The Bramson delay in the non-local Fisher-KPP equation

Bouin, Emeric; Henderson, Christopher; Ryzhik, Lenya

doi:10.1016/j.anihpc.2019.07.001

Bouin, Emeric ¹ ; Henderson, Christopher ² ; Ryzhik, Lenya ³

¹ CEREMADE - Université Paris-Dauphine, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France
² Department of Mathematics, University of Arizona, Tucson, AZ 85721, United States of America
³ Department of Mathematics, Stanford University, Stanford, CA 94305, United States of America

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 51-77.

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Résumé
Abstract

Dans cet article, nous considérons l'équation de Fisher-KPP non locale, qui modélise la dynamique d'une population ou la force de compétition pour les ressources dépend de la distance entre les individus. Nous obtenons une asymptotique précise en temps long de la position d'une population qui est initialement localisée en espace. Selon la décroissance à l'infini du noyau de compétition, la position du front est soit $2 t - (3 / 2) \log t + O (1)$ , comme dans le cas de l'équation locale, soit $2 t - O (t^{β})$ , pour un $β \in (0, 1)$ calculé explicitement. Les outils les plus importants utilisés dans cet article sont une version locale en temps d'une inégalité de Harnack parabolique ainsi qu'une analyse fine du problème linéarisé avec une condition de bord de Dirichlet dynamique. Notre analyse donne aussi, pour tout $β \in (0, 1)$ , des exemples de non-linéarités de type Fisher-KPP pour lesquelles le front se trouve en $2 t - O (t^{β})$ .

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either $2 t - (3 / 2) \log t + O (1)$ , as in the local case, or $2 t - O (t^{β})$ for some explicit $β \in (0, 1)$ . Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any $β \in (0, 1)$ , examples of Fisher-KPP type non-linearities $f_{β}$ such that the front for the local Fisher-KPP equation with reaction term $f_{β}$ is at $2 t - O (t^{β})$ .

MR Zbl

DOI : 10.1016/j.anihpc.2019.07.001

Classification : 35K57, 35Q92, 45K05, 35C07
Mots-clés : Reaction-diffusion equations, Logarithmic delay, Parabolic Harnack inequality

Affiliations des auteurs :

Bouin, Emeric ¹ ; Henderson, Christopher ² ; Ryzhik, Lenya ³

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     title = {The {Bramson} delay in the non-local {Fisher-KPP} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {51--77},
     publisher = {Elsevier},
     volume = {37},
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     year = {2020},
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     zbl = {1436.35239},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2019.07.001/}
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Bouin, Emeric; Henderson, Christopher; Ryzhik, Lenya. The Bramson delay in the non-local Fisher-KPP equation. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 1, pp. 51-77. doi : 10.1016/j.anihpc.2019.07.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.07.001/

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