On systems of parabolic variational inequalities with multivalued terms

Abstract

AbstractIn this paper we present an analytical framework for the following system of multivalued parabolic variational inequalities in a cylindrical domain $$Q=\varOmega \times (0,\tau )$$ Q = Ω × ( 0 , τ ) : For $$k=1,\dots , m$$ k = 1 , ⋯ , m , find $$u_k\in K_k$$ u k ∈ K k and $$\eta _k\in L^{p'_k}(Q)$$ η k ∈ L p k ′ ( Q ) such that $$\begin{aligned}&u_k(\cdot ,0)=0\ \text{ in } \varOmega ,\ \ \eta _k(x,t)\in f_k(x,t,u_1(x,t), \dots , u_m(x,t)), \\&\langle u_{kt}+A_k u_k, v_k-u_k\rangle +\int _Q \eta _k\, (v_k-u_k)\,dxdt\ge 0,\ \ \forall \ v_k\in K_k, \end{aligned}$$ u k ( · , 0 ) = 0 in Ω , η k ( x , t ) ∈ f k ( x , t , u 1 ( x , t ) , ⋯ , u m ( x , t ) ) , ⟨ u kt + A k u k , v k - u k ⟩ + ∫ Q η k ( v k - u k ) d x d t ≥ 0 , ∀ v k ∈ K k , where $$K_k $$ K k is a closed and convex subset of $$L^{p_k}(0,\tau ;W_0^{1,p_k}(\varOmega ))$$ L p k ( 0 , τ ; W 0 1 , p k ( Ω ) ) , $$A_k$$ A k is a time-dependent quasilinear elliptic operator, and $$f_k:Q\times \mathbb {R}^m\rightarrow 2^{\mathbb {R}}$$ f k : Q × R m → 2 R is an upper semicontinuous multivalued function with respect to $$s\in {\mathbb R}^m$$ s ∈ 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 $$K_k$$ K k , systems of evolutionary variational-hemivariational inequalities are shown to be a subclass of the above system of multivalued parabolic variational inequalities.