Semilinear Parabolic Equations with Preisach Hysteresis

A coupled system consisting of a nonlinear parabolic partial differential equation and a family of ordinary differential equations is realized as an abstract Cauchy problem. We establish coercivity and accretiveness estimates for the multi-valued operator in the abstract Cauchy problem, and then apply the Crandall-Liggett theorem to recover the integral solution for the abstract Cauchy problem. A special case of the coupled system corresponds to the Super -Stefan problem, i.e. the delayed phase transition model for a material subjected to super-heating and super-cooling. In this case, the nonlinearity in the parabolic partial differential equation appears as the time derivative of a simple relay hysteresis functional produced by one member of the family of ordinary differential equations. Another special case of the coupled system corresponds to a one-dimensional derivation from Maxwell's equations for a ferromagnetic body under slowly varying field conditions. In this case, the nonlinearity is the time derivative of the classical Preisach hysteresis functional, and the family of ordinary differential equations produce a family of simple relay hysteresis functionals present in the construction of the Preisach hysteresis functional.