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

Shu Ji; Li Linyan; Huang Xin; Zhang Jian

doi:10.3934/mcrf.2020044

Article Contents

2021, Volume 11, Issue 4: 715-737. Doi: 10.3934/mcrf.2020044 open access

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

1.
School of Mathematical Sciences and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, Sichuan 610068, China
2.
School of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan 610068, China
3.
Department of Basic Courses, Sichuan Vocational College of Finance and Economics, Chengdu, Sichuan 610101, China
4.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

^* Corresponding author: Jian Zhang
^* Corresponding author: Jian Zhang

Received: April 30, 2020

Revised: June 30, 2020

Early access: October 2020

Published: December 2021

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Abstract

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 $ \mathbb{R}^{3} $ 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.

Keywords:

Random attractor,
stochastic integro-differential equation with memory,
unbounded domains,
asymptotic compactness,
upper semicontinuity.

Mathematics Subject Classification: Primary: 37L55, 60H15; Secondary: 35Q56.

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