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

Francois Hamel (1) , James Nolen (2) , Jean-Michel Roquejoffre (3) , Lenya Ryzhik (4)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 Department of Mathematics [Durham]
3 IMT - Institut de Mathématiques de Toulouse UMR5219
4 Department of Mathematics [Stanford]

In this paper, we explain in simple PDE terms a famous result of Bramson about the loga- rithmic delay of the position of the solutions u(t, x) of Fisher-KPP reaction-diffusion equations in 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.

Domaines: Equations aux dérivées partielles [math.AP]

Francois Hamel, James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik. A short proof of the logarithmic Bramson correction in Fisher-KPP equations. Networks and Heterogeneous Media, 2013, 8 (1), pp.275-279. ⟨10.3934/nhm.2013.8.275⟩. ⟨hal-00815553⟩