Communications in Mathematical Sciences

Volume 14 (2016)

Number 8

Asymptotic stability and semi-classical limit for bipolar quantum hydrodynamic model

Pages: 2331 – 2371



Haifeng Hu (School of Mathematics and Statistics, Northeast Normal University, Changchun, China; and the Center for Partial Differential Equations, East China Normal University, Shanghai, China)

Ming Mei (Department of Mathematics, Champlain College, Saint-Lambert, Quebec, Canada; and Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)

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


In this paper, the initial-boundary value problem of a 1-D bipolar quantum semiconductor hydrodynamic model is investigated under a non-linear boundary condition which means the quantum effect vanishes on the boundary. First of all, the existence and uniqueness of the corresponding stationary solution are established. Then the exponentially asymptotic stability of the stationary solution and the semi-classical limits are further studied. The adopted approach is the elementary energy method but with some new developments.


bipolar quantum hydrodynamic model, stationary solution, energy estimates, asymptotic stability, semi-classical limit

2010 Mathematics Subject Classification

35A01, 35B35, 35B40, 35B45, 35M33, 82D37

Published 26 October 2016