Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors

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.