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

and Applied Analysis 3 introducing a new inner product over the resolvent of Laplacian, we surmount this obstacle and obtain the estimate of the solution in the Sobolev space V0, which is weaker than V , see Lemma 4.2 in Section 4. Here some basic results about Dirichlet forms of Laplacian are used. For details on the Dirichlet forms of a negative definite and self-adjoint operator please refer to 13 . The existence and uniqueness of solution, which ensure the existence of continuous RDS, are proved by employing the standard in 14 . If a restrictive assumption is imposed on the monotonicity coefficient in 1.6 we obtain a compact attractor consisting of a single point which attracts every deterministic bounded subset of H. The organization of this paper is as follows. In the next section, we present some notions and results on the theory of RDS and Dirichlet forms which are necessary to our discussion. In Section 3, we prove the existence and uniqueness of the solution to the pLaplacian-type equation with additive noise and obtain the corresponding RDS. In Section 4, we give some estimates for the solution satisfying 1.1 – 1.6 in given Hilbert space and then prove the existence of a random attractor for this RDS. In the last section, we prove the existence of the single point attractor under the given condition. 2. Preliminaries In this section, we first recall some notions and results concerning the random attractor and the random flow, which can be found in 5, 6 . For more systematic theory of RDS we refer to 15 . We then list the Sobolev spaces, Laplacian and its semigroup and Dirichlet forms. The basic notion in RDS is a measurable dynamical system MSD . The form Ω,F,P, θs is called a MSD if Ω,F,P is a complete probability space and {θs : Ω → Ω, s ∈ R} is a family of measure-preserving transformations such that s,w → θsw is measurable, θ0 id and θt s θtθs for all s, t ∈ R. A continuous RDS on a complete separable metric space X, d with Borel sigmaalgebra B X over MSD Ω,F,P, θs is by definition a measurable map φ : R ×Ω ×X −→ X, t,w, x −→ φ t,w x 2.1 such that P-a.s. w ∈ Ω i φ 0, w id on X, ii φ t s,w φ t, θsw φ s,w for all s, t ∈ R cocycle property , iii φ t,w : X → X is continuous for all t ∈ R . A continuous stochastic flow is by definition a family of measurable mapping S t, s;w : X → X,−∞ ≤ s ≤ t ≤ ∞, such that P-a.s. w ∈ Ω S t, r;w S r, s;w x S t, s;w x, x ∈ X, S t, s;w x S t − s, 0.; θsw x, x ∈ X, 2.2 for all s ≤ r ≤ t, and s → S t, s;w x is continuous in X for all s ≤ t and x ∈ X. A random compact set {K w }w∈Ω is a family of compact sets indexed by w such that for every x ∈ X the mapping w → d x,K w is measurable with respect to F. 4 Abstract and Applied Analysis Let A w be a random set. One says that A w is attracting if for P-a.s. w ∈ Ω and every deterministic bounded subset B ⊂ X lim t→∞ dist ( φ t, θ−tw B,A w ) 0, 2.3 where dist ·, · is defined by dist A,B supx∈Ainfy∈Bd x, y . We say thatA w absorbs B ⊂ X if P-a.s.w ∈ Ω, there exists tB w > 0 such that for all t ≥ tB w , φ t, θ−tw B ⊂ A w . 2.4 Definition 2.1. Recall that a random compact setw → A w is called to be a random attractor for the RDS φ if for P-a.s. w ∈ Ω i A w is invariant, that is, φ t,w A w A θtw , for all t ≥ 0; ii A w is attracting. Theorem 2.2 see 5 . Let φ t,w be a continuous RDS over a MDS Ω,F,P; θt with a separable Banach Space X. If there exists a compact random absorbing set K w absorbing every deterministic bounded subset of X, then φ possesses a random attractorA w defined by A w ⋃ B∈B X ⋂ s≥0 ⋃ t≥s φ t, θ−tw B, 2.5 where B X denotes all the bounded subsets of X. Let L D be the p-times integrable functions space on D with norm denoted by ‖ · ‖p, V D be the space consisting of infinitely continuously differential real-valued-functions with a compact support in D. We use V to denote the norm closure of V D in Sobolev space W2,p D , that is, V W 0 D . Since D is a bounded smooth domain in R , we can endow the Sobolev space V with equivalent norm see 1, page 166