Resumo

Hysteresis naturally appears as a mechanism of self-organization and is often used in control theory. Important features of hysteresis operator are discontinuity and memory. We consider reaction-diffusion equations with hysteresis. Such equations describe processes in which diffusive and non-diffusive instances interact according to a hysteresis law. Due to the discontinuity of hysteresis, questions of well-posedness of such equations are highly non trivial.

For so-called transverse initial data it is possible to establish existence and uniqueness of the solution. Important part of the proof is the free boundary problem and fixed point theorem.

For non transverse initial data we consider a spatial discretization of the problem and present a new mechanism of pattern formation, which we call rattling. The profile of the solution forms two hills propagating with non-constant velocity. The profile of hysteresis forms a highly oscillating quasiperiodic pattern, which explains mechanism of illposedness of the original problem and suggests a possible regularization. Rattling is very robust and persists in arbitrary dimension and in systems acting on different time scales. We expect that it could be explained via Young measures – this is subject of future research.