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We are concerned with the nonlinear problem $u_t=u_{xx}+f(u)$, where $f$ isof 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 somepositive constant $C$ and all large time $t$. In this paper, making use of twodifferent approaches, we establish more accurate upper and lower boundestimates on $h(t)$ for the transition solution, which suggest that thenonlinearity $f$ can essentially influence the propagation speed.

Refined estimates for the propagation speed of the transition solution to a free boundary problem with a nonlinearity of combustion type