A generalized phase relaxation model with hysteresis

A nonlinear system of PDEs describing a phase transition phenomenon is introduced. The energy balance equation takes into account the action of the interior dissipative forces driven by the microscopic movements of particles. Moreover, it is assumed that the phase transition process is characterized by some hysteresis effect. This gives rise to a particular form of the kinetic equation for the order parameter. The existence of a local in time solution to the coupled system is proved via a Schauder fixed point argument. Global existence holds under suitable monotonicity, contractivity, and boundedness properties of the hysteresis operators characterizing the process. It is worth noting that such assumptions are fulfilled in the physically relevant case. The extension argument relies on maximum principle techniques together with a comparison procedure.