Large time behavior of solutions to multi-dimensional bipolar hydrodynamic model of semiconductors with vacuum

In this paper, a multi-dimensional hydrodynamic model for the bipolar semiconductor device with insulating boundary conditions and a non-flat doping profile is investigated. The model is assumed to be spherically symmetrical. A large time behavior framework is builded for any L∞L∞ bounded radial symmetry weak solutions, that is, the weak entropy solutions are shown to converge to the corresponding stationary solutions in L2L2 norm and an exponential decay rate is also derived. No smallness and regularity conditions are assumed and the doping profile are permitted to be of large variation.