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\((L^2,H^1)\)-random attractors for stochastic reaction-diffusion equation on unbounded domains. (English) Zbl 1343.37080

Summary: We study the random dynamical system generated by a stochastic reaction-diffusion equation with additive noise on the whole space \(\mathbb R^n\) and prove the existence of an \((L^2,H^1)\)-random attractor for such a random dynamical system. The nonlinearity \(f\) is supposed to satisfy the growth of arbitrary order \(p-1(p\geq 2)\). The \((L^2,H^1)\)-asymptotic compactness of the random dynamical system is obtained by using an extended version of the tail estimate method introduced by B. Wang [Physica D 128, No. 1, 41–52 (1999; Zbl 0953.35022)] and the cut-off technique.

MSC:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
37L55 Infinite-dimensional random dynamical systems; stochastic equations
35K57 Reaction-diffusion equations
35R60 PDEs with randomness, stochastic partial differential equations

Citations:

Zbl 0953.35022
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Full Text: DOI

References:

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