Asymptotic behaviour of stochastic heat equations in materials with memory on thin domains

Abstract

This paper deals with the dynamical behaviour of a stochastic integro-differential equation driven by additive noise defined on thin domains. We prove the existence and uniqueness of random attractors for the equation in an $(n+1)$-dimensional narrow domain. Due to the fact that the memory term includes the whole past history of the phenomenon, we are not able to prove compactness of the generated RDS, but its asymptotic compactness can be proved by the splitting method.

Keywords:

• Random dynamical system
• thin domain
• stochastic heat equation with memory
• random attractor
• asymptotic compactness

2010 Mathematics Subject Classifications:

• 37L55
• 60H15

Acknowledgments

The authors would like to thank the reviewers for their helpful comments. This work was partially supported by the National Natural Science Foundation of China (No.11871138), joint research project of Laurent Mathematics Center of Sichuan Normal University and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification, the funding of V.C. & V.R. Key Lab of Sichuan Province.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China [grant number 11871138], joint research project of Laurent Mathematics Center of Sichuan Normal University and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification, the funding of V.C. & V.R. Key Lab of Sichuan Province.