Parabolic equations with hysteresis

We consider a nonlinear parabolic equation with spatially distributed (discontinuous) hysteresis. Due to discontinuous nature of hysteresis well-posedness of the problem is not trivial.

Depending on the initial data we distinguish two cases: "transverse" and "non-transverse".

For the transverse case we show a connection of the above problem with free boundary problems. This allows us to find sufficient conditions that guarantee the well-posedness (existence, uniqueness and continuous dependence on initial data), which is generally not typical for systems with discontinuous hysteresis.

For the non-transverse case we consider spatial discretization of the problem, which leads us to a lattice differential equation. In that case we observe a strange "rattling" phenomenon, which leads to non-trivial pattern formation. The main technique for this case is careful estimate of the discrete parabolic Green function.