Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models

We prove asymptotic stability results for nonlinear bipolar drift-diffusion-Poisson Systems arising in semiconductor device modeling and plasma physics in one space dimension. In particular, we prove that, under certain structural assumptions on the external potentials and on the doping profile, all solutions match for large times with respect to all q-Wasserstein distances. We also prove exponential convergence to stationary solutions in relative entropy via the so called entropy dissipation (or Bakry-Emery) method.