Regularity of pullback attractors and equilibrium for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains

Abstract

A theory on bi-spatial random attractors developed recently by Li \emph{et al.} is extended to study stochastic Fitzhugh-Nagumo system driven by a non-autonomous term as well as a general multiplicative noise. By using the so-called notions of uniform absorption and uniformly pullback asymptotic compactness, it is showed that every generated random cocycle has a pullback attractor in $L^l({\mathbb{R}}^N)\times L^2({\mathbb{R}}^N)$ with $l\in (2,p]$, and the family of obtained attractors is upper semi-continuous at any intensity of noise. Moreover, if some additional conditions are added, then the system possesses a unique equilibrium and is attracted by a single point.