J. Carrillo, S. Hittmeir, A. Jüngel:
"Cross diffusion and nonlinear diffusion preventing blow up in the Keller-Segel model";
Mathematical Models & Methods in Applied Sciences, 22 (2012), S. 1 - 35.

Kurzfassung deutsch:
Siehe englisches Abstract.

Kurzfassung englisch:
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^\infty 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.

Cell biology, Keller-Segel model

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