Improved energy methods for nonlocal diffusion problems

Abstract

We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations: L u ( x ) := ∫ R N K ( x , y ) ( u ( y ) − u ( x ) ) d y , where L acts on a real function u defined on R N , and we assume that K ( x , y ) is uniformly strictly positive in a neighbourhood of x = y . The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation ∂ t u = L u as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the L p norms of u and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases.