# American Institute of Mathematical Sciences

December  2019, 39(12): 7265-7290. doi: 10.3934/dcds.2019303

## Sharp large time behaviour in $N$-dimensional Fisher-KPP equations

 1 Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse; CNRS, Université Toulouse Ⅲ, 118 route de Narbonne, 31062 Toulouse, France 2 Centre d'Analyse et de Mathématique Sociales; UMR 8557, Paris Sciences et Lettres; CNRS, EHESS, 54 Bv. Raspail, 75006 Paris, France 3 Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse; CNRS, INSA Toulouse, 135 av. Rangueil, 31077 Toulouse, France

* Corresponding author

Dedicated to L. Caffarelli, as a sign of friendship, admiration and respect

Received  February 2019 Revised  August 2019 Published  September 2019

Fund Project: The first and second authors are supported by the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 321186 - ReaDi - "Reaction-Diffusion Equations, Propagation and Modelling". The third author is supported by the ANR project NONLOCAL ANR-14-CE25-0013.

We study the large time behaviour of the Fisher-KPP equation
 $\partial_t u = \Delta u +u-u^2$
in spatial dimension
 $N$
, when the initial datum is compactly supported. We prove the existence of a Lipschitz function
 $s^\infty$
of the unit sphere, such that
 $u(t, x)$
approaches, as
 $t$
goes to infinity, the function
 $U_{c_*}\bigg(|x|-c_*t + \frac{N+2}{c_*} \mathrm{ln}t + s^\infty\Big(\frac{x}{|x|}\Big)\bigg),$
where
 $U_{c*}$
is the 1D travelling front with minimal speed
 $c_* = 2$
. This extends an earlier result of Gärtner.
Citation: Jean-Michel Roquejoffre, Luca Rossi, Violaine Roussier-Michon. Sharp large time behaviour in $N$-dimensional Fisher-KPP equations. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7265-7290. doi: 10.3934/dcds.2019303
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