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 L2(RN)×L2(RN) 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 (Lp(RN)∩H1(RN))×L2(RN) with respect to the initial values belonging to L2(RN)×L2(RN). As an application of the result on the continuity, the existence of a pullback attractor in (Lp(RN)∩H1(RN))×L2(RN) is proved for arbitrary N≥1 and p>2.