Global Existence Result for Pair Diffusion Models

Abstract

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 nonsmooth data and kinetic coefficients additionally depending on the state variables. Our proof is based on regularizations, on a priori estimates which are obtained by estimates of the free energy and by 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.