Steady Hydrodynamic Model of Semiconductors with Sonic Boundary: (II) Supersonic Doping Profile

Abstract

This is the second part of our series of studies concerning the well-posedness/ill-posedness and regularity of stationary solutions to the hydrodynamic model of semiconductors represented by Euler--Poisson equations with sonic boundary condition. In this paper, we consider the case of a supersonic doping profile, and prove that the system does not hold any interior subsonic solution; furthermore, the system doesn't admit any interior supersonic solution and any transonic solution if such a supersonic doping profile is small enough or the relaxation time is small, but it has at least one interior supersonic solution and infinitely many transonic solutions if the supersonic doping profile is close to the sonic line and the relaxation time is large. The nonexistence of any type of solutions in the case of a small doping profile or small relaxation time indicates that the semiconductor effect for the system is remarkable and cannot be ignored.