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. (C) 2019 Elsevier Ltd. All rights reversed.