QUARTERLY OF APPLIED MATHEMATICS VOLUME LI, NUMBER 1 MARCH 1993, PAGES 43-53 EXISTENCE THEOREMS FOR A FREE BOUNDARY PROBLEM IN COMBUSTION THEORY By ROBERTO GIANNI and PAOLA MANNUCCI Universita di Firenze, Firenze, Italy 0. Introduction. In this paper we analyze a one-dimensional parabolic partial dif- ferential equation which models a large number of physical phenomena. These are phenomena which, viewed on an appropriate time-scale, exhibit a switch-like be- haviour. For this reason we have a source term in the equation that is discontinuous as a function of the dependent variable with a jump discontinuity. This kind of problem was investigated by Norbury and Stuart in [1] where the equation was derived to model a combustion problem in a porous medium. In further papers, [4, 6, 7], Norbury and Stuart studied some mathematical aspects of such an equation (travelling waves, steady solutions, stability, and asymptotics). In [9], Friedman and Tzavaras proved the existence of a weak solution for the complete system of equations proposed in [ 1 ] to describe combustion problems. The aim of this paper is to prove a global existence theorem of a classical solution having a "regular" free boundary, that is, a curve x = s(t) which separates the domain in which we study the problem into two regions and through which the source term exhibits a jump. The problem that we study, in the standard functional space, is: ut - (K{u)ux)x = f(u)M(u - 1) inQ, u(x, 0) = <t>(x) in (0, 1), t) +&ux{0, t) = g^t) in(0,r), Wu(l, t) +3tux{\, t) = g2(t) in (0, T), where ^ = (0, l)x(0, 7"); (s/ , 38), (£?, 3$) are either (0, 1) or (1,0). In the problem studied by Norbury and Stuart, which will be our reference model, u(x, t) is the temperature of a porous medium. H(*) is the Heaviside function, that we define to be zero when its argument is nonpositive, and one otherwise. The interface is the level set 6 = {(x, t): u(x, t) = 1}. We look for a solution for which 6 is a regular curve x = s(t) such that ux{x, t) and ut(x, t) are continuous across {x — s(/)}; this means that the discontinuity, due to the presence of a discontinuous source term in the equation, appears only on Received January 7, 1991. 1991Mathematics Subject Classification. Primary 35K57, 35R35, 80A25. Work partially supported by the Strategic Project "Mathematical Models for Industry" of the Italian C.N.R. ©1993 Brown University 43