Convexity and uniqueness in a free boundary problem arising in combustion theory

We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain O I QT = Rn x (0,T) for some T > 0 and a function u > 0 in O such thatut = ?u, in O,u = 0 and |Nu| = 1, on G := ?O n QT,u(·,0) = u0, on O0,where O0 is a given domain in Rn and u0 is a positive and continuous function in O0, vanishing on ?O0. If O0 is convex and u0 is concave in O0, then we show that (u,O) is unique and the time sections Ot are convex for every t I (0,T), provided the free boundary G is locally the graph of a Lipschitz function and the fixed gradient condition is understood in the classical sense.