Continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system on RN

In this article, we use the so-called difference estimate method to investigate the continuity and random dynamics of the non-autonomous stochastic FitzHugh–Nagumo system with a general nonlinearity. Firstly, under weak assumptions on the noise coefficient, we prove the existence of a pullback attractor in $L^2\left({\mathbb{R}}^N\right)\times L^2\left({\mathbb{R}}^N\right)$ by using the tail estimate method and a certain compact embedding on bounded domains. Secondly, although the difference of the first component of solutions possesses at most $p$-times integrability where $p$ is the growth exponent of the nonlinearity, we overcome the absence of higher-order integrability and establish the continuity of solutions in $(L^p\left({\mathbb{R}}^N\right)\cap H^1\left({\mathbb{R}}^N\right))\times L^2\left({\mathbb{R}}^N\right)$ with respect to the initial values belonging to $L^2\left({\mathbb{R}}^N\right)\times L^2\left({\mathbb{R}}^N\right)$. As an application of the result on the continuity, the existence of a pullback attractor in $(L^p\left({\mathbb{R}}^N\right)\cap H^1\left({\mathbb{R}}^N\right))\times L^2\left({\mathbb{R}}^N\right)$ is proved for arbitrary $N\ge 1$ and $p>2$.