X. Chen, E. Daus, A. Jüngel:

"Global existence analysis of cross-diffusion population systems for multiple species";

in: "ASC Report 18/2016", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2016, ISBN: 978-3-902627-09-4, 1 - 30.

The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homo-geneous Neumann boundary conditions. The existence proof is based on a reﬁned entropy method and a new approximation scheme. Global existence follows under a detailed bal-ance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsagerīs principle in thermodynamics. Under detailed balance (and without reaction), the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold.

Population dynamics, Shigesada-Kawasaki-Teramoto system, competition model, detailed balance, entropy method, global existence of weak solutions, Onsagerīs principle.

http://www.asc.tuwien.ac.at/preprint/2016/asc18x2016.pdf

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