On large time asymptotics for drift-diffusion-Poisson systems

In this paper we analyze the convergence rate of solutions of certain drift-diffusion-Poisson systems to their unique steady state. These bi-polar equations model the transport of two populations of charged particles and have applications for semiconductor devices and plasmas. When prescribing a confinement potential for the particles we prove exponential convergence to the equilibrium. Without confinement the solution decays with an algebraic rate towards a self-similar state. The analysis is based on a relative entropy type functional and it uses logarithmic Sobolev inequalities.