No formulation of a new phase for a free boundary problem in combustion theory

TL;DR

This paper investigates a free boundary problem in combustion theory, demonstrating non-existence of solutions under certain conditions and exploring potential solutions with unbounded external temperature.

Contribution

It establishes conditions for the non-existence of solutions and introduces a self-similar solution when external temperature is unbounded.

Findings

No solutions when initial temperature equals fixed temperature with bounded heat source.

Existence of a self-similar solution with unbounded external temperature as time approaches zero.

Solution behavior depends on the boundedness of the external heat source.

Abstract

We consider a free boundary problem for the heat equation with a given non-negative external heat source. On the free boundary, we impose the zero Dirichlet condition and the fixed normal derivative so that heat escapes from the boundary. In various settings, we show that there exist no solutions when the initial temperature equals the fixed temperature no matter where the initial location of the free boundary is given provided that the external heat source is bounded from above. We also note that there is a chance to have a solution when the external temperature is unbounded as time tends to zero by giving a self-similar solution.