Relaxation limit in bipolar semiconductor hydrodynamic model with non-constant doping profile

The relaxation limit from bipolar semiconductor hydrodynamic (HD) model to drift-diffusion (DD) model is shown under the non-constant doping profile assumption for both stationary solutions and global-in-time solutions, which satisfy the general form of the Ohmic contact boundary condition. The initial layer phenomenon will be analyzed because the initial data is not necessarily in the momentum equilibrium. Due to the bipolar coupling structure, the analysis is hard and different from the previous literature on unipolar model or bipolar model with zero doping profile restriction. We first construct the non-constant uniform stationary solutions by the operator method for both HD and DD models in a unified procedure. Then we prove the global existence of DD model and uniform global existence of HD model by the elementary energy method but with some new developments. Based on the above existence results, we further calculate the convergence rates in relaxation limits.