On Parabolic Variational Inequalities with Multivalued Terms and Convex Functionals

Abstract

Abstract In this paper, we consider the following parabolic variational inequality containing a multivalued term and a convex functional: Find u ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) {u\in L^{p}(0,T;W^{1,p}_{0}(\Omega))} and f ∈ F ⁢ ( ⋅ , ⋅ , u ) {f\in F(\cdot,\cdot,u)} such that u ⁢ ( ⋅ , 0 ) = u 0 {u(\cdot,0)=u_{0}} and 〈 u t + A ⁢ u , v - u 〉 + Ψ ⁢ ( v ) - Ψ ⁢ ( u ) ≥ ∫ Q f ⁢ ( v - u ) ⁢ 𝑑 x ⁢ 𝑑 t for all ⁢ v ∈ L p ⁢ ( 0 , T ; W 0 1 , p ⁢ ( Ω ) ) , \langle u_{t}+Au,v-u\rangle+\Psi(v)-\Psi(u)\geq\int_{Q}f(v-u)\,dx\,dt\quad% \text{for all }v\in L^{p}(0,T;W^{1,p}_{0}(\Omega)), where A is the principal term; F is a multivalued lower-order term; Ψ ⁢ ( u ) = ∫ 0 T ψ ⁢ ( t , u ) ⁢ 𝑑 t {\Psi(u)=\int_{0}^{T}\psi(t,u)\,dt} is a convex functional. Moreover, we study the existence and other properties of solutions of this inequality assuming certain growth conditions on the lower-order term F.