Multi-dimensional bipolar hydrodynamic model of semiconductor with insulating boundary conditions and non-zero doping profile

In this paper, we are concerned with the global existence and asymptotic behavior of smooth solution to the multi-dimensional bipolar hydrodynamic model of semiconductor with insulating boundary conditions and non-zero doping profile. For any space dimension, we prove the global solutions exist and converge to the corresponding stationary with an exponential decay rate when the initial data are close to a certain steady state, which may not be constants. We also get a L2 decay estimate which can be valid for any L∞ weak entropy solution. Moreover, we give specific conditions on initial data and doping profile when space dimension d=1.