Forward-backward parabolic equations and hysteresis

A forward-backward parabolic problem is obtained by coupling the equation (partial derivative)/(partial derivativet) (u + w) - Deltau = f with a nonmonotone relation u = a(w). in the framework of a two-scale model, we replace the latter condition by a relaxation dynamics which converges to a hysteresis relation. We provide a suitable formulation of the hysteresis law, approximate it by the relaxation dynamics, couple it with the P.D.E., derive uniform estimates via an L-1-technique, and then pass to the limit as the relaxation parameter vanishes. This yields existence of a solution for the modified problem. This procedure is also applied to other equations.