2015/02/19 by Nicola Zamponi, Ansgar Jüngel, Zamponi, Nicola +1 · 3 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #35K51 #35K65 #35Q92 #92D25 #Analysis of PDEs (math.AP) #Evolution and Genetic Dynamics #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.1502.05617
openalex publication_date 2015/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A class of parabolic cross-diffusion systems modeling the interaction of an\narbitrary number of population species is analyzed in a bounded domain with\nno-flux boundary conditions. The equations are formally derived from a\nrandom-walk lattice model in the diffusion limit. Compared to previous results\nin the literature, the novelty is the combination of general degenerate\ndiffusion and volume-filling effects. Conditions on the nonlinear diffusion\ncoefficients are identified, which yield a formal gradient-flow or entropy\nstructure. This structure allows for the proof of global-in-time existence of\nbounded weak solutions and the exponential convergence of the solutions to the\nconstant steady state. The existence proof is based on an approximation\nargument, the entropy inequality, and new nonlinear Aubin-Lions compactness\nlemmas. The proof of the large-time behavior employs the entropy estimate and\nconvex Sobolev inequalities. Moreover, under simplifiying assumptions on the\nnonlinearities, the uniqueness of weak solutions is shown by using the H-1\nmethod, the E-monotonicity technique of Gajewski, and the subadditivity of\nthe Fisher information.\n