Cross diffusion and nonlinear diffusion preventing blow up in the keller-segel model

José Antonio Carrillo, Sabine Hittmeir, Ansgar Jüngel

Research output: Contribution to journalArticleResearchpeer-review

28 Citations (Scopus)


A parabolic-parabolic (Patlak-)Keller-Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical concentration. For arbitrarily small cross-diffusion coefficients and for suitable exponents of the nonlinear diffusion terms, the global-in-time existence of weak solutions is proved, thus preventing finite-time blow up of the cell density. The global existence result also holds for linear and fast diffusion of the cell density in a certain parameter range in three dimensions. Furthermore, we show L∞ bounds for the solutions to the parabolic-elliptic system. Sufficient conditions leading to the asymptotic stability of the constant steady state are given for a particular choice of the nonlinear diffusion exponents. Numerical experiments in two and three space dimensions illustrate the theoretical results. © 2012 World Scientific Publishing Company.
Original languageEnglish
Article number1250041
JournalMathematical Models and Methods in Applied Sciences
Issue number12
Publication statusPublished - 22 Oct 2012


  • Blow up
  • Chemotaxis
  • Cross-diffusion
  • Degenerate diffusion
  • Global existence of solution
  • Keller-Segel model


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