Improved energy methods for nonlocal diffusion problems
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American Institute of Mathematical Sciences (AIMS)
Nonlocal diffusionEntropy methodsEnergy methodsAsymptotic behaviour
José A. Cañizo, Alexis Molino. Improved energy methods for nonlocal diffusion problems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1405-1425. doi: 10.3934/dcds.2018057.
SponsorshipJ. 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).
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.