  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.
