# A gradient flow generated by a nonlocal model of a neural field in an unbounded domain

## TL;DR

This paper analyzes a nonlocal neural field model in an unbounded domain, establishing the existence of a global attractor, a Lyapunov function, and demonstrating the flow's properties in weighted function spaces.

## Contribution

It introduces a framework for the nonlocal neural field equation in unbounded domains, proving the existence of attractors and a Lyapunov function, which were not previously established.

## Key findings

- Existence of a continuous flow in specified function spaces.
- Presence of a global compact attractor under certain conditions.
- Construction of a Lyapunov function characterizing the attractor.

## Abstract

In this paper we consider the non local evolution equation $$ \frac{\partial u(x,t)}{\partial t} + u(x,t)= \int_{\mathbb{R}^{N}}J(x-y)f(u(y,t))\rho(y)dy+ h(x). %\,\,\, h \geq 0. $$ We show that this equation defines a continuous flow in both the space $C_{b}(\mathbb{R}^{N})$ of bounded continuous functions and the space $C_{\rho}(\mathbb{R}^{N})$ of continuous functions $u$ such that $u \cdot \rho$ is bounded, where $\rho $ is a convenient "weight function"'. We show the existence of an absorbing ball for the flow in $C_{b}(\mathbb{R}^{N})$ and the existence of a global compact attractor for the flow in $C_{\rho}(\mathbb{R}^{N})$, under additional conditions on the nonlinearity. We then exhibit a continuous Lyapunov function which is well defined in the whole phase space and continuous in the $C_{\rho}(\mathbb{R}^{N})$ topology, allowing the characterization of the attractor as the unstable set of the equilibrium point set. We also illustrate our result with a concrete example.