Multi-valued parabolic variational inequalities and related variational-hemivariational inequalities

Abstract

In this paper we study multi-valued parabolic variational inequalities involving quasilinearparabolic operators, and multi-valued integral terms over the underlying parabolic cylinderas well as over parts of the lateral parabolic boundary, where the multi-valued functionsinvolved are assumed to be upper semicontinuous only. Note, since lower semicontinuousmulti-valued functions allow always for a Carath ́eodory selection, this case can be consid-ered as the trivial case, and therefore will be omitted. Our main goal is threefold: First,we provide an analytical frame work and an existence theory for the problems under con-sideration. Unlike in recent publications on multi-valued parabolic variational inequalities,the closed convex setKrepresenting the constraints is not required to possess a nonemptyinterior. Second, we prove enclosure and comparison results based on a recently developedsub-supersolution method due to the authors. Third, we consider classes of relevant gen-eralized parabolic variational-hemivariational inequalities that will be shown to be specialcases of the multi-valued parabolic variational inequalities under consideration.