An application of the monotone iterative method to a combustion problem in porous media

Problems related to combustion fronts in porous media have been studied by many authors recently, see e.g. [Y. Akkutlu, Y.C. Yortsos, The dynamics of in-situ combustion fronts in porous media, Combust. Flame 134 (2003) 239–247; J.C. da Mota, W. Dantas, D. Marchesin, Combustion fronts in porous media, SIAM J. Appl. Math. 62 (2002) 2175–2198; D.A. Schult, B.J. Matkowsky, V.A. Volpert, A.C. Fernandez-Pello, Forced forward smolder combustion, Combust. Flame 104 (1996) 1–26]. Most of this interest is due to the combustion process for oil recovery.In this paper we construct monotone iteractions for a Cauchy problem arising from a combustion model in a porous medium derived in [J.C. da Mota, S. Schecter, Combustion fronts in a porous medium with two layers, J. Dynam. Differential Equations 18 (3) (2006) 615–665]. We conclude that the monotone iteractions converge to a unique solution of this Cauchy problem, globally in time.