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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

Nobuyuki Kenmochi
Faculty of Education, Chiba University, Chiba, Japan

Marek Niezgódka
CNT Center, Cardinal Stefan Wyszynski University, Warsaw, Poland

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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