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.J. A. Cañizo was supported by the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF/FEDER), project MTM2014-52056-P. A. Molino was partially supported by MINECO - FEDER GrantMTM2015-68210-P(Spain), Junta de Andalucía FQM-116 (Spain) andMINECO Grant BES-2013-066595 (Spain)