Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions. (English) Zbl 1112.35023

Summary: 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.


35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35K15 Initial value problems for second-order parabolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
92C17 Cell movement (chemotaxis, etc.)
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