  We are concerned with the nonlinear problem $u_t=u_{xx}+f(u)$, where $f$ is
of combustion type, coupled with the Stefan-type free boundary $h(t)$.
According to [4,5], for some critical initial data, the transition solution $u$
locally uniformly converges to $\theta$, which is the ignition temperature of
$f$, and the free boundary satisfies $h(t)=C\sqrt{t}+o(1)\sqrt{t}$ for some
positive constant $C$ and all large time $t$. In this paper, making use of two
different approaches, we establish more accurate upper and lower bound
estimates on $h(t)$ for the transition solution, which suggest that the
nonlinearity $f$ can essentially influence the propagation speed.
