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A. Jüngel, N. Zamponi:
"Boundedness of weak solutions to cross-diffusion systems from population dynamics";
in: "ASC Report 21/2014", issued by: Institute for Analysis and Scientific Computing; Institut für Angewandte und Numerische Mathematik, Vienna University of Technology, Wien, 2014, ISBN: 978-3-902627-07-0, 1 - 10.

The global-in-time existence of nonnegative bounded weak solutions to a class of cross-diffusion systems for two population species is proved. The diffusivities are assumed to depend linearly on the population densities in such a way that a certain formal
gradient-flow structure holds. The main feature of these systems is that the diffusion matrix may be neither symmetric nor positive definite. The key idea of the proof is to employ the boundedness-by-entropy principle which yields at the same time the existence
of global weak solutions and their boundedness. In particular, the uniform boundedness of weak solutions to the population model of Shigesada, Kawasaki, and Teramoto in several space dimensions under certain conditions on the diffusivities is shown for the first time.

Strongly coupled parabolic systems, population dynamics, uniform boundedness of weak solutions, gradient-flow structure, entropy method

http://www.asc.tuwien.ac.at/preprint/2014/asc21x2014.pdf