Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises

Abstract

In this article, we study the dynamical behaviour of solutions of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ with both multiplicative noises and non-autonomous forces, where the nonlinearity is a polynomial-like growth function of arbitrary order. An asymptotic smoothing effect of this system is demonstrated, namely, that the random pullback attractor in the initial space $L^2(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$ is actually a compact, measurable and attracting set in $H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$. A difference estimates method, rather than the usual truncation estimate and spectrum decomposition technique, is employed to overcome the lack of Sobolev compact embedding in $H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$, despite of the loss of the high-order integrability of the difference of solutions for this system.