2014/04/24 by Ansgar Jüngel, Jüngel, Ansgar, Nicola Zamponi +1 · 1 citation
Medicine · Mathematics · Biochemistry, Genetics and Molecular Biology · #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematical Biology Tumor Growth #Evolution and Genetic Dynamics
paper · pdf · doi:10.48550/arxiv.1404.6054
The global-in-time existence of nonnegative bounded weak solutions to a class\nof cross-diffusion systems for two population species is proved. The\ndiffusivities are assumed to depend linearly on the population densities in\nsuch a way that a certain formal gradient-flow structure holds. The main\nfeature of these systems is that the diffusion matrix may be neither symmetric\nnor positive definite. The key idea of the proof is to employ the\nboundedness-by-entropy principle which yields at the same time the existence of\nglobal weak solutions and their boundedness. In particular, the uniform\nboundedness of weak solutions to the population model of Shigesada, Kawasaki,\nand Teramoto in several space dimensions under certain conditions on the\ndiffusivities is shown for the first time.\n