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(R-N) x L-2(R-N) 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(R-N)boolean AND H-1(R-N))x L-2(R-N) with respect to the initial values belonging to L-2(R-N) x L-2 (R-N). As an application of the result on the continuity, the existence of a pullback attractor in (L-P(R-N)boolean AND H-1(R-N)) x L-2(R-N) is proved for arbitrary N >= 1 and p > 2. (C) 2018 Elsevier Ltd. All rights reserved.