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Solvability of a class of phase field systems related to a sliding mode control problem. (English) Zbl 1413.35265
Summary: 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 the existence of strong solutions. Moreover, under further assumptions, we show the continuous dependence on the initial data and the uniqueness of the solution.

MSC:
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35K25 Higher-order parabolic equations
35B25 Singular perturbations in context of PDEs
35D30 Weak solutions to PDEs
80A22 Stefan problems, phase changes, etc.
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