Communications in Mathematical Sciences

Volume 20 (2022)

Number 2

The Bramson correction for integro-differential Fisher–KPP equations

Pages: 563 – 596



Cole Graham (Division of Applied Mathematics, Brown University, Providence, Rhode Island, U.S.A.)


We consider integro-differential Fisher–KPP equations with nonlocal diffusion. For typical equations, we establish the logarithmic Bramson delay for solutions with step-like initial data. That is, these solutions resemble a front at position $c_\ast t - \frac{3}{2\lambda_\ast} \operatorname{log} t + \mathcal{O}(1)$ for explicit constants $c_\ast$ and $\lambda_\ast$. Certain strongly asymmetric diffusions exhibit more exotic behaviour.


reaction-diffusion, integro-differential equation, Fisher–KPP, asymptotics

2010 Mathematics Subject Classification

35B40, 35K57, 60J80

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Received 10 May 2020

Received revised 23 August 2021

Accepted 24 August 2021

Published 28 January 2022