Semiconductor full quantum hydrodynamic model with non-flat doping profile: (i) stability of steady state

This is the first part of our series of studies concerning the full quantum hydrody-namic model for semiconductors with non-flat doping profile. In this paper, we are concerned with the existence, uniqueness , asymptotic stability of subsonic steady states to the model in a bounded interval, which is subject to physical boundary conditions. The main results are proved by Stampac-chia's truncation method, the Leray-Schauder Fixed Point Theorem, Schauder's Fixed Point Theorem and intricate energy estimates.