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

We are concerned with the nonlinear problem ut = uxx+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 θ, which is the ignition temperature off, and the free boundary satisfies h(t) =C√t+o(1)√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.