EXTINCTION AND FOCUSING BEHAVIOR OF SPHERICAL AND ANNULAR FLAMES DESCRIBED BY A FREE-BOUNDARY PROBLEM

We consider a free-boundary problem for the heat equation which arises in the description of premixed equi-diffusional flames in the limit o f high activation energy. It consists of the heat equation u(t) = Delt a u, u > 0, posed in an a priori unknown set Omega subset of Q(T) = R- N x (0, T) for some T > 0 with boundary conditions on the free lateral boundary Gamma = partial derivative Omega boolean AND Q(T) (the flame front): u = 0 and partial derivative u/partial derivative nu = -1. We impose initial condition u(0)(x) greater than or equal to 0 on the kn own initial domain Omega(0) = <(Omega)over bar> boolean AND {t = 0}. T he paper establishes a theory of existence, uniqueness and regularity for radial symmetric solutions having bounded support. We remark that such solutions vanish in finite time (extinction phenomenon). In the p aper we analyze the different types of possible extinction behaviour. We also investigate the focusing behaviour for solutions whose support expands in finite time to fill a hole. In all the cases the asymptoti c behaviour is shown to be self-similar.