Singular Limit and Homogenization for Flame Propagation in Periodic Excitable Media

This paper is concerned with a class of singular equations modelling the combustion of premixed gases in periodic media. The model involves two parameters: the period of the medium |L| and a singular parameter ɛ related to the activation energy. The existence of pulsating travelling fronts for fixed ɛ and |L| was proved by Berestycki & Hamel in [BH]. In the present paper, we investigate the behaviour of such solutions when [inline-graphic not available: see fulltext] More precisely, we establish that pulsating travelling fronts behave like travelling waves, when the period |L| is small and [inline-graphic not available: see fulltext]. We also study the convergence of the solution, as ɛ goes to zero (and |L| is fixed), toward a solution of a free boundary problem.