Existence and Asymptotic Behavior of Multi-Dimensional Quantum Hydrodynamic Model for Semiconductors
This paper is devoted to the study of the existence and the time-asymptotic of multi-dimensional quantum hydrodynamic equations for the electron particle density, the current density and the electrostatic potential in spatial periodic domain. The equations are formally analogous to classical hydrodynamics but differ in the momentum equation, which is forced by an additional nonlinear dispersion term, (due to the quantum Bohm potential) and are used in the modelling of quantum effects on semiconductor devices. We prove the local-in-time existence of the solutions, in the case of the general, nonconvex pressure-density relation and large and regular initial data. Furthermore we propose a ``subsonic'' type stability condition related to one of the classical hydrodynamical equations. When this condition is satisfied, the local-in-time solutions exist globally in-time and converge time exponentially toward the corresponding steady-state. Since for this problem classical methods like, for instance, the Friedrichs theory for symmetric hyperbolic systems cannot be used, we investigate via an iterative procedure an extended system, which incorporates the one under investigation as a special case. In particular the dispersive terms appear in the form of a fourth-order wave type equation.