Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions




Blanchet, Adrien
Dolbeault, Jean
Perthame, Benoit

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Texas State University-San Marcos, Department of Mathematics


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.



Keller-Segel model, Existence, Weak solutions, Free energy, Entropy method, Logarithmic Hardy-Littlewood-Sobolev inequality, Critical mass, Aubin-Lions compactness method, Hypercontractivity, Large time behavior, Time-dependent rescaling, Self-similar variables, Intermediate asymptotics


Blanchet, 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.


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