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

TL;DR

This paper classifies solutions to a one-dimensional Stefan problem with nonlinear boundary conditions, identifying conditions for exponential decay, non-exponential decay, or finite-time blow-up, and describes their behavior at blow-up.

Contribution

It provides a complete classification of solution behaviors based on initial data size for this nonlinear heat equation problem.

Findings

Solutions are classified into three types: exponential decay, non-exponential decay, and blow-up.

The classification depends on the initial function's size.

Behavior at blow-up time is characterized.

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.