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arXiv · 0812.4876

Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model

Adrien Blanchet·Jean Dolbeault·Miguel Escobedo·Javier Fernández
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Abstract

The Keller-Segel system describes the collective motion of cells that 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. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less $8\pi $, a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables.

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Adrien Blanchet, Jean Dolbeault, Miguel Escobedo, Javier Fernández. 2008-12-29. Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model. https://doi.org/10.1016/j.jmaa.2009.07.034

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