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

TL;DR

This paper extends the theory of bi-spatial random attractors to the stochastic FitzHugh-Nagumo system with non-autonomous and multiplicative noise, establishing existence, upper semi-continuity, and conditions for unique equilibrium.

Contribution

It develops a new framework for analyzing pullback attractors in non-autonomous stochastic PDEs, specifically applying it to the FitzHugh-Nagumo system with broader noise considerations.

Findings

Existence of pullback attractors in specified function spaces.

Upper semi-continuity of attractors with respect to noise intensity.

Conditions ensuring the system has a unique equilibrium.

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.