Limiting dynamics of stochastic heat equations with memory on thin domains

Abstract

This paper is concerned with the limiting behavior 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. We also establish the upper-semicontinuity of these attractors when a family of $(n+1)$-dimensional thin domains collapses onto an n-dimensional domain. The main difficulty of this paper is the non-compactness of the generated RDS based on the fact that the memory term includes the whole past history of the phenomenon. To solve this, a splitting method is employed to prove the asymptotic compactness.

Keywords:

• Thin domain
• stochastic heat equation with memory
• random pullback attractor
• additive noise
• upper semicontinuity

2010 Mathematics Subject Classification:

• 60H15

Acknowledgments

The authors would like to thank the reviewers for their helpful comments.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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), the funding of V.C. & V.R. Key Lab of Sichuan Province.