Global existence result for pair diffusion models

In this paper we prove a global existence result for pair diffusion models in two dimensions. Such models describe the transport of charged particles in semiconductor heterostructures. The underlying model equations are continuity equations for mobile and immobile species coupled with a nonlinear Poisson equation. The continuity equations for the mobile species are nonlinear parabolic PDEs involving drift, diffusion and reaction terms, the corresponding equations for the immobile species are ODEs containing reaction terms only. Forced by applications to semiconductor technology these equations have to be considered with non-smooth data and kinetic coefficients additionally depending on the state variables. Our proof is based on regularizations, on a priori estimates which are obtained by energy estimates and Moser iteration as well as on existence results for the regularized problems. These are obtained by applying the Banach Fixed Point Theorem for the equations of the immobile species, and the Schauder Fixed Point Theorem for the equations of the mobile species

Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling

We consider a model system consisting of two reaction-diffusion equations, where one species diffuses in a volume while the other species diffuses on the surface which surrounds the volume. The two equations are coupled via a nonlinear reversible Robin-type boundary condition for the volume species and a matching reversible source term for the boundary species. As a consequence of the coupling, the total mass of the two species is conserved. The considered system is motivated for instance by models for asymmetric stem cell division. Firstly we 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, our main result shows explicit exponential convergence to equilibrium via an entropy method after deriving a suitable entropy entropy-dissipation estimate for the considered nonlinear volume-surface reaction-diffusion system.Comment: 31 page