On behavior of free boundaries to generalized two-phase Stefan problems for parabolic partial differential equation systems

TL;DR

This paper analyzes the behavior of free boundaries in generalized two-phase Stefan problems for parabolic PDEs, focusing on existence, uniqueness, and maximal interval of solutions under certain conditions.

Contribution

It improves solution regularity to overcome previous difficulties and establishes local existence, uniqueness, and maximal interval results for the problem.

Findings

Established local in time existence and uniqueness of solutions.

Improved regularity conditions to handle boundary dependencies.

Derived conditions for the maximal interval of solution existence.

Abstract

Recently, we have proposed a new free boundary problem representing the bread baking process in a hot oven. Unknown functions in this problem are the position of the evaporation front, the temperature field and the water content. For solving this problem we observed two difficulties that the growth rate of the free boundary depends on the water content and the boundary condition for the water content contains the temperature. In this paper, by improving the regularity of solutions, we overcome these difficulties and establish existence of a solution locally in time and its uniqueness. Moreover, under some sign conditions for initial data, we derive a result on the maximal interval of existence to solutions.