Long-time Behaviour of the Non-autonomous Stochastic FitzHugh-Nagumo Systems on Thin Domains

Abstract

We study the long-time behavior of non-autonomous stochastic FitzHugh-Nagumo systems on thin domains. As the $(n+1)$-dimensional thin domains collapses onto an n-dimensional domain, an n-dimensional limiting FitzHugh-Nagumo system is derived. This n-dimensional limiting system encodes the defining geometry of the $(n+1)$-dimensional system. To justify this limiting process, we show that the pullback measure attractors of the FitzHugh-Nagumo systems on thin domains are upper semi-continuous as the height of thin direction tends to zero.