Abstract
We consider the Fisher–KPP equation with a nonlocal 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 nonlocal 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 

Pages (fromto)  32593302 
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
 Nonlocal equation
 Spreading speed
ASJC Scopus subject areas
 Analysis
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Sarah Penington
 Department of Mathematical Sciences  Royal Society Research Fellow (and Proleptic Reader)
 Probability Laboratory at Bath
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching, Researcher