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Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 629–646 | Cite as

Convergence to a single wave in the Fisher-KPP equation

  • James Nolen
  • Jean-Michel Roquejoffre
  • Lenya Ryzhik
Article

Abstract

The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t−(3/2)log t+x , the solution of the equation converges as t → +∞ to a translate of the traveling wave corresponding to the minimal speed c * = 2. The constant x depends on the initial condition u(0, x). The proof is elaborate, and based on probabilistic arguments. The purpose of this paper is to provide a simple proof based on PDE arguments.

Keywords

Traveling waves KPP Front propagation Asymptotic analysis Reaction-diffusion 

2000 MR Subject Classification

35K57 35C07 35B40 

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Notes

Acknowledgments

Lenya Ryzhik and Jean-Michel Roquejoffre thank the Labex CIMI for a PDE-probability quarter in Toulouse, in Winter 2014, out of which the idea of this paper grew and which provided a stimulating scientific environment for this project.

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Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • James Nolen
    • 1
  • Jean-Michel Roquejoffre
    • 2
  • Lenya Ryzhik
    • 3
  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Institut de Mathématiques (UMR CNRS 5219)Université Paul SabatierToulouse cedexFrance
  3. 3.Department of MathematicsStanford UniversityStanfordUSA

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