A one-dimensional Stefan problem for the heat equation with a nonlinear boundary condition

TL;DR

This work analyzes the one-dimensional, one-phase Stefan problem for the heat equation with a nonlinear boundary condition at $x=0$ and a moving boundary $s(t)$, given by $\partial_t u=\partial_x^2 u$, $-\partial_x u(t,0)=u(t,0)^p$, $u(t,s(t))=0$, and $s'(t)=-\partial_x u(t,s(t))$, with initial data $(s_0,\varphi)$. Using energy methods, blow-up analysis, and a scaling approach in the initial data, the authors establish a trichotomy of global-existence with exponential decay, global-existence with non-exponential decay, and finite-time blow-up, depending on the size of $\varphi$. A key tool is the energy functional $E(s(t),u(t))$ whose sign governs the global behavior, and the paper also characterizes the blow-up behavior of $(s,u)$ at the blow-up time. In the global-decay regime, the paper proves convergence $\|u(t)\|_{L^{\infty}}\to0$ and provides a small-data condition ensuring exponential decay, along with stability results under perturbations of the initial data. The results advance understanding of Stefan-type problems with nonlinear boundary interactions and offer explicit growth/decay rates for the moving boundary, including lower bounds like $s_\lambda(t) \ge C t^{(p-1)/(2p-1)}$ in certain regimes.

Abstract

We study the one-dimensional one-phase Stefan problem for the heat equation with a nonlinear boundary condition. We show that all solutions fall into one of three distinct types: global-in-time solutions with exponential decay, global-in-time solutions with non-exponential decay, and finite-time blow-up solutions. The classification depends on the size of the initial function. Furthermore, we describe the behavior of solutions at the blow-up time.