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The variational formulation of the nonlinear problem of steady heat conduction in an arbitrary configuration domain is applied to investigate the conditions for the existence of a steady-state temperature distribution in a stationary medium with the intensity of volumetric energy release rising with increasing temperature. Based on the relations of the time-independent theory of thermal explosion, a variational model of this phenomenon is constructed, which makes it possible to obtain estimates of the critical values of the parameters that determine the temperature state of the medium preceding the thermal explosion. The examples of a comparative analysis of such values for a solid and a hollow cylinder of finite length are given.

The purpose of the study was to build a nonlinear mathematical model governing the steady-state one-dimensional temperature distribution in the electrical insulation layer made in the form of a long hollow circular cylinder whose surfaces are given a constant electric field potential difference. By means of this model, integral ratios, which connecting the parameters of the temperature state of this layer, the heat transfer conditions on its surfaces and the temperature-dependent heat conduction coefficient and the electrical resistivity of the electrical insulating material with a given electrical potential difference, were built. A quantitative assay of the integral ratios is carried out with regard to the electrical insulation layer of a superconducting cable cooled by liquid nitrogen.

A mathematical heat transfer model in variational form is constructed for steady-state heat transfer through an inhomogeneous anisotropic solid of arbitrary configuration. This model is used to obtain two-sided thermal heat transfer resistance estimates for such a solid. These estimates permit calculating the maximum possible error when approximating the thermal resistance by half the sum of these estimates.