# American Institute of Mathematical Sciences

March  2013, 8(1): 275-289. doi: 10.3934/nhm.2013.8.275

## 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, France 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  May 2012 Revised  November 2012 Published  April 2013

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.
Citation: François Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Networks & Heterogeneous Media, 2013, 8 (1) : 275-289. doi: 10.3934/nhm.2013.8.275
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