Abstract
We consider a free-boundary problem for the heat equation which arises in the description of premixed equi-diffusional flames in the limit of high activation energy. It consists of the heat equation u t = Δu, u > o, posed in an a priori unknown set Ω ⊂ Q T = R N × (0, T) for some T > 0 with boundary conditions on the free lateral boundary τ = ∂Ω∩ Q T (the flame front): u = o and ∂u ∂v = − 1. We impose initial condition u 0( x) ≥ 0 on the known initial domain Ω 0 = Ω ∩ {t = 0} . The 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 paper 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 asymptotic behaviour is shown to be self-similar.
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