Bifurcation analysis of the Degond-Lucquin-Desreux-Morrow model for gas discharge

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Suzuki, Masahiro¹  Tani, Atusi² 

[1] Nagoya Inst Technol, Dept Comp Sci & Engn, Showa Ku, Gokiso Cho, Nagoya, Aichi 4668555, Japan

[2] Keio Univ, Dept Math, Yokohama, Kanagawa 2238522, Japan

关键词: Hyperbolic-parabolic-elliptic coupled system;    Stationary solutions;    Nonlinear stability;    Nonlinear instability;    Stable manifold;    Spectrum analysis;   

DOI  :  10.1016/j.jde.2019.10.039

来源: Elsevier

【 摘 要 】

The main purpose of this paper is to investigate mathematically gas discharge. Townsend discovered alpha- and gamma-mechanisms which are essential for ionization of gas, and then derived a threshold of voltage at which gas discharge can happen. In this derivation, he used some simplification such as discretization of time. Therefore, it is an interesting problem to analyze the threshold by using the Degond-Lucquin-Desreux-Morrow model and also to compare the results of analysis with Townsend's theory. Note that gas discharge never happens in Townsend's theory if gamma-mechanism is not taken into account. In this paper, we study an initial-boundary value problem to the model with alpha-mechanism but no gamma-mechanism. This problem has a trivial stationary solution of which the electron and ion densities are zero. It is shown that there exists a threshold of voltage at which the trivial solution becomes unstable from stable. Then we conclude that gas discharge can happen for a voltage greater than this threshold even if gamma-mechanism is not taken into account. It is also of interest to know the asymptotic behavior of solutions to this initial-boundary value problem for the case that the trivial solution is unstable. To this end, we establish bifurcation of nontrivial stationary solutions by applying Crandall and Rabinowitz's Theorem, and show the linear stability and instability of those non-trivial solutions. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free