Parabolic Quasi-Variational Inequalities. I: Semimonotone Operator Approach

Variational inequalities, formulated on unknown-dependent convex sets, are called quasi-variational inequalities (QVI). This paper is concerned with an abstract approach to a class of parabolic QVIs arising in many biochemical/mechanical problems. The approach is based on a compactness theorem for parabolic variational inequalities shown previously by the authors [A new compactness theorem for variational inequalities of parabolic type, Houston J. Math. 44 (2018) 319--350]. 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.