### Abstract

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]logt+o(logt), 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 language | English |
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Pages (from-to) | 3259-3302 |

Number of pages | 44 |

Journal | Journal of Functional Analysis |

Volume | 275 |

Issue number | 12 |

Early online date | 10 Oct 2018 |

DOIs | |

Publication status | Published - 15 Dec 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis