Asymptotic behavior of stochastic p-Laplacian-type equation with multiplicative noise
© Zhao; licensee Springer 2012
Received: 28 December 2011
Accepted: 13 April 2012
Published: 22 June 2012
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 . 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.
Keywordsrandom dynamical systems stochastic p-Laplacian-type equation random attractors
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  obtained the global attractor for (1.1) with exterior forcing term , 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  in nonautonomous case. In random case, Zhao  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 satisfying some growth conditions. The multiplicative noise 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 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 with high order differentials and the forcing term with 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 , which is weaker than V. Here some basic results about the Laplacian are used. We refer to  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 .
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.
The basic notion in RDS is a metric dynamical system (MSD) , which is a probability space with a group , of measure preserving transformations of . MSD θ is said to be ergodic under if for any θ-invariant set we have either or , where the θ-invariant set is in the sense for and all .
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 , i.e., the smallest σ-algebra on X which contains all open subsets.
- (2)A continuous stochastic flow is a family of measurable mappings , , such that for -a.e. ,
A random compact set is a family of compact sets indexed by ω such that for every the mapping is measurable with respect to .
- (4)A random set is an attracting set if for every deterministic bounded subset and -a.e. ,
- (5)A random set is an absorbing set if for every deterministic bounded subset and -a.e. , there exists such that for all ,
Definition 2.2 A random compact set is called to be a random attractor for the RDS φ if is an attracting set and for and all .
Theorem 2.3 (see )
wheredenotes all the bounded subsets of X.
where the injections are continuous and each space is dense in the following one.
Moreover, for and , where is the domain of Δ.
3 Existence and uniqueness of RDS
Since , by our assumption (3.4)-(3.6) and , it is easy to check that for given , the operator mapping into is well defined, where .
Then is an ergodic MDS.
where the Ornstein-Uhlenbeck constant equals to 1.
Note that is a Gaussian process with mathematical expectation and variance , see , whereas . Furthermore, from [2, 15, 18], the random variable is continuous in t for -a.e. and grows sublinearly, i.e., .
where satisfies (3.4)-(3.6) and f is given in , .
Note that System (3.1)-(3.3) and System (3.10)-(3.12) are equivalent by (3.9). Let and 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 is continuous in H for the initial value in H, then the mapping is also continuous in H, vice verse.
We now show the existence and uniqueness of solution to System (3.1)-(3.6).
for alland-a.e. . Furthermore, the mappingfrom H into H is continuous for all.
Proof We first show that for every there exists a unique solution . By Theorem 4.2.4 and Exercise 4.1.2 in , it suffices to show that for every fixed possesses Hemi-continuity, Monotonicity, Coercivity, and Bounded-ness properties (for the definitions of these notions please refer to p.56 of ). But the proofs are an analogy of the corresponding works in . So we omit them here.
We then show that the solution is in . By our assumptions that and , we can check that maps to for . Thus if , then (3.14) implies that . Now by the general fact (see p.164 of ) it follows that v is almost everywhere equal to a function belonging to . Hence by the transformation (3.9) and the continuous property of Ornstein-Uhlenbeck process, is almost everywhere equal to a function belonging to .
We finally prove the continuity of the mapping from H into H. It suffices to prove that the mapping is continuous from H into H.
Then, the continuity of the mapping from H into H is followed from the contraction property (3.18). This finishes the total proofs of Theorem 3.1. □
with , then by Theorem 3.1 ψ is a continuous RDS associated with System (3.1)-(3.3).
That is to say, can be interpreted as the position of the trajectory at time 0, which was in at time −t (see ).
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 and . Note that in the following ; the results will hold for -a.e. and the generic constants c or , are independent of in the context, where .
whereis the solution to Equation (3.10) with.
Thus the right-hand side of (4.13) gives an expression for . □
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 and function in Equation (3.10), it seems difficult to derive the V-norm estimate as in , where the author only deals with a linear case, i.e., . So we relax to estimate the solution in a weaker Sobolev with equivalent norms denoted by for . Here, just as stated in the introduction, we use the properties of Dirichlet forms for the Laplacian Δ.
whereis the solution to (3.10) with.
with , which gives an expression for . This completes the proof. □
By Theorem 2.3 and Lemma 4.2, we have obtained our main result in this section.
wheredenotes all the bounded subsets of H and the closure is the H-norm.
5 The single point attractor
In this section, we consider a special case, that is, 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.
for everybelonging to the bounded subset B of H. Furthermore, the convergence in (5.1) is uniform with respect to all.
as . This shows that is an attracting set, and thus we complete the proof. □
ZW carried out all studies in this article.
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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