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The method of hysteresis operators in modelling phase transitions is applied here to the problem of asymptotic stabilization of solutions to a phase-eld system with hysteresis. While it is known that a unique global strong solution exists for every initial data and that the evolution process described by this system is thermodynam-ically consistent in the sense that the absolute temperature remains positive f o r a l l times and the Clausius-Duhem inequality holds almost everywhere, we studyfatcat:geyxofcsf5clbpeekqmq6ulsii