Lower Semicontinuity of Global Attractors for a Class of Evolution Equations of Neural Fields Type in a Bounded Domain

Abstract

In this work we consider the nonlocal evolution equation $$\displaystyle \frac{\partial u(w,t)}{\partial t}=-u(w,t)+ \int _{S^{1}}J(wz^{-1})f(u(z,t))\mathrm{d}z+ h, \quad h > 0,$$ which arises in models of neuronal activity, in \(L^{2}(S^{1})\), where \(S^{1}\) denotes the unit sphere. We obtain more interesting results on existence of global attractors and the associate Lypaunov functional than the already existing in the literature. Furthermore, we prove the result, not yet known in the literature, of lower semicontinuity of global attractors with respect to connectivity function J.