A singular limit in a fractional reaction-diffusion equation with periodic coefficients

TL;DR

This paper analyzes the long-term behavior of a fractional reaction-diffusion equation with periodic coefficients, showing that the stable state invades the unstable one at an exponentially increasing speed over time.

Contribution

It provides an asymptotic analysis of a non-local Fisher-KPP equation with periodic media and fractional diffusion, revealing the exponential invasion speed.

Findings

Stable state invades at exponential speed

Asymptotic behavior characterized for fractional operators

Long time-long range scaling applied

Abstract

We provide an asymptotic analysis of a non-local Fisher-KPP type equation in periodic media and with a non-local stable operator of order $\alpha$ $\in$ (0, 2). We perform a long time-long range scaling in order to prove that the stable state invades the unstable state with a speed which is exponential in time.