A short proof of the logarithmic Bramson correction in Fisher-KPP equations

François Hamel; James Nolen; Jean-Michel Roquejoffre; Lenya Ryzhik; François Hamel; James Nolen; Jean-Michel Roquejoffre; Lenya Ryzhik

doi:10.3934/nhm.2013.8.275

Networks and Heterogeneous Media

2013, Volume 8, Issue 1: 275-289. doi: 10.3934/nhm.2013.8.275

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A short proof of the logarithmic Bramson correction in Fisher-KPP equations

1.
Université d'Aix-Marseille, LATP, 39 rue F. Joliot-Curie, 13453 Marseille Cedex 13
2.
Department of Mathematics, Duke University, Box 90320, Durham, NC, 27708-0320
3.
Institut de Mathématiques, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse Cedex 4
4.
Department of Mathematics, Stanford University, Stanford, CA 94305

Received: 01 May 2012 Revised: 01 November 2012
Primary: 35K57, 35C07; Secondary: 35B40.

In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions $u(t,x)$ of Fisher-KPP reaction-diffusion equations in $\mathbb{R}$, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of $u$ to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions $u$ along their level sets to the profile of the minimal travelling front.
- Fisher-KPP equations,
- travelling fronts,
- logarithmic correction
Citation: François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations[J]. Networks and Heterogeneous Media, 2013, 8(1): 275-289. doi: 10.3934/nhm.2013.8.275

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Abstract

In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay of the position of the solutions $u(t,x)$ of Fisher-KPP reaction-diffusion equations in $\mathbb{R}$, with respect to the position of the travelling front with minimal speed. Our proof is based on the comparison of $u$ to the solutions of linearized equations with Dirichlet boundary conditions at the position of the minimal front, with and without the logarithmic delay. Our analysis also yields the large-time convergence of the solutions $u$ along their level sets to the profile of the minimal travelling front.

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