Large time behavior of a bipolar hydrodynamic model with large data andvacuum

In this paper, it is considered a hydrodynamic model for the bipolar semiconductor devicein the case of a pressure with the exponent γ = 2. The model has a non-flat doping profile andinsulating boundary conditions. Firstly, the existence and uniqueness of the corresponding steadysolutions which satisfy some bounded estimates are proved. Then,using a technical energy method andan entropy dissipation estimate,we present a framework for the large time behavior of bounded weakentropy solutions with vacuum. It is shown that the weak solutions converge to the stationary solutionsin L2 norm with exponential decay rate. No smallness and regularity conditions are assumed