# Dissipative property for a class of non local evolution equations

## TL;DR

This paper studies a class of non-local evolution equations, proving well-posedness, smoothness of solutions, existence of a global attractor, and establishing a Lyapunov functional indicating gradient flow behavior.

## Contribution

It introduces a new analysis of non-local evolution equations with a Lyapunov functional and demonstrates the gradient property of the flow.

## Key findings

- Solutions are well-posed and smooth with respect to initial data.
- Existence of a global attractor for the evolution problem.
- Identification of a Lyapunov functional indicating gradient flow.

## Abstract

In this work we consider the non local evolution problem \[ \begin{cases} \partial_t u(x,t)=-u(x,t)+g(\beta K(f\circ u)(x,t)+\beta h), ~x \in\Omega, ~t\in[0,\infty[;\\ u(x,t)=0, ~x\in\mathbb{R}^N\setminus\Omega, ~t\in[0,\infty[;\\ u(x,0)=u_0(x),~x\in\mathbb{R}^N, \end{cases} \] where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N, ~g,f: \mathbb{R}\to\mathbb{R}$ satisfying certain growing condition and $K$ is an integral operator with symmetric kernel, $ Kv(x)=\int_{\mathbb{R}^{N}}J(x,y)v(y)dy.$ We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Futhermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has a gradient property.