Optimal Estimates on the Propagation of Reactions with Fractional Diffusion

We study the reaction-fractional-diffusion equation u_t+(-Δ )^s u=f(u) with ignition and monostable reactions f , and s∈ (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.