Existence and uniqueness of the traveling front in premixed combustion of porous media

We study a mathematical model of combustion processes in an inert porous media filled with a combustible gaseous mixture. We focus on the phenomenon of a combustion wave driven by a local pressure elevation. In this article, we are concerned with subsonic pressure-driven flames and with the case of a quadratic dependence of the friction force on the velocity of the gaseous mixture. After a suitable non-dimensionalization, the resulting mathematical model includes three nonlinear ordinary differential equations (ODEs). The system contains an unknown parameter V that represents the traveling wave speed. The existence of the traveling wave is proven in this study. It means that the parameter V can be chosen so that the corresponding phase trajectory satisfies the boundary conditions. Moreover, under reasonable assumptions about the monotonicity of the flame front, we prove the uniqueness of the pressure-driven wave.