no. 4
Partial differential equations
p-Laplacian Keller–Segel equation: Fair competition and diffusion-dominated cases
[Équation d'agrégation et diffusion avec un p-Laplacien : cas de la compétition équitable et de la diffusion dominante]

Présenté par : Benoît Perthame

Lafleche, Laurent1, 2 ; Salem, Samir1
1 CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, place du Maréchal-de-Lattre-de-Tassigny, 75775 Paris cedex 16, France
2 CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau cedex, France
Comptes Rendus. Mathématique, Tome 357 (2019) no. 4, pp. 360-365.

Ce travail concerne l'étude d'une famille d'équations d'agrégation diffusion

∂tρ=Δpρ+λdiv((Ka⁎ρ)ρ),
où Ka(x)=x|x|a est un champ d'attraction et Δp est le p-Laplacien. On démontre que le domaine a<p(d+1)−2d est sous-critique du point de vue de la compétition entre l'agrégation et la diffusion en montrant l'existence d'une solution, quelle que soit la masse. Dans le cas critique, on montre l'existence d'une solution dans un régime de petite masse pour une condition Lln⁡L.

This work deals with the aggregation diffusion equation

∂tρ=Δpρ+λdiv((Ka⁎ρ)ρ),
where Ka(x)=x|x|a is an attraction kernel and Δp is the so called p-Laplacian. We show that the domain a<p(d+1)−2d is subcritical with respect to the competition between the aggregation and diffusion by proving the existence of a solution unconditionally with respect to the mass. In the critical case, we show the existence of a solution in a small mass regime for an Lln⁡L initial condition.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.03.002
Lafleche, Laurent 1, 2 ; Salem, Samir 1

1 CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, place du Maréchal-de-Lattre-de-Tassigny, 75775 Paris cedex 16, France
2 CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau cedex, France 
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     title = {\protect\emph{p}-Laplacian {Keller{\textendash}Segel} equation: {Fair} competition and diffusion-dominated cases},
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Lafleche, Laurent; Salem, Samir. p-Laplacian Keller–Segel equation: Fair competition and diffusion-dominated cases. Comptes Rendus. Mathématique, Tome 357 (2019) no. 4, pp. 360-365. doi : 10.1016/j.crma.2019.03.002. http://www.numdam.org/articles/10.1016/j.crma.2019.03.002/

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