burning firm fuel, a crack-visible cavity with bondings between margins in the end zones, cohesive forces, a gas current in a cavity.

The flat stationary problem about local destruction of the burning fuel weakened by a rectilinear crack-visible cavity the surface of which burns is solved. The fuel is strengthened with the protective cylinder of other elastic material. The mathematical description of settlement model about the development of crack-visible cavity in the fuel is given. The definition of the unknown parametres characterising the disclosing and growth of a crack-visible cavity, comes to the solution of the singular integrated equation. The growth condition of crack-visible cavities is formulated with consideration of the criterion of a limiting extract of the material bondings. The local condition of stability (safety) of a mode “burning-destruction” is established.