Transition Between Linear and Exponential Propagation in Fisher-KPP Type Reaction-Diffusion Equations

We study the Fisher-KPP equation with a fractional Laplacian of order α ∈ (0, 1). We know that the stable state invades the unstable one at constant speed for α = 1, and at an exponential in time velocity for α ∈ (0, 1). The transition between these two different speeds is examined in this paper. We prove that during a time of the order -ln (1 ? α), the propagation is linear, and becomes exponential as soon as the time exceeds a large multiple of -ln (1 ? α).