The Cauchy problem for a combustion model in porous media

Abstract

We prove the existence of a global solution to the Cauchy problem for a nonlinear reaction-diffusion system coupled with a system of ordinary differential equations. The system models the propagation of a combustion front in a porous medium with two layers, as derived by J. C. da Mota and S. Schecter in Combustion fronts in a porous medium with two layers, Journal of Dynamics and Differential Equations, 18(3) (2006). For the particular case, when the fuel concentrations in both layers are known functions, the Cauchy problem was solved by J. C. da Mota and M. M. Santos in An application of the monotone iterative method to a combustion problem in porous media, Nonlinear Analysis: Real World Application, 12 (2010). For the full system, in which the fuel concentrations are also unknown functions, we construct an iterative scheme that contains a sequence which converges to a solution of the system, locally in time, under the conditions that the initial data are Holder continuous, bounded and nonnegative functions. We also show the existence of a global solution, if the initial date are additionally in the Lebesgue space $L^p$, for some $p\in(1,\infty)$. Furthermore, for a broad class of reaction-diffusion systems we show that the non negative quadrant is a positively invariant region, and, as a consequence, that classical solutions of similar systems, with the reactions functions being non decreasing in one unknown and semi-lipscthitz continuous in the other, are bounded by lower and upper solutions for any positive time if so they are at time zero.