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

Fernandez, Javier; Escobedo, Miguel; Dolbeault, Jean; Blanchet, Adrien

Fernandez, Javier; Escobedo, Miguel; Dolbeault, Jean; Blanchet, Adrien (2010), Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model, Journal of Mathematical Analysis and Applications, 361, 2, p. 533-542. http://dx.doi.org/10.1016/j.jmaa.2009.07.034

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00349216/fr/

Date

2010

Journal name

Journal of Mathematical Analysis and Applications

Volume

361

Number

Publisher

Elsevier

Pages

533-542

Abstract (EN)

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.

Subjects / Keywords

Intermediate Asymptotics; Self-similar Solution; Drift-diffusion; Chemotaxis; Keller-Segel Model; Entropy; Free Energy; Rate of Convergence; Heat Kernel