Parabolic Free Boundary Problems Arising in Porous Medium Combustion

Abstract

A parabolic partial differential equation approximating the evolution of temperature in highly exothermic porous-medium combustion at low driving velocities is examined. The equation is of reaction-diffusion type with a reaction term which is discontinuous as a function of the dependent variable. Firstly the variation of the steady solution set with the scaled heat of the reaction is described and the related time-dependent behaviour analysed. The stability results follow from characterizing the ends of the solution branch and fold points explicitly and deducing global stability results about the whole of the continuous solution branch. The results are used to indicate the parameter regimes and temporal scales on which the small driving velocity approximation ceases to be valid. Secondly the behaviour of the discontinuous partial differential equation is compared with that of a continuous equation which it approximates. This provides justification for the approximation of reaction terms possessing steep gradients by discontinuous functions; the large activation energy limit in porous-medium combustion involves such a process.