Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions
Blanchet, Adrien; Dolbeault, Jean; Perthame, Benoît (2006), Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electronic Journal of Differential Equations, 44, p. 1-33
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00021782/en/
Journal nameElectronic Journal of Differential Equations
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Abstract (EN)The 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.
Subjects / KeywordsKeller-Segel model; Intermediate asymptotics; Self-similar variables; Time-dependent rescaling; Large time behavior; Hypercontractivity; Aubin-Lions compactness method; Critical Mass; logarithmic Hardy-Littlewood-Sobolev inequality; Entropy method; Free energy; Weak solutions; Existence
Showing items related by title and author.
Fernandez, Javier; Escobedo, Miguel; Dolbeault, Jean; Blanchet, Adrien (2010) Article accepté pour publication ou publié