| Abstract: | The paper is devoted to a nonlinear analysis of superheating 1, 2] instability of an electric discharge stabilized by electrodes 3] in the framework of a thermal model 4] where the stability of the discharge relative to the long-wave and short-wave perturbations is proved in a linear approximation. Similar boundary-value problems arise in the theories of chemically and biologically reacting mixtures 5–7], thermal breakdown of dielectrics 8], thermal explosion 9], in the investigation of nonlinear waves in semiconductors and superconductors 10, 11], and in the investigation of Couette flow with variable viscosity 12]. The uniqueness of the one-dimensional steady solutions of the thermal model of discharge and the stability relative to the small spatial perturbations, respectively, for the exponential and step dependence of the electrical conductivity on the temperature are proved in 3, 13]. The uniqueness of the solutions in the one-dimensional case for the same electrode temperature and arbitrary dependences of the electrical and thermal conductivity on the temperature is established in paper 14]. In the present paper, the existence and uniqueness of steady solutions of the thermal model of discharge in a three-dimensional formulation for arbitrary fairly smooth electrical and thermal conductivity functions of the temperature in the case of isothermal isopotential electrodes are proved analytically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 140–145, January–February, 1986.The author expresses his gratitude to A. G. Kulikovskii and A. A. Barmin for the formulation of the problem and their discussions. |
