A Compactness Tool for the Analysis of Nonlocal Evolution Equations

In this paper we analyze the long time behavior of the solutions of a nonlocal diffusion-convection equation. 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 expansion of the solutions. Previous results of [J. Bourgain, H. Brezis, and P. Mironescu, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001] are used to obtain a compactness result in the spirit of the Aubin--Lions--Simon lemma.