Communications in Mathematical Sciences

Volume 21 (2023)

Number 5

Semiconductor full quantum hydrodynamic model with non-flat doping profile: I) Stability of steady state

Pages: 1215 – 1246



Haifeng Hu (School of Science, Changchun University, Changchun, China; and Department of Mathematics and Statistics, McGill University, Montreal, Québec, Canada)

Kaijun Zhang (School of Mathematics and Statistics, Northeast Normal University, Changchun, China)


This is the first part of our series of studies concerning the full quantum hydrodynamic model for semiconductors with non-flat doping profile. In this paper, we are concerned with the existence, uniqueness and 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 Stampacchia’s truncation method, the Leray–Schauder Fixed Point Theorem, Schauder’s Fixed Point Theorem and intricate energy estimates.


full quantum hydrodynamic model, dispersive velocity term, non-flat doping profile, asymptotic stability, semiconductor

2010 Mathematics Subject Classification

35A01, 35B40, 35M33, 76Y05

This research was initiated when the first author was a postdoc in the Center for Partial Differential Equations of ECNU; then it is now finally completed when the first author is visiting the Department of Mathematics and Statistics of McGill University; the financial support and kind hospitality from both institutes are gratefully acknowledged.

The first author’s research was supported by the National Natural Science Foundation of China (Grant No.11801039), China Scholarship Council (File No.202007535001) and Programme of Scientific Research of Changchun University (No.ZKP202013).

The second author’s research work was partially supported by the National Natural Science Foundation of China (Grant No.11771071).

Received 1 December 2021

Accepted 14 September 2022

Published 30 August 2023