Singular limit and homogenization for flame propagation in periodic excitable media

This paper is concerned with a class of singular equations modeling the combustion of premixed gas in periodic media. The model involves two parameters: the period of the medium |L| and a singular parameter ε related to the activation energy. The existence of pulsating travelling fronts for fixed ε and |L| was proved by H. Berestycki and F. Hamel in [BH]. In the present paper, we investigate the behaviour of such solutions when ε ≤ ε|L| 1. More precisely, we establish that Pulsating Travelling Fronts behave like Travelling Waves, when the period |L| is small and ε ≤ ε|L|. We also study the convergence as ε goes to zero (and |L| is fixed) of the solution toward a solution of a free boundary problem.