The Fisher-KPP Equation with Nonlinear Fractional Diffusion

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 contrast to the traveling wave behavior 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, $u_{t} + (-\Delta)^s u^m=0$.