A free boundary problem arising in flame propagation

We study a free boundary problem describing the propagation of laminar flames. The problem arises as the limit of a singular perturbation problem. We introduce the notion of viscosity solutions for the problem to show the maximum principle-type property of the solutions. Using this property we show the uniform convergence of the approximating solutions and the uniqueness of the viscosity solution under several geometric conditions on the initial data. 0 Introduction In this paper we consider a free boundary problem for the heat equation. The classical formulation is as the following. Consider Ω0: an open subset of IR n and a nonnegative initial data u0 ∈ C(IR) with its nonempty bounded positivity set {u0 > 0} = Ω0. The problem consists in finding a nonnegative continuous function u in Q = IR × (0,∞) such that