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Milan Journal of Mathematics

, Volume 72, Issue 1, pp 1–28 | Cite as

Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions

  • L. Corrias
  • B. Perthame
  • H. Zaag
Original Paper

Abstract.

We consider two simple conservative systems of parabolic-elliptic and parabolic-degenerate type arising in modeling chemotaxis and angiogenesis. Both systems share the same property that when the \(L^{\frac{d} {2}} \) norm of initial data is small enough, where d ≥ 2 is the space dimension, then there is a global (in time) weak solution that stays in all the L p spaces with max \(\left\{ {\left. {1;\frac{d} {2} - 1} \right\} \leq p < \infty .} \right.\) This result is already known for the parabolic-elliptic system of chemotaxis, moreover blow-up can occur in finite time for large initial data and Dirac concentrations can occur. For the parabolic-degenerate system of angiogenesis in two dimensions, we also prove that weak solutions (which are equi-integrable in L1) exist even for large initial data. But break-down of regularity or propagation of smoothness is an open problem.

Mathematics Subject Classification (2000).

35B60 35Q80 92C17 92C50 

Keywords.

Chemotaxis angiogenesis degenerate parabolic equations global weak solutions blow-up 

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Copyright information

© Birkhäuser Verlag, Basel 2004

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité d’Evry Val d’EssonneEvry Cedex
  2. 2.Département de Mathématiques et ApplicationsÉcole Normale Supérieure, CNRS UMR8553Paris Cedex 05

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