Electron. J. Diff. Equ., Vol. 2015 (2015), No. 282, pp. 1-22.

Dynamics of stochastic nonclassical diffusion equations on unbounded domains

Wenqiang Zhao, Shuzhi Song

Abstract:
This article concerns the dynamics of stochastic nonclassical diffusion equation on $\mathbb{R}^N$ perturbed by a $\epsilon$-random term, where $\epsilon\in(0,1]$ is the intension of noise. By using an energy approach, we prove the asymptotic compactness of the associated random dynamical system, and then the existence of random attractors in $H^1(\mathbb{R}^N)$. Finally, we show the upper semi-continuity of random attractors at $\epsilon=0$ in the sense of Hausdorff semi-metric in $H^1(\mathbb{R}^N)$, which implies that the obtained family of random attractors indexed by $\epsilon$ converge to a deterministic attractor as $\epsilon$ vanishes.

Submitted April 27, 2015. Published November 10, 2015.
Math Subject Classifications: 60H15, 35B40, 35B41.
Key Words: Stochastic nonclassical diffusion equation; random attractor; asymptotic compactness; weak continuity; upper semi-continuity.

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Wenqiang Zhao
School of Mathematics and Statistics
Chongqing Technology and Business University
Chongqing 400067, China
email: gshzhao@sina.com
Shuzhi Song
School of Mathematics and Statistics
Chongqing Technology and Business University
Chongqing 400067, China
email: 13718903@qq.com

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