Asymptotic behavior of fractional stochastic heat equations in materials with memory

Abstract

This paper deals with the asymptotic behavior of solutions for a fractional stochastic integro-differential equation driven by additive noise with $\alpha \in (0,1)$. We first apply the Galerkin method to prove the existence and uniqueness of solutions for the equation, then establish the existence and uniqueness of tempered pullback random attractors for the equation in an appropriate Hilbert space, which is different from previous works [Liu L, Caraballo T. Well-posedness and dynamics of a fractional stochastic integro-differential equation. Phys D. 2017;355:45–57].

COMMUNICATED BY:

• Ming Mei

Keywords:

• Random dynamical system
• random attractor
• fractional Laplacian
• integro-differential equation
• additive noise

2010 Mathematics Subject Classifications:

• 37L55
• 60H15

Acknowledgements

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

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China [grant numbers 11571245 and 11871138], the funding of V. C. & V. R. Key Laboratory of Sichuan Province.