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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:361

Article

Blanchet, Adrien^1,2  Dolbeault, Jean³  Escobedo, Miguel⁴  Fernandez, Javier⁵ 

[1] Univ Toulouse, GREMAQ, CNRS, UMR 5604, F-31000 Toulouse, France

[2] Univ Toulouse, INRA 1291, F-31000 Toulouse, France

[3] Univ Paris 09, CNRS, CEREMADE, UMR 7534, F-75775 Paris 16, France

[4] Univ Basque Country, Fac Ciencias & Tecnol, Dept Matemat, Lejona 48940, Vizcaya, Spain

[5] Univ Publ Navarra, Dept Automat & Computac, Pamplona 31006, Spain

关键词: Keller-Segel model;    Chemotaxis;    Drift-diffusion;    Self-similar solution;    Intermediate asymptotics;    Entropy;    Free energy;    Rate of convergence;    Heat kernel;   

DOI  :  10.1016/j.jmaa.2009.07.034

来源: Elsevier

【 摘 要 】

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 chemoattractant 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. (c) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free