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| We consider a model system consisting of two reaction-diffusion equations, where one specie diffuses in a volume while the other specie diffuses on the surface which surrounds the volume. The two equations are coupled via nonlinear reversible Robin-type boundary conditions for the volume specie and a matching reversible source term for the boundary specie. As a consequence the total mass of the species is conserved.
The considered system is motivated by models for asymmetric stem cell division. We first prove the existence of a unique weak solution via an iterative method of converging upper and lower solutions to overcome the difficulties of the nonlinear boundary terms. Secondly, we show explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate. |
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