We study the existence and the asymptotic stability of a
stationary solution to the initial boundary value problem
for a one-dimensional hydrodynamic model of semiconductors.
This problem is considered, in the previous researches [2]
and [11], under the assumption that a doping profile is flat,
which makes the stationary solution also flat. However, this
assumption is too narrow to cover the doping profile in actual
diode devices. Thus, the main purpose of the present paper
is to prove the asymptotic stability of the stationary solution
without this assumption on the doping profile. Firstly, we
prove the existence of the stationary solution. Secondly,
the stability is shown by an elementary energy method, where
the equation for an energy form plays an essential role.