Blanchet, AdrienDolbeault, JeanPerthame, Benoit2021-07-162021-07-162006-04-06Blanchet, A., Dolbeault, J., & Perthame, B. (2006). Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions. <i>Electronic Journal of Differential Equations, 2006</i>(44), pp. 1-33.1072-6691https://hdl.handle.net/10877/13917The 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.Text33 pages1 file (.pdf)enAttribution 4.0 InternationalKeller-Segel modelExistenceWeak solutionsFree energyEntropy methodLogarithmic Hardy-Littlewood-Sobolev inequalityCritical massAubin-Lions compactness methodHypercontractivityLarge time behaviorTime-dependent rescalingSelf-similar variablesIntermediate asymptoticsArticle