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Journal of Mathematical Biology

, Volume 63, Issue 1, pp 1–32 | Cite as

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

  • Piotr Biler
  • Lucilla CorriasEmail author
  • Jean Dolbeault
Article

Abstract

In two space dimensions, the parabolic–parabolic Keller–Segel system shares many properties with the parabolic–elliptic Keller–Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold M c . However, this threshold is not as clear in the parabolic–parabolic case as it is in the parabolic–elliptic case, in which solutions with mass above M c always blow up. Here we study forward self-similar solutions of the parabolic–parabolic Keller–Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above M c , which is forbidden in the parabolic–elliptic case.

Keywords

Keller–Segel model Chemotaxis Self-similar solution Nonlocal parabolic equations Critical mass Existence Blowup 

Mathematics Subject Classification (2000)

35B30 35K40 35K57 35J60 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland
  2. 2.Département de MathématiquesUniversité d’Évry Val d’EssonneÉvry CédexFrance
  3. 3.Ceremade (UMR CNRS 7534)Université Paris-DauphineParis Cédex 16France

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