  We study a parabolic free boundary problem, arising from a model for the
propagation of equi-diffusional premixed flames with high activation energy. If
an initial data is compactly supported, then the solution vanishes in a finite
time, called the extinction time. In this paper, we give a quantitative
estimate on the flatness of the free boundary and prove that the free boundary
is asymptotically spherical and the solution is asymptotically self-similar.
