UNIQUENESS OF LIMIT SOLUTIONS TO A FREE BOUNDARY PROBLEM FROM COMBUSTION J. FERN ´ ANDEZ BONDER AND N. WOLANSKI Abstract. We investigate the uniqueness of limit solutions for a free boundary problem in heat propagation that appears as a limit of a parabolic system that arises in flame propagation. 1. Introduction In this paper we consider the following problem arising in combustion theory (1.1)  Δu ε − u ε t = Y ε f ε (u ε ) in D, ΔY ε − Y ε t = Y ε f ε (u ε ) in D, where D⊂ R N +1 . This model appears in combustion theory in the analysis of the propagation of curved flames. It is derived in the framework of the theory of equidiffusional premixed flames analyzed in the relevant limit of high activation energy for Lewis number 1. In this application, Y ε represents the fraction of some reactant (and hence it is assumed to be nonnegative), and u ε is minus the temperature (more precisely, u ε = λ(T f − T ε ) where T f is the flame temperature and λ is a normalization factor). Observe that the term Y ε f ε (u ε ) acts as an absorption term in the equation (1.1). Since T ε = T f − (u ε /λ), it is in fact a reaction term for the temperature. In the flame model, such a term represents the effect of the exothermic chemical reaction and f has accordingly a number of properties: it is a nonnegative Lipschitz continuous function which is positive in an interval (−∞,ε) and vanishes otherwise (i.e., reaction occurs only when T>T f − ε λ ). The parameter ε is essentially the inverse of the activation energy of the chemical reaction. For the sake of simplicity we will assume that f ε (s)= 1 ε 2 f ( s ε ), where f is a nonnegative, Lipschitz continuous function with support in (−∞, 1]. For the derivation of the model, we cite [1]. 1991 Mathematics Subject Classification. Primary 35K05, 35K60, 80A25. Key words and phrases. Free-boundary problem, combustion, heat equation, uniqueness, classical so- lution, limit solution. Partially supported by grant BID1201/OC-AR PICT 03-00000-05009 and CONICET grant PIP0660/98. The second author is a member of CONICET (Consejo Nacional de Investigaciones Cient´ ıficas y T´ ecnicas of Argentina). 1