Comptes Rendus
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

Laurent Lafleche1, 2 ; Samir Salem1
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.

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
Laurent Lafleche 1, 2 ; Samir Salem 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},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {360--365},
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Laurent Lafleche; Samir Salem. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.03.002/

[1] G. Aronsson; L. Evans; Y. Wu Fast/slow diffusion and growing sandpiles, J. Differ. Equ., Volume 131 (1996) no. 2, pp. 304-335

[2] T. Aubin Problèmes isopérimétriques et espaces de Sobolev, J. Differ. Geom., Volume 11 (1976) no. 4, pp. 573-598

[3] A. Blanchet; J. Dolbeault; B. Perthame Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differ. Equ., Volume 32 (2006)

[4] V. Calvez; J.A. Carrillo; F. Hoffmann Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Anal., Volume 159 (2017), pp. 85-128

[5] L.C. Evans; M. Feldman Fast/slow diffusion and collapsing sandpiles, J. Differ. Equ., Volume 137 (1997) no. 1, pp. 166-209

[6] F.K.O. Hoffmann Keller–Segel-Type Models and Kinetic Equations for Interacting Particles: Long-Time Asymptotic Analysis, University of Cambridge, UK, July 2017 (PhD thesis)

[7] T. Horiuchi Some remarks on Kato's inequality, J. Inequal. Appl., Volume 6 (2001) no. 1, pp. 29-36

[8] L. Lafleche; S. Salem Fractional Keller–Segel equation: global well-posedness and finite time blow-up, Sept. 2018 | arXiv

[9] E.H. Lieb Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2), Volume 118 (1983) no. 2, pp. 349-374

[10] P. Lindqvist Notes on the p-Laplace Equation, Department of Mathematics and Statistics, University of Jyväskylä, Finland, 2006 (Report No. 102)

[11] J.-G. Liu; W. Cong A degenerate p-Laplacian Keller–Segel model, Kinet. Relat. Models, Volume 9 (2016) no. 4, pp. 687-714

[12] X. Liu; T. Horiuchi Remarks on Kato's inequality when Δpu is a measure, Math. J. Ibaraki Univ., Volume 48 (2016), pp. 45-61

[13] S. Salem Propagation of chaos for some 2 dimensional fractional Keller Segel equations in diffusion dominated and fair competition cases, Dec. 2017 | arXiv

[14] G. Talenti Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), Volume 110 (1976) no. 1, pp. 353-372

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