Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains

Abstract

<p style='text-indent:20px;'>We consider the dynamical behavior of fractional stochastic integro-differential equations with additive noise on unbounded domains. The existence and uniqueness of tempered random attractors for the equation in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{3} $\end{document}</tex-math></inline-formula> are proved. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. The main difficulty is to show the pullback asymptotic compactness due to the lack of compactness on unbounded domains and the fact that the memory term includes the whole past history of the phenomenon. We establish such compactness by the tail-estimate method and the splitting method.</p>