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We consider a generalization of the Gelfand problem arising inFrank-Kamenetskii theory of thermal explosion. This generalization is a naturalextension of the Gelfand problem to two phase materials, where, in contrast tothe classical Gelfand problem which utilizes single temperature approach, thestate of the system is described by two different temperatures. We show thatsimilar to the classical Gelfand problem the thermal explosion occursexclusively due to the absence of stationary temperature distribution. We alsoshow that the presence of inter-phase heat exchange delays a thermal explosion.Moreover, we prove that in the limit of infinite heat exchange between phasesthe problem of thermal explosion in two phase porous media reduces to theclassical Gelfand problem with renormalized constants.

Gelfand type problem for two phase porous media