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

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We consider the Fisher–KPP equation with a non-local interaction term. In [16], Hamel and Ryzhik showed that in solutions of this equation, the front location at a large time t is 2t+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 −[Formula presented]log⁡t+o(log⁡t), which is the same correction as found by Bramson in [7] 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 [7], although some of the ideas used are similar.

Original languageEnglish
Pages (from-to)3259-3302
Number of pages44
JournalJournal of Functional Analysis
Issue number12
Early online date10 Oct 2018
Publication statusPublished - 15 Dec 2018


  • Feynman–Kac formula
  • Fisher–KPP equation
  • Non-local equation
  • Spreading speed

ASJC Scopus subject areas

  • Analysis


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