Communications in Mathematical Sciences

Volume 19 (2021)

Number 5

Large-time behavior of solutions to Cauchy problem for bipolar Euler–Poisson system with time-dependent damping in critical case

Pages: 1207 – 1231



Liping Luan (Materials Genome Institute, Shanghai 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)

Bruno Rubino (Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Italy)

Peicheng Zhu (Department of Mathematics, Shanghai University, Shanghai, China)


This paper is concerned with the Cauchy problem of a bipolar hydrodynamic model for semiconductor device, a system of one dimensional Euler–Poisson equations with time-dependent damping effect in the critical case. The global existence and uniqueness of the solutions to the Cauchy problem are proved by the technical time-weighted energy method, when the initial perturbation around the constant states are small enough. Particularly, the algebraic time-convergence-rates for the solutions to their constant states are also derived.


Euler–Poisson equations, time-dependent damping, time-weighted energy method, asymptotic behavior, global solutions, Cauchy problem

2010 Mathematics Subject Classification

35B40, 35L60, 35L67

The research by the first and fourth authors is partly supported by the Key project No. 2017YFB0701502, of the Ministry of Sci. Tech., China. The research of second author was supported in part by National Sciences and Engineering Research Council of Canada under NSERC grant RGPIN 354724-2016 and Fonds de recherche du Qu´ebec nature et technologies under Fqrnt grant 256440.

Received 17 August 2020

Accepted 21 December 2020

Published 7 July 2021