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A. Jüngel, S. Hittmeir, J. Carrillo:
"Cross diffusion and nonlinear diffusion preventing blow up in the Keller-Segel model";
in: "ASC Report 36/2011", issued by: Instute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2011, ISBN: 978-3-902627-04-9.

A parabolic-parabolic (Patlak-) Keller-Segel model in up to three space dimensions
with nonlinear cell di usion and an additional nonlinear cross-di usion 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-di usion coe cients and for suitable exponents of the nonlinear di usion
terms, the global-in-time existence of weak solutions is proved, thus preventing nite-time
blow up of the cell density. The global existence result also holds for linear and fast di usion
of the cell density in a certain parameter range in three dimensions. Furthermore, we
show L1 bounds for the solutions to the parabolic-elliptic system. Su cient conditions
leading to the asymptotic stability of the constant steady state are given for a particular
choice of the nonlinear di usion exponents. Numerical experiments in two and three space
dimensions illustrate the theoretical results.