Forward–backward parabolic equations and hysteresis

Abstract.

A forward-backward parabolic problem is obtained by coupling the equation ${\partial\over\partial t}(u+w) -\Delta u= f$ with a nonmonotone relation $u=\alpha(w)$. 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.