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dc.contributor.authorBlanchet, Adrien
dc.contributor.authorDolbeault, Jean
HAL ID: 87
ORCID: 0000-0003-4234-2298
dc.contributor.authorPerthame, Benoît
HAL ID: 739446
ORCID: 0000-0002-7091-1200
dc.subjectKeller-Segel modelen
dc.subjectIntermediate asymptotics
dc.subjectSelf-similar variables
dc.subjectTime-dependent rescaling
dc.subjectLarge time behavior
dc.subjectAubin-Lions compactness method
dc.subjectCritical Mass
dc.subjectlogarithmic Hardy-Littlewood-Sobolev inequality
dc.subjectEntropy method
dc.subjectFree energy
dc.subjectWeak solutions
dc.titleTwo-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutionsen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherCNRS - Ecole Normale Supérieure de Paris - ENS Paris;France
dc.contributor.editoruniversityotherINRIA - Ecole Nationale des Ponts et Chaussées;France
dc.description.abstractenThe Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial mass, there is global existence of solutions and for large initial mass blow-up occurs. In this paper we complete this picture and give a detailed proof of the existence of weak solutions below the critical mass, above which any solution blows-up in finite time in the whole euclidean space. Using hypercontractivity methods, we establish regularity results which allow us to prove an inequality relating the free energy and its time derivative. For a solution with sub-critical mass, this allows us to give for large times an ``intermediate asymptotics'' description of the vanishing. In self-similar coordinates, we actually prove a convergence result to a limiting self-similar solution which is not a simple reflect of the diffusion.en
dc.relation.isversionofjnlnameElectronic Journal of Differential Equations
dc.relation.isversionofjnlpublisherTexas State University
dc.subject.ddclabelProbabilités et mathématiques appliquéesen

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