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Combustion occurring in porous media has various practical applications, such as in in-situ combustion processes in oil reservoirs, the combustion of biogas in sanitary landfills, and many others. A porous medium where combustion takes place can consist of layers with different physical properties. This study demonstrates that the initial value problem for a combustion model in a multi-layer porous medium has a unique solution, which is continuous with respect to the initial data and parameters in $\mathtt{L}^2(\mathbb{R})^n$. In summary, it establishes that the initial value problem is well-posed in $\mathtt{L}^2(\mathbb{R})^n$. The model is governed by a one-dimensional reaction-diffusion-convection system, where the unknowns are the temperatures in the layers. Previous studies have addressed the same problem in $\mathtt{H}^2(\mathbb{R})^n$. However, in this study, we solve the problem in a less restrictive space, namely $\mathtt{L}^2(\mathbb{R})^n$. The proof employs a novel approach to combustion problems in porous media, utilizing an evolution operator defined from the theory of semigroups in Hilbert space and Kato's theory for a well-posed associated initial value problem.

This study proved that the Cauchy problem for a one-dimensional reaction-diffusion-convection system is locally and globally well-posed in $\mathtt{H}^2(\mathbb{R})$. The system modeled a gasless combustion front through a multi-layer porous medium when the fuel concentration in each layer was a known function. Combustion has critical practical porous media applications, such as in in-situ combustion processes in oil reservoirs and several other areas. Earlier studies considered physical parameters (e.g., porosity, thermal conductivity, heat capacity, and initial fuel concentration ) constant. Here, we consider a more realistic model where these parameters are functions of the spatial variable rather than constants. Furthermore, in previous studies, we did not consider the continuity of the solution regarding the initial data and parameters, unlike the current study. This proof uses a novel approach to combustion problems in porous media. We follow the abstract semigroups theory of operators in the Hilbert space and the well-known Kato's theory for a well-posed associated initial value problem., Comment: 39 pages

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.