Partial differential equations
p-Laplacian Keller–Segel equation: Fair competition and diffusion-dominated cases

Presented by: 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, Volume 357 (2019) no. 4, pp. 360-365.

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.

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.

Received:
Accepted:
Published online:
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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Lafleche, Laurent; Salem, Samir. p-Laplacian Keller–Segel equation: Fair competition and diffusion-dominated cases. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 360-365. doi : 10.1016/j.crma.2019.03.002. http://archive.numdam.org/articles/10.1016/j.crma.2019.03.002/

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