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Random attractors for partly dissipative stochastic lattice dynamical systems. (English) Zbl 1143.37050

Summary: We consider the long term behavior for the stochastic lattice dynamical systems with some partly dissipative nonlinear term in \(l^{2}\times l^{2}\). 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 the 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 \(l^{2}\times l^{2}\).

MSC:

37L55 Infinite-dimensional random dynamical systems; stochastic equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37H10 Generation, random and stochastic difference and differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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