Lower semicontinuity of global attractors for a class of evolution equations type neural fields in a bounded domain

TL;DR

This paper studies the stability and continuity properties of global attractors for a class of nonlocal evolution equations modeling neural activity, establishing new results on their existence, Lyapunov functionals, and lower semicontinuity.

Contribution

It introduces stronger existence results for global attractors, constructs Lyapunov functionals, and proves lower semicontinuity of attractors with respect to the connectivity function J.

Findings

Existence of global attractors is established with improved results.

A Lyapunov functional for the system is constructed.

Global attractors are shown to be lower semicontinuous with respect to J.

Abstract

In this work we consider the nonlocal evolution equation $$ \frac{\partial u(w,t)}{\partial t}=-u(w,t)+ \int_{S^{1}}J(wz^{-1})f(u(z,t))dz+ h, \,\,\, h > 0 $$ which arises in models of neuronal activity, in $L^{2}(S^{1})$, where $S^{1}$ denotes the unit sphere. We obtain stronger results on existence of global attractors and 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 $J$.