DOI: 10.1007/s10884-007-9079-9 Journal of Dynamics and Differential Equations, Vol. 19, No. 4, December 2007 (© 2007) Traveling Waves in Porous Media Combustion: Uniqueness of Waves for Small Thermal Diffusivity Anna Ghazaryan, 1,3 Peter Gordon, 2 and Christopher K. R. T Jones 1 Received May 30, 2006 We study traveling wave solutions arising in Sivashinsky’s model of subsonic detonation which describes combustion processes in inert porous media. Sub- sonic (shockless) detonation waves tend to assume the form of a reaction front propagating with a well defined speed. It is known that traveling waves exist for any value of thermal diffusivity [5]. Moreover, it has been shown that, when the thermal diffusivity is neglected, the traveling wave is unique. The question of whether the wave is unique in the presence of thermal diffusivity has remained open. For the subsonic regime, the underlying phys- ics might suggest that the effect of small thermal diffusivity is insignificant. We analytically prove the uniqueness of the wave in the presence of non-zero diffusivity through applying geometric singular perturbation theory. KEY WORDS: Geometric singular perturbation theory; traveling waves; sub- sonic detonation; porous media combustion. 1. INTRODUCTION Gaseous detonation is one of the classical topics of combustion theory. In the past decade, there has been significant progress in the study of the key features of this phenomenon. However, it is still far from completely understood. Recently, Sivashinsky proposed a model of subsonic detona- tion that describes propagation of the combustion fronts in highly resist- ible media [7]. The assumption of high resistance of the media provides a Dedicated to Mr. Brunovsky in honor of his 70th birthday. 1 Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA. E-mail: ghazarya@email.unc.edu 2 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA. 3 To whom correspondence should be addressed. 951 1040-7294/07/1200-0951/0 © 2007 Springer Science+Business Media, LLC