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

## TL;DR

This paper refines the estimates for the propagation speed of transition solutions in a combustion-type free boundary problem, showing how nonlinearity influences the speed with more precise bounds.

## Contribution

It introduces two approaches to establish sharper upper and lower bounds on the free boundary's propagation speed, highlighting the impact of nonlinearity.

## Key findings

- More accurate bounds on $h(t)$ for transition solutions.
- Nonlinearity $f$ significantly influences propagation speed.
- Transition solutions converge to ignition temperature $ heta$.

## Abstract

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.