Solvability of a class of phase field systems related to a sliding mode   control problem
  We consider a phase-field system of Caginalp type perturbed by the presence of an additional maximal monotone nonlinearity. Such a system arises from a recent study of a sliding mode control problem. We prove existence and regularity of the solutions. Moreover, under further assumptions, we show the continuous dependence on the initial data and the uniqueness of the solution. 
Michele Colturato
math.AP