Optimal Estimates on the Propagation of Reactions with Fractional Diffusion

TL;DR

This paper derives optimal bounds on the spread of reaction-diffusion fronts governed by fractional diffusion equations, especially in scenarios lacking traveling wave solutions, extending understanding of such systems.

Contribution

It provides the first optimal bounds for front propagation in reaction-fractional-diffusion equations without traveling fronts, covering most cases and initial data types.

Findings

Established optimal bounds on front propagation

Extended results to cases without traveling fronts

Applicable to localized initial data propagation

Abstract

We study the reaction-fractional-diffusion equation $u_t+(-\Delta)^{s} u=f(u)$ with ignition and monostable reactions $f$, and $s\in(0,1)$. We obtain the first optimal bounds on the propagation of front-like solutions in the cases where no traveling fronts exist. Our results cover most of these cases, and also apply to propagation from localized initial data.