Properties of the phase boundary in the parabolic problem with hysteresis

TL;DR

This paper analyzes the properties of the interface boundary in solutions to parabolic equations with hysteresis, establishing regularity results for different initial data and showing non-existence in certain cases.

Contribution

It provides new regularity results for the interface boundary in parabolic hysteresis problems and clarifies conditions for existence and regularity based on initial data.

Findings

Interface boundary is Holder continuous with exponent 1/2 for certain initial data.

Interface boundary is Lipschitz continuous for smoother initial data.

Solutions with interface boundary do not exist for non-transversal initial data.

Abstract

We study solutions of parabolic equations with a discontinuous hysteresis operator, described by a free interface boundary. It is established that for spatially transverse initial data from the space $W^{2-2/q}_q$ with $q > 3$, there exists a solution in the space $W^{2,1}_q$, where the interface boundary exhibits Holder continuity with an exponent $1/2$. Furthermore for initial data from the space $W^2_\infty$, it is proven that the interface boundary satisfies the Lipschitz condition. It is shown that for non-transversal initial data, solutions with an interface boundary do not exist.