Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay

Abstract

This paper is concerned with the asymptotic behavior of the solutions to a class of non-autonomous nonlocal fractional stochastic parabolic equations with delay defined on bounded domain. We first prove the existence of a continuous non-autonomous random dynamical system for the equations as well as the uniform estimates of solutions with respect to the delay time and noise intensity. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors by utilizing the Arzela-Ascoli theorem and the uniform estimates of solutions in fractional Sobolev space \begin{document}$ H^\alpha(\mathbb{R}^n) $\end{document} with \begin{document}$ \alpha\in (0,1) $\end{document} as well as their time derivatives in \begin{document}$ L^2(\mathbb{R}^n) $\end{document} . Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively.