On systems of parabolic variational inequalities with multivalued terms

Abstract

In this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain Q=Ω×(0,τ): For k=1,…,m, find uk∈Kk and ηk∈Lp′k(Q) such that uk(⋅,0)=0 in Ω, ηk(x,t)∈fk(x,t,u1(x,t),…,um(x,t)),⟨ukt+Akuk,vk−uk⟩+∫Qηk(vk−uk)dxdt≥0, ∀ vk∈Kk,

where Kk is a closed and convex subset of Lpk(0,τ;W1,pk0(Ω)), Ak is a time-dependent quasilinear elliptic operator, and fk:Q×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.