The spreading speed of solutions of the non-local Fisher-KPP equation

2017/08/26 by Penington, Sarah
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1708.07965

Abstract

We consider the Fisher-KPP equation with a non-local interaction term. Hamel and Ryzhik showed that in solutions of this equation, the front location at a large time t is √ 2 t +o(t). We study the asymptotics of the second order term in the front location. If the interaction kernel ϕ(x) decays sufficiently fast as x→ ∞ then this term is given by -(3)/(2√ 2 )log t +o(log t), which is the same correction as found by Bramson for the local Fisher-KPP equation. However, if ϕ has a heavier tail then the second order term is -t^β+o(1), where β∈ (0,1) depends on the tail of ϕ. The proofs are probabilistic, using a Feynman-Kac formula. Since solutions of the non-local Fisher-KPP equation do not obey the maximum principle, the proofs differ from those in Bramson's work, although some of the ideas used are similar.

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