The fisher-kpp equation with nonlinear fractional diffusion

Abstract

Abstract.We study the propagation properties of nonnegative and bounded solutions of theclass of reaction-diffusion equations with nonlinear fractional diffusion:ut+(−Δ)s(um)=f(u). Forall 0<s<1andm>mc=(N−2s)+/N, we consider the solution of the initial-value problemwith initial data having fast decay at infinity and prove that its level sets propagate exponentiallyfast in time, in contrast to the traveling wave behavior of the standard KPP case, which correspondsto puttings=1,m=1,andf(u)=u(1−u). The proof of this fact uses as an essential ingredientthe recently established decay properties of the self-similar solutions of the purely diffusive equation,ut+(−Δ)sum=0