Journal of Convex Analysis 29 (2022), No. 2, 531--558
Copyright Heldermann Verlag 2022
Parabolic Quasi-Variational Inequalities. I: Semimonotone Operator Approach
Maria Gokieli
Faculty of Mathematics and Natural Sciences, School of Exact Sciences, Cardinal Stefan Wyszynski University, Warsaw, Poland
m.gokieli@uksw.edu.pl
Nobuyuki Kenmochi
Faculty of Education, Chiba University, Chiba, Japan
nobuyuki.kenmochi@gmail.com
Marek Niezgódka
CNT Center, Cardinal Stefan Wyszynski University, Warsaw, Poland
m.niezgodka@uksw.edu.pl
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.
Keywords: Variational inequalities, convex analysis, set-valued monotone operators, parabolic inequalities, superconductivity model.
MSC: 34G25, 35G45, 35K51, 35K57, 35K59.
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