Upper semi-continuity of random attractors for a non-autonomous dynamical system with a weak convergence condition

• Published in: Acta mathematica scientia Vol. 40; no. 4; pp. 921 - 933
• Main Authors: Zhao, Wenqiang, Zhang, Yijin
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.07.2020; Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China%Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Science,Chongqing University of Posts and Telecommunications, Chongqing 400065, China
• Subjects: Analysis; Mathematics; Mathematics and Statistics; non-autonomous random dynamical system; upper semi-continuity; 35R60; random attractor; 35B41; 35B40; weak convergence
• ISSN: 0252-9602; 1572-9087
• DOI: 10.1007/s10473-020-0403-3

In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity. These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.