  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.
