Abstract. We study front solutions of a system that models combustion in highly hydraulically resistant porous media. The spectral stability of the fronts is tackled by a combination of energy estimates and numerical Evans function computations. Our results suggest that there is a parameter regime for which there are no unstable eigenvalues. We use recent works about partially parabolic systems to prove that in the absence of unstable eigenvalues the fronts are convectively stable.Key words. traveling waves, combustion fronts, Evans function, spectral stability, nonlinear stability, partly parabolic systems 1. Introduction. This paper is devoted to the stability analysis of a traveling front in a combustion model. A combustion front is a coherent flame structure that propagates with a constant velocity. In a physical system it represents an intermediate asymptotic behavior of a nonstationary combustion wave [5]. To capture combustion waves, mathematical models are often cast as systems of partial differential equations (PDEs) posed on infinite domains, with multiple spatially homogeneous equilibria that correspond to cold states and completely burned states. Traveling fronts then are exhibited as a result of a competition of two different states of the system. To know the stability stability of a combustion front is of importance for the general understanding of the underlying phenomena, and has implications, for example, for chemical technology and fire and explosion safety. From the mathematical point of view, combustion models are of interest because they are formulated as systems of coupled nonlinear PDEs of high complexity and as such require refined mathematical techniques for their treatment.We consider here a system that is obtained from a model for combustion in highly hydraulically resistant porous media [35]. The original model [6,35] is considered to adequately capture the rich physical dynamics of the combustion in hydraulically resistant porous media and at the same time be approachable from the mathematical point of view. The model consists of a partly parabolic system of two PDEs coupled through a nonlinearity. Partly parabolic systems are systems where some but not all quantities diffuse. A reduced system can be obtained from the porous media combustion model suggested in [6,35] for a special value of the ratio of pressure and molecular diffusivities or a special value of the Lewis number, which is the ratio of the thermal diffusivity to mass diffusivity [15].The existence and uniqueness of combustion front solutions of the system have been studied extensively [8,11,16,17]. The stability of the fronts has not been studied yet to the authors' knowledge. We present here an attempt to provide the fullest stability analysis possible that would allow for physical interpretation of the results. Our strategy is to perform the stability study for the reduced system. This has the advantage of simplifying the computations. We then show that when the fronts are

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In this paper we study the stability of fronts in a reduction of a well-known PDE system that is used to model the combustion in hydraulically resistant porous media. More precisely, we consider the original PDE system under the assumption that one of the parameters of the model, the Lewis number, is chosen in a specific way and with initial conditions of a specific form. For a class of initial conditions, then the number of unknown functions is reduced from three to two. For the reduced system, the existence of combustion fronts follows from the existence results for the original system. The stability of these fronts is studied here by a combination of energy estimates and numerical Evans function computations and nonlinear analysis when applicable. We then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the Lewis number. The fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level.