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We study the reaction-fractional-diffusion equation $u_t+(-\Delta)^{s}u=f(u)$ with ignition and monostable reactions $f$, and $s\in(0,1)$. We obtainthe first optimal bounds on the propagation of front-like solutions in thecases where no traveling fronts exist. Our results cover most of these cases,and also apply to propagation from localized initial data.

Optimal Estimates on the Propagation of Reactions with Fractional Diffusion