Improved energy methods for nonlocal diffusion problems
José A. Cañizo, Alexis Molino · Discrete and Continuous Dynamical Systems · 2018
We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations:\begin{document}\begin{equation*} Lu (x) := \int_{\mathbb{R}^N} K(x, y) (u(y) -u(x)) \,\mathrm{d} y, \end{equation*}\end{document}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.