convergence in (5.11)-(5.13) is uniform with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M385">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M386">View MathML</a> belonging to every bounded subset of H. Then (5.10) implies that for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M388">View MathML</a> is a Cauchy sequence in H with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M389">View MathML</a>. Therefore, by the completeness of H, for every fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M39">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M392">View MathML</a> has a limit in H denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M357">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M394">View MathML</a>

 □

Theorem 5.2Assume thatgsatisfies (3.4)-(3.6) andfis given in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M348">View MathML</a>. Then the RDS<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M45">View MathML</a>generated by the solution to (3.10)-(3.12) possesses a single point attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62">View MathML</a>, i.e., there exists a single point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M399">View MathML</a>inHsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M400">View MathML</a>

Proof Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M401">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M402">View MathML</a> is a stochastic flow associated with System (3.10)-(3.12) and the RDS <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M403">View MathML</a>. By Lemma 5.1 we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M404">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M257">View MathML</a>. Then we need to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M406">View MathML</a> is a compact attractor. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M407">View MathML</a> is a compact random set. Hence by Definition 2.2 it suffices to prove the invariance and attracting property for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M407">View MathML</a>. Since by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M45">View MathML</a> and the flow properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M410">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M411">View MathML</a>

That is to say, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M412">View MathML</a>. On the other hand, by the uniform convergence of (5.1), it follows from (3.22) that for every bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M413">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M414">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M415">View MathML</a>. This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/61/mathml/M62">View MathML</a> is an attracting set, and thus we complete the proof. □

Competing interests

The author declares that he has no competing interests.

Author’s contributions

ZW carried out all studies in this article.

Acknowledgements

The author is indebted to the referee for giving some valuable suggestions that improved the presentations of this article. This work was supported by the China NSF Grant (no. 10871217), the Science and Technology Funds of Chongqing Educational Commission (no. KJ120703), the Fundamental Funds of the Central Universities (no. XDJK2009C100) and the Doctor Funds of Southwest University (no. SWU111068).

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