2014/04/10 by Tang Quoc Bao, Bao, Tang Quoc, Klemens Fellner +4
Computer Science · Engineering · Mathematics · #35A01 #35B40 #35K57 #35K61 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1404.2809
openalex publication_date 2014/04/10 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We consider a model system consisting of two reaction-diffusion equations,\nwhere one species diffuses in a volume while the other species diffuses on the\nsurface which surrounds the volume. The two equations are coupled via a\nnonlinear reversible Robin-type boundary condition for the volume species and a\nmatching reversible source term for the boundary species. As a consequence of\nthe coupling, the total mass of the two species is conserved. The considered\nsystem is motivated for instance by models for asymmetric stem cell division.\n Firstly we prove the existence of a unique weak solution via an iterative\nmethod of converging upper and lower solutions to overcome the difficulties of\nthe nonlinear boundary terms. Secondly, our main result shows explicit\nexponential convergence to equilibrium via an entropy method after deriving a\nsuitable entropy entropy-dissipation estimate for the considered nonlinear\nvolume-surface reaction-diffusion system.\n