Contributions to Books:

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.

English abstract:
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.

Created from the Publication Database of the Vienna University of Technology.