Well-posedness for a two-phase Stefan problem with radiation

TL;DR

This paper establishes the local and global well-posedness of a one-dimensional two-phase Stefan problem involving conduction and radiation, with a non-local integral operator in the temperature equation.

Contribution

It introduces a novel analysis of a Stefan problem with radiation and a non-local term, proving well-posedness using fixed-point and parabolic theory methods.

Findings

Proved local well-posedness of the problem.

Developed a global well-posedness theory for broad initial data.

Utilized sub- and supersolutions to analyze solution behavior.

Abstract

In this paper we consider a free boundary problem for the melting of ice where we assume that the heat is transported by conduction in both the liquid and the solid part of the material and also by radiation in the solid. Specifically, we study a one-dimensional two-phase Stefan-like problem which contains a non-local integral operator in the equation describing the temperature distribution of the solid. We will prove the local well-posedness of this free boundary problem combining the Banach fixed-point theorem and classical parabolic theory. Moreover, constructing suitable stationary sub- and supersolutions we will develop a global well-posedness theory for a large class of initial data.