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

Electronic Journal of Differential Equations

关键词: Keller-Segel model;    existence;    weak solutions;    free energy;    entropy method;    logarithmic Hardy-Littlewood-Sobolevinequality;    critical mass;    Aubin-Lions compactness method;    hypercontractivity;    large time behavior;    time-dependent rescaling;    self-similar variables;    intermediate asymptotics.;   

DOI  :  

来源: DOAJ

【 摘 要 】

The Keller-Segel system describes the collective motion of cellswhich are attracted by a chemical substance and are able to emit it.In its simplest form it is a conservative drift-diffusion equationfor the cell density coupled to an elliptic equation for thechemo-attractant concentration. It is known that, in two spacedimensions, for small initial mass, there is global existence ofsolutions and for large initial mass blow-up occurs. In this paperwe complete this picture and give a detailed proof of the existenceof weak solutions below the critical mass, above which any solutionblows-up in finite time in the whole Euclidean space.Using hypercontractivity methods, we establish regularity resultswhich allow us to prove an inequality relating the free energy andits time derivative. For a solution with sub-critical mass, thisallows us to give for large times an ``intermediate asymptotics''description of the vanishing. In self-similar coordinates, weactually prove a convergence result to a limiting self-similarsolution which is not a simple reflect of the diffusion.

【 授权许可】

Unknown