Communications in Mathematical Sciences

Volume 17 (2019)

Number 2

Time periodic solutions to the full hydrodynamic model to semiconductors

Pages: 413 – 445



Ming Cheng (College of Mathematics, Jilin University, Changchun, China)

Yong Li (College of Mathematics, Jilin University, Changchun, China; and Center for Mathematics and Interdisciplinary Sciences, School of Mathematics and Statistics, Northeast Normal University, Changchun, China)


In this paper, a full hydrodynamic semiconductor model with a time periodic external force is concerned. First, we regularize the system under consideration and prove the existence of time periodic solutions to the linearized approximate system by applying Tychonoff fixed point theorem combined with the energy method and the decay estimates. This idea is from the Massera-type criteria for linear periodic evolution equations. Then, the existence of a strong time periodic solution under some smallness assumptions is established by using the topological degree theory and an approximation scheme. The uniqueness of time periodic solutions is proved basing on the energy estimates. Also, the existence of the stationary solution is obtained.


hydrodynamic model to semiconductors, time periodic solutions, fixed point theorem and topological degree

2010 Mathematics Subject Classification

35B10, 35Q31, 76N15, 82D37

The work is partially supported by Natural Science Foundation for Young Scientists of Jilin Province (No. 20170520047JH), National Basic Research Program of China (Grant No. 2013CB834100), the Scientific and Technological Project of Jilin Provinces Education Department in Thirteenth Five-Year (Grant No. JJKH20190180KJ) and National Natural Science Foundation of China (Grant No. 11571065, 11171132 and 11201173).

Received 28 June 2018

Received revised 6 December 2018

Accepted 6 December 2018

Published 8 July 2019