Nonlocal phase-field models for non-isothermal phase transitions and hysteresis

In this paper a nonlocal phase-field model for non-isothermal phase transitions with a non-conserved order parameter is studied. The paper complements recent investigations by S. Zheng and the second author and treats the case when the part of the free energy density forcing the order parameter to attain values within the physically meaningful range [0,1] is not given by a logarithmic expression but by the indicator function of [0,1]. The resulting field equations form a system of integro-partial differential inclusions that are highly nonlinearly coupled. For this system, results concerning global existence, uniqueness and large-time asymptotic behaviour are derived. The main results are proved by first transforming the system of inclusions into an equivalent system of equations in which hysteresis operators occur, and then employing techniques similar to those recently developed by the authors for phase-field systems involving hysteresis operators