Rate of propagation for the Fisher-KPP equation with nonlocal diffusion and free boundaries

Abstract

. In this paper, we obtain sharp estimates for the rate of propagation of the Fisher-KPP equation with nonlocal diffusion and free boundaries. The nonlocal diffusion operator is given by (cid:82) R J ( x − y ) u ( t, y ) dy − u ( t, x ), and our estimates hold for some typical classes of kernel functions J ( x ). For example, if for | x | (cid:29) 1 the kernel function satisfies J ( x ) ∼ | x | − γ with γ > 1, then it follows from [17] that there is a finite spreading speed when γ > 2, namely the free boundary x = h ( t ) satisfies lim t →∞ h ( t ) /t = c 0 for some uniquely determined positive constant c 0 depending on J , and when γ ∈ (1 , 2], lim t →∞ h ( t ) /t = ∞ ; the estimates in the current paper imply that, for t (cid:29) 1, c 0 t − h ( t ) ∼   1 when γ > 3 ln t when γ = 3 , t 3 − γ when γ ∈ (2 , 3) , and h ( t ) ∼ (cid:26) t ln t when γ = 2 , t 1 / ( γ − 1) when γ ∈ (1 , 2) . Our approach is based on subtle integral estimates and constructions of upper and lower solutions, which rely crucially on guessing correctly the order of growth of the term to be estimated. The techniques developed here lay the ground for extensions to more general situations.