A compactness tool for the analysis of nonlocal evolution equations

TL;DR

This paper introduces a new compactness criterion in Lebesgue spaces and applies it to analyze the asymptotic behavior of solutions to nonlocal convection diffusion equations, extending classical compactness results.

Contribution

It develops a novel compactness criterion inspired by the Aubin-Lions-Simon Lemma and applies it to nonlocal evolution equations for the first time.

Findings

Established a new compactness criterion in Lebesgue spaces.

Derived the first term in the asymptotic behavior of solutions.

Extended classical compactness results to nonlocal equations.

Abstract

In this paper we give a new compactness criterion in the Lebesgue spaces $L^p((0,T)\times \Omega)$ and use it to obtain the first term in the asymptotic behaviour of the solutions of a nonlocal convection diffusion equation. We use previous results of Bourgain, Brezis and Mironescu to give a new criterion in the spirit of the Aubin-Lions-Simon Lemma.