Symmetric periodic solutions of parabolic problems with hysteresis

TL;DR

This paper investigates symmetric periodic solutions in a multidimensional heat equation with hysteresis boundary control, analyzing their stability, coexistence, and the mechanisms behind their emergence and disappearance.

Contribution

It constructs all two-switching periodic solutions, studies their stability, and explores how multiple solutions coexist and evolve in hysteresis-controlled heat equations.

Findings

Multiple stable and unstable periodic solutions coexist.

A detailed mechanism for the appearance and disappearance of solutions.

Construction of all solutions with exactly two switches per period.

Abstract

We consider the heat equation in a multidimensional domain with nonlocal hysteresis feedback control in a boundary condition. Thermostat is our prototype model. We construct all periodic solutions with exactly two switching on the period and study their stability. Coexistence of several periodic solutions with different stability properties is proved to be possible. A mechanism of appearance and disappearance of periodic solutions is investigated.