A singular limit arising in combustion theory: Fine properties of the free boundary

The equation where converges to the Dirac measure concentrated at with mass has been used as a model for the propagation of flames with high activation energy. For initial data that are bounded in and have a uniformly bounded support, we study non-negative solutions of the Cauchy problem in as We show that each limit of is a solution of the free boundary problem in on (in the sense of domain variations and in a more precise sense). For a.e. time t the graph of u(t) has a unique tangent cone at -a.e. The free boundary is up to a set of vanishing measure the sum of a countably n-1-rectifiable set and of the set on which vanishes in the mean. The non-degenerate singular set is for a.e. time a countably n-1-rectifiable set. As key tools we introduce a monotonicity formula and, on the singular set, an estimate for the parabolic mean frequency. Received: 8 August 2001 / Accepted: 8 May 2002 / Published online: 5 September 2002 RID="a" ID="a" Partially supported by a Grant-in-Aid for Scientific Research, Ministry of Education, Japan.