Dynamics of stochastic nonclassical diffusion equations on unbounded domains

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.

Keywords

• Stochastic nonclassical diffusion equation
• random attractor
• asymptotic compactness
• weak continuity
• upper semi-continuity