Random attractors for partly dissipative stochastic lattice dynamical systems1

Abstract

In this paper, we consider the long term behavior for the stochastic lattice dynamical systems with some partly dissipative nonlinear term in l 2 × l 2. 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 × l 2.