Traveling waves for a two-phase Stefan problem with radiation

TL;DR

This paper proves the existence of traveling wave solutions in a two-phase Stefan problem with radiation, showing different behavior from classical models and analyzing their properties using advanced mathematical methods.

Contribution

It introduces and demonstrates the existence of traveling wave solutions in a radiation-influenced Stefan problem, contrasting with classical self-similar solutions.

Findings

Traveling waves exist with solid expansion.

Solutions differ from classical parabolic scaling.

Mathematical analysis confirms properties of these solutions.

Abstract

In this paper we study the existence of traveling wave solutions for a free-boundary problem modeling the phase transition of a material where the heat is transported by both conduction and radiation. Specifically, we consider a one-dimensional two-phase Stefan problem with an additional non-local non-linear integral term describing the situation in which the heat is transferred in the solid phase also by radiation, while the liquid phase is completely transparent, not interacting with radiation. We will prove that there are traveling wave solutions for the considered model, differently from the case of the classical Stefan problem in which only self-similar solutions with the parabolic scale $ x\sim \sqrt{t} $ exist. In particular we will show that there exist traveling waves for which the solid expands. The properties of these solutions will be studied using maximum-principle methods, blow-up limits and Liouville-type Theorems for non-linear integral-differential equations.