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arXiv · 2510.26088

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

Kensho Araya·Kazuhiro Ishige
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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.

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Kensho Araya, Kazuhiro Ishige. 2025-10-30. A one-dimensional Stefan problem for the heat equation with a nonlinear boundary condition. https://arxiv.org/abs/2510.26088

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