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Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model |
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| Authors: | Adrien Blanchet Jean Dolbeault |
| Institution: | a GREMAQ (UMR CNRS no. 5604 et INRA no. 1291), Université de Toulouse, Manufacture des Tabacs, Aile J.J. Laffont, 21 allée de Brienne, 31000 Toulouse, France b CEREMADE (UMR CNRS no. 7534), Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France c Departamento de Matemáticas, Facultad de Ciencias y Tecnología, Universidad del País Vasco, Barrio Sarriena s/n, 48940 Lejona (Vizcaya), Spain d Departamento Automática y Computación, Universidad Pública de Navarra, Campus Arrosadía s/n, 31.006 Pamplona, Spain |
| 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 8π, a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables. |
| Keywords: | Keller-Segel model Chemotaxis Drift-diffusion Self-similar solution Intermediate asymptotics Entropy Free energy Rate of convergence Heat kernel |
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