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**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.

### MSC:

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.) |