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: $Lu(x):={\int }_{{\mathbb{R}}^N}K(x,y)(u(y)-u(x))dy$, where $L$ acts on a real function $u$ defined on ${\mathbb{R}}^N$, 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 ${\partial }_{t}u=Lu$ 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 (particularly, and surprisingly, in dimensions $N=1,2$).

References in corpus (2)

• Nonlocal heat equations: decay estimates and Nash inequalities
• A Rosenau-type approach to the approximation of the linear Fokker--Planck equation

Cited by in corpus (1)

• Parabolic equations with natural growth approximated by nonlocal equations