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S. Hittmeir, A. Jüngel:
"Cross diffusion preventing blow up in the two-dimensional Keller-Segel model";
in: "ASC Report 26/2010", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2010, ISBN: 978-3-902627-03-2.

A (Patlak-) Keller-Segel model in two space dimensions with an additional crossdi
usion term in the equation for the chemical signal 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 substance. This allows one to prove, for arbitrarily small cross di usion, the global existence
of weak solutions to the parabolic-parabolic model as well as the global existence of bounded weak
solutions to the parabolic-elliptic model, thus preventing blow up of the cell density. Furthermore,
the long-time decay of the solutions to the parabolic-elliptic model is shown and nite-element
simulations are presented illustrating the in
uence of the regularizing cross-di usion term.

Chemotaxis, Keller-Segel model, cross-di usion, global existence of solutions, longtime

http://www.asc.tuwien.ac.at/preprint/2010/asc26x2010.pdf