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

Pengyu Chen; Xuping Zhang

doi:10.3934/dcdsb.2020290

Article Contents

2021, Volume 26, Issue 8: 4325-4357. Doi: 10.3934/dcdsb.2020290

This issue Previous Article On initial value and terminal value problems for subdiffusive stochastic Rayleigh-Stokes equation Next Article Density function analysis for a stochastic SEIS epidemic model with non-degenerate diffusion

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

Pengyu Chen^, and
Xuping Zhang

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

^* Corresponding author: Pengyu Chen
^* Corresponding author: Pengyu Chen

Received: June 30, 2020

Early access: October 2020

Published: August 2021

Chen was supported by the National Natural Science Foundation of China (No. 12061063), Project of NWNU-LKQN2019-3 and China Scholarship Council (No. 201908625016). Zhang was supported by Science Research Project for Colleges and Universities of Gansu Province (No. 2019B-047), Project of NWNU-LKQN2019-13 and Doctoral Research Fund of Northwest Normal University (No. 6014/0002020209)

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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 $ H^\alpha(\mathbb{R}^n) $ with $ \alpha\in (0,1) $ as well as their time derivatives in $ L^2(\mathbb{R}^n) $. Finally, we establish the upper semi-continuity of the random attractors when noise intensity and time delay approaches zero, respectively.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.

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