The Fisher-KPP equation with nonlinear fractional diffusion

TL;DR

This paper investigates how solutions to nonlinear fractional reaction-diffusion equations propagate, revealing exponential speed of level set movement, contrasting with classical wave-like propagation in standard KPP models.

Contribution

It demonstrates exponential propagation in fractional diffusion equations with nonlinearities, extending understanding beyond classical reaction-diffusion models.

Findings

Level sets propagate exponentially fast in time.

Contrasts with traveling wave behavior in classical KPP equations.

Utilizes decay properties of self-similar solutions for proof.

Abstract

We study the propagation properties of nonnegative and bounded solutions of the class of reaction-diffusion equations with nonlinear fractional diffusion: $u_{t} + (-\Delta)^s (u^m)=f(u)$. For all $0<s<1$ and $m> m_c=(N-2s)_+/N $, we consider the solution of the initial-value problem with initial data having fast decay at infinity and prove that its level sets propagate exponentially fast in time, in contradiction to the traveling wave behaviour of the standard KPP case, which corresponds to putting $s=1$, $m=1$ and $f(u)=u(1-u)$. The proof of this fact uses as an essential ingredient the recently established decay properties of the self-similar solutions of the purely diffusive equation, $u_{t} + (-\Delta)^s u^m=0$.