  We study a two-point free boundary problem in a sector for a quasilinear
parabolic equation. The boundary conditions are assumed to be spatially and
temporally "self-similar" in a special way. We prove the existence, uniqueness
and asymptotic stability of an expanding solution which is self-similar at
discrete times. We also study the existence and uniqueness of a shrinking
solution which is self-similar at discrete times.
