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

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Abstract

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.

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Authors and Affiliations

  1. Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

    Wenqiang Zhao

  2. Chongqing Key Laboratory of Social Economy and Applied Statistics, School of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Yijin Zhang

Authors
  1. Wenqiang Zhao
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  2. Yijin Zhang
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Correspondence to Wenqiang Zhao.

Additional information

This work was supported by CTBU (KFJJ2018101), CTBU ZDPTTD201909, Chongqing NSF (2019jcyj-msxmX0115) and NSFC (11871122).

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Zhao, W., Zhang, Y. Upper semi-continuity of random attractors for a non-autonomous dynamical system with a weak convergence condition. Acta Math Sci 40, 921–933 (2020). https://doi.org/10.1007/s10473-020-0403-3

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  • DOI: https://doi.org/10.1007/s10473-020-0403-3

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