Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

  • Adrien Blanchet
  • José A. Carrillo
  • Philippe Laurençot
Article

Abstract

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass Mc such that if \({M \in (0, M_c]}\) solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for \({M \in (0, M_c)}\) . While characterising the possible infinite time blowing-up profile for M  =  Mc, we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two.

Mathematics Subject Classification (2000)

35K65 35B45 35J20 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Adrien Blanchet
    • 1
  • José A. Carrillo
    • 2
  • Philippe Laurençot
    • 3
  1. 1.DAMTP, Centre for Mathematical SciencesCambridgeUK
  2. 2.ICREA (Institució Catalana de Recerca i Estudis Avançats) and Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  3. 3.Institut de Mathématiques de ToulouseCNRS UMR 5219 and Université de ToulouseToulouse Cedex 9France

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