A. Jüngel, N. Zamponi:

"Qualitative behavior of solutions to cross-diffusion systems from population dynamics";

in: "ASC Report 37/2015", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2015, ISBN: 978-3-902627-08-7, 1 - 17.

A general class of cross-diﬀusion systems for two population species in a bounded domain with no-ﬂux boundary conditions and Lotka-Volterra-type source terms is analyzed. Although the diﬀusion coeﬃcients are assumed to depend linearly on the population densities, the equations are strongly coupled. Generally, the diﬀusion matrix is neither symmetric nor positive deﬁnite. Three main results are proved: the existence of global uniformly bounded weak solutions, their convergence to the constant steady state in the weak competition case, and the uniqueness of weak solutions. The results hold under appropriate conditions on the diﬀusion parameters which are made explicit and which contain simpliﬁed Shigesada-Kawasaki-Teramoto population models as a special case. The proofs are based on entropy methods, which rely on convexity properties of suitable Lyapunov functionals.

Strongly coupled parabolic systems, population dynamics, boundedness of weak

http://www.asc.tuwien.ac.at/preprint/2015/asc37x2015.pdf

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