Improved energy methods for nonlocal diffusion problems

TL;DR

This paper establishes a nonlocal Nash inequality for diffusion operators with positive kernels, enabling precise decay estimates for solutions of nonlocal diffusion equations, especially in low dimensions.

Contribution

It introduces a simplified proof of a nonlocal Nash inequality and extends decay results to dimensions one and two.

Findings

Derived a nonlocal Nash inequality analogous to the classical one.

Provided explicit decay rates for solutions in various L^p norms.

Extended decay results to low-dimensional cases N=1,2.

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 $\partial_t u = L u$ 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 (particularly, and surprisingly, in dimensions $N = 1, 2$).