Fast Propagation for Fractional KPP Equations with Slowly Decaying Initial Conditions

In this paper we study the large-time behavior of solutions of one-dimensional fractional Fisher-KPP reaction-diffusion equations, when the initial condition is asymptotically frontlike and it decays at infinity more slowly than a power x−b, where b < 2α and α ∈ (0, 1) is the order of the fractional Laplacian. We prove that the level sets of the solutions move exponentially fast as time goes to infinity. Moreover, a quantitative estimate of motion of the level sets is obtained in terms of the decay of the initial condition.