Symmetric Periodic Solutions of Parabolic Problems With Discontinuous Hysteresis

We develop a framework for treating the long-term behavior of solutions for parabolic equations in multidimensional domains with discontinuous hysteresis. Bearing in mind the thermostat model, we concentrate in this paper on the prototype heat equation with hysteresis in the boundary condition. We provide an algorithm for constructing all periodic solutions with exactly two switchings 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.