Global existence of smooth solutions of the N-dimensional Euler-Poisson model

The global existence of smooth solutions of the Cauchy problem for the N-dimensional Euler--Poisson model for semiconductors is established, under the assumption that the initial data is a perturbation of a stationary solution of the drift-diffusion equations with zero electron velocity, which is proved to be unique. The resulting evolutionary solutions converge asymptotically in time to the unperturbed state. The singular relaxation limit is also discussed.