Comptes Rendus
no. 6
Partial differential equations
On the stability of the state 1 in the non-local Fisher–KPP equation in bounded domains

Presented by: Haïm Brézis

Camille Pouchol1, 2
1 Sorbonne Université, UPMC Université Paris-6, CNRS UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
2 INRIA Team Mamba, INRIA Paris, 2, rue Simone- Iff, CS 42112, 75589 Paris, France
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 644-647.

We consider the non-local Fisher–KPP equation on a bounded domain with Neumann boundary conditions. Thanks to a Lyapunov function, we prove that, under a general hypothesis on the kernel involved in the non-local term, the homogenous steady state 1 is globally asymptotically stable. This assumption happens to be linked to some conditions given in the literature, which ensure that travelling waves link 0 to 1.

Nous considérons l'équation de Fisher–KPP non locale en domaine borné, avec conditions de Neumann au bord. À l'aide d'une fonction de Lyapunov, nous montrons que, sous une hypothèse générale sur le noyau présent dans le terme non local, l'état stationnaire 1 est globalement asymptotiquement stable. Cette hypothèse se trouve être reliée à certaines conditions données dans la littérature, qui assurent que les fronts de propagation relient 0 et 1.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.016

Camille Pouchol 1, 2

1 Sorbonne Université, UPMC Université Paris-6, CNRS UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
2 INRIA Team Mamba, INRIA Paris, 2, rue Simone- Iff, CS 42112, 75589 Paris, France 
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Camille Pouchol. On the stability of the state 1 in the non-local Fisher–KPP equation in bounded domains. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 644-647. doi : 10.1016/j.crma.2018.04.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.016/

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