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

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 Hölder 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 ∈ (1, ∞). Our proof of the local existence relies on a careful analysis on the construction of the fundamental solution for parabolic equations obtained by the parametrix method. In particular, we show the continuous dependence of the fundamental solution for parabolic equations with respect to the coefficients of the equations. To obtain the global existence, we employ the "method of auxiliary functions" as used by O. A. Oleinik and S. N. Kruzhkov in Quasi-linear second-order parabolic equations with many independent variables, Russian Mathematical Surveys, 16(5) (1961). 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.

Existência e Unicidade de Solução para um Problema de Combustão num Meio Poroso com n Camadas

“…Mais recentemente, alguns autores naárea de Matemática Aplicada vem trabalhando com modelos de combustão em meios porosos simplificados, de tal modo que são passíveis de serem analisados com as teorias matemáticas hoje disponíveis. Como exemplos, citamos [1,2,[5][6][7][8][9][10][11][12]. No presente trabalho, estudamos a existência e unicidade de solução global no tempo para um problema de valor inicial e de contorno associado a um sistema de reação-difusão-convecção, com n equações parabólicas não lineares, acoplado com n 2 equações diferenciais ordinárias.…”