Wong-Zakai approximations and attractors for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains

Abstract

In this paper, we study the Wong-Zakai approximations and long term behavior of the stochastic FitzHugh-Nagumo system driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation system. Then we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic FitzHugh-Nagumo system for both additive and multiplicative noise on unbounded domains. The tail estimates and decomposition method are employed to derive the pullback asymptotic compactness of solutions in order to overcome the obstacles caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the absence of regularity of one component of the solutions.

Keywords:

• Wong-Zakai approximation
• random attractor
• FitzHugh-Nagumo system
• tail estimate
• upper semicontinuity

Acknowledgement

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).