On Systems of Parabolic Variational Inequalities with Multivalued Terms

In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain [Formula] where Kk is a closed and convex subset of [Formula], Ak is a time-dependent quasilinear elliptic operator, and fk: Q x ℝm → 2ℝ is an upper semicontinuous multivalued function with respect to s ∈ ℝm. We provide an existence theory for the above system under certain coercivity assumptions. In the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence and enclosure results. As an application, a multivalued parabolic obstacle system is treated. Moreover, under a lattice condition on the constraints Kk, systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities