Fractional decay bounds for nonlocal zero order heat equations

TL;DR

This paper derives decay bounds for solutions to a nonlocal heat equation with kernels exhibiting polynomial tails, extending understanding of long-time behavior in such nonlocal diffusion models.

Contribution

It provides new decay rate estimates for solutions with kernels having polynomial tails, a case less studied than rapidly decaying kernels.

Findings

Decay rate bounds for solutions in L^q norms

Explicit polynomial decay estimates for large times

Applicable to kernels with polynomial tail lower bounds

Abstract

In this paper we obtain bounds for the decay rate for solutions to the nonlocal problem $\partial_t u(t,x) = \int_{\R^n} J(x,y)[u(t,y) - u(t,x)] dy$. Here we deal with bounded kernels $J$ but with polynomial tails, that is, we assume a lower bound of the form $J(x,y) \geq c_1|x-y|^{-(n + 2\sigma)}$, for $|x - y| > c_2$. Our estimates takes the form $\|u(t)\|_{L^q(\R^n)} \leq C t^{-\frac{n}{2\sigma} (1 - \frac{1}{q})}$ for $t$ large.