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=varOmega times (0,tau ): For k=1,dots , m, find u_kin K_k and eta _kin L^{p'_k}(Q) such that uk(·,0)=0inΩ,ηk(x,t)∈fk(x,t,u1(x,t),⋯,um(x,t)),⟨ukt+Akuk,vk-uk⟩+∫Qηk(vk-uk)dxdt≥0,∀vk∈Kk,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}&u_k(\\cdot ,0)=0\\ \\text{ in } \\varOmega ,\\ \\ \\eta _k(x,t)\\in f_k(x,t,u_1(x,t), \\dots , u_m(x,t)), \\\\&\\langle u_{kt}+A_k u_k, v_k-u_k\\rangle +\\int _Q \\eta _k\\, (v_k-u_k)\\,dxdt\\ge 0,\\ \\ \\forall \\ v_k\\in K_k, \\end{aligned}$$\\end{document}where K_k is a closed and convex subset of L^{p_k}(0,tau ;W_0^{1,p_k}(varOmega )), A_k is a time-dependent quasilinear elliptic operator, and f_k:Qtimes mathbb {R}^mrightarrow 2^{mathbb {R}} is an upper semicontinuous multivalued function with respect to sin {mathbb R}^m. 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 K_k, systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.