  We are concerned with the global existence and large time behavior of entropy
solutions to the one dimensional unipolar hydrodynamic model for semiconductors
in the form of Euler-Poisson equations in a bounded interval. In this paper, we
first prove the global existence of entropy solution by vanishing viscosity and
compensated compactness framework. In particular, the solutions are uniformly
bounded with respect to space and time variables by introducing modified
Riemann invariants and the theory of invariant region. Based on the uniform
estimates of density, we further show that the entropy solution converges to
the corresponding unique stationary solution exponentially in time. No any
smallness condition is assumed on the initial data and doping profile.
Moreover, the novelty in this paper is about the unform bound with respect to
time for the weak solutions of the isentropic Euler-Possion system.

Analysis of PDEs