Large time behavior of solutions to a bipolar hydrodynamic model with big data and vacuum

In this note, we consider a physically relevant hydrodynamic model for the bipolar semiconductor device with insulating boundary conditions and a non-flat doping profile. We prove that the corresponding steady solutions are unique and satisfy some bounded estimates, which are essential in the following consideration. For the hydrodynamic model, by means of a technical energy method and a proper entropy dissipation estimate, the large time behavior framework for any uniformly bounded weak entropy solutions with vacuum is presented. The solutions are shown to converge to the stationary solutions in L2 norm and an exponential decay rate is also derived. No smallness and regularity conditions are assumed and the doping profile is permitted to be of big variation.