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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.
Acknowledgements
The authors would like to thank the reviewers and the editor for their valuable suggestions and comments which have helped improve the presentation of the paper.
Notes
1. This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G, the National Natural Science Foundation of China under Grant 60774073, the Natural Science Foundation of Jiangsu Province of China under Grant BK2007075, the Natural Science Foundation of Jiangsu Education Committee of China under Grant 06KJD110206, the Scientific Innovation Fund of Yangzhou University of China under Grant 2006CXJ002, and the Alexander von Humboldt Foundation of Germany.