Monatshefte für Mathematik

, Volume 188, Issue 1, pp 131–162

# The Cauchy problem for a combustion model in a porous medium with two layers

• J. C. da Mota
• M. M. Santos
• R. A. Santos
Article

## Abstract

We prove the existence of local and global in time solutions of the Cauchy problem for a combustion model in a porous medium with two layers. The model is a system of four equations, consisting of two nonlinear reaction–convection–diffusion equations coupled with two ordinary differential equations, with the coupling occurring in both the reaction functions and in the differential operator coefficients. To obtain the local solution, we first construct an iteration scheme of approximate solutions to the system. Using the continuous dependence of solutions for parabolic equations with respect to the coefficients of the equations, we show that the constructed iteration scheme contains a sequence which converges to a local solution of the system, under the assumption that the initial data are Lipschitz continuous, bounded and non negative. We show that this solution can be extended globally, if the initial data are additionally in the Lebesgue space $$L^p$$, for some $$p\in (1,\infty )$$.

## Keywords

Reaction–diffusion system Combustion model Iterative monotone method Parametrix method Auxiliary functions

## Mathematics Subject Classification

35K15 35K45 35K57 35K60 80A25

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