Random Attractors For Partly Dissipative Stochastic Lattice Dynamical Systems

In this paper, we are concerned with the long time behavior of the following partly dissipative stochastic lattice differential equations given by with the initial data: ui (0)= ui,0, vi (0)= vi,0,i∈Z , (3) where Z denotes the integer set, u = (u i ) i∈Z ,v = ( vi )i∈Z∈l2,λ,σ,υ,α,βare positive constants, f i are nonlinear smooth functions satisfying some conditions, h = ( hi )i∈Z,g = ( g i )i∈Z,a = ( ai )i∈Z∈l 2 are given, { wi :i∈Z} are independent Brownian motions.The main purpose of this paper is to establish the existence of a compact global random attractor. The uniqueness and existence is first proved for the solution of an infinite dimensional random dynamical system, and a priori estimate is obtained on the solutions. The existence of a random absorbing set is then discussed for the systems, and an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of solutions. Finally, the global random attractor is proved to exist within the set of tempered random bounded sets rather than all bounded deterministic sets, i.e., the stochastic lattice system has a global random attractor in l 2×l2. In Section 1, the background and history about the related are given.In Section 2, we give some preliminary results that are necessary in this dissertation.Section 3 is devoted to the existence of an infinite-dimensional random lattice dynamical system generated by equations (1) ? (3) under the given assumptions. We will derive a priori estimate on the solutions to equations, and then prove the existence of random lattice dynamical system on one-dimensional lattice Z .In Section 4, we get our major result, i.e., the existence of a random global attractor of stochastic lattice dynamical system. The proof is mainly composed of two important results, one is the existence of an random absorbing set, and the other is the estimate on the asymptotic compactness of solutions.