The phenomenon of the spontaneous insulation fires is investigated. An original physical model is based on the classical cell approach of multiphase media. The real complex media is modelled by a set of the identical parallel capillaries containing the thin film of the combustible fluid spread over their inner surface. The dynamics of the thermal explosion in this multiphase medium is studied. An advanced mathematical technique (integral manifold method) is used for qualitative analysis of the system dynamics. The classification of the possible scenarios of the thermal runaway is obtained analytically. Peculiarities of the dynamical regimes are investigated and their dependence on physical system parameters is analyzed. The research is focused on the regime with delay. Lower and upper analytical estimates for the delay time have been gained. The theoretical delay times estimations were verified when compared to those found experimentally. This comparison demonstrates sufficient qualitative and quantitative agreement.