Large Time Behavior of the Solutions to a Hydrodynamic Model for Semiconductors

We establish the global existence of smooth solutions to the Cauchy problem for the one-dimensional isentropic Euler--Poisson (or hydrodynamic) model for semiconductors for small initial data. In particular we show that, as t→∞, these solutions converge to the stationary solutions of the drift-diffusion equations. The existence and uniqueness of stationary solutions to the drift-diffusion equations are proved without the smallness assumption.