Parabolic Quasi-Variational Inequalities (I) -- Semimonotone Operator Approach

TL;DR

This paper develops an abstract framework for parabolic quasi-variational inequalities involving semimonotone operators, addressing problems with unknown convex constraints in biochemical and mechanical systems.

Contribution

It introduces a novel class of parabolic QVIs with unknown convex constraints, formulated via semimonotone operators in reflexive Banach spaces.

Findings

Established a compactness theorem for parabolic variational inequalities.

Formulated a prototype model involving feedback systems and unknown convex constraints.

Provided a precise mathematical formulation for a new class of QVIs.

Abstract

Variational inequalities, formulated on unknown dependent convex sets, are called quasi-variational inequalities (QVI). This paper is concerned with the abstract approach to a class of parabolic QVIs arising in many biochemical/mechanical problems, based on a compactness theorem for parabolic variational inequalities (cf. [9]). The prototype of our model for QVIs of parabolic type is formulated in a reflexive Banach space as the sum of the time-derivative operator under unknown convex constraints and a semimonotone operator, including a feedback system which selects a convex constraint. The main objective of this work is to specify a class of unknown-state dependent convex constraints and to give a precise formulation of QVIs.