A body moving in a planet atmosphere is under the influence of aerodynamic loads, the forces of inertia, and heat flux. As a result, the body undergoes ablation and even could be completely destroyed. First of all, the stressed state within the body at any time is determined through an accurate solution of the Lamé equations. Based on the solution, one can investigate the nature of the destruction of the body and evaluate the altitude of destruction for well-known meteoroids if their composition and space velocities are known. During the flight of small fragments thermoelastic forces become significant. Unlike large fragments smaller fragments heat up in a short time resulting in accelerating the process of destruction, which also contributes to the ablation, i. e., reduces the fragment's size. The state of stress in the hot object of decreasing radius is under consideration in this paper. Finally, "thermal explosion" due to the rapid evaporation of small fragments in a typical size range of the fragments was considered. Assessment of the length of run and time of evaporation of small particles allows one to speak about the explosive outbreak and disappearance of the asteroid in the final stages of its demise.