Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Maria Gokieli

Faculty of Mathematics and Natural Sciences, School of Exact Sciences, Cardinal Stefan Wyszynski University, Warsaw, Poland

m.gokieli@uksw.edu.pl

Keywords

  • Variational inequalities
  • convex analysis
  • set-valued monotone operators
  • parabolic inequalities
  • superconductivity model

Mathematics Subject Classification

34G25, 35G45, 35K51, 35K57, 35K59

References

  1. G. Akagi: Convergence of functionals and its applcations to parabolic equations, Abstract Appl. Analysis 11 (2004) 907–933.https://doi.org/10.1155/S1085337504403030Google Scholar
  2. A. Azevedo, L. Santos: A diffusion problem with gradient constraint depending on the temperature, Adv. Math. Sci. Appl. 20 (2010) 151–166.Google Scholar
  3. A. Bensoussan, J. L. Lions: Contrôle impulsionnel et contrôle continu. Méthode des inéquations quasi variationnelles non linéaires, C. R. Acad. Sci. Paris 278 (1974) 675–679.Google Scholar
  4. H. Brézis: Opérateurs Maximaux Monotones et Semi-Groupes de Contraction dans les Espaces de Hilbert, Mathematical Studies 5, North-Holland, Amsterdam (1973).Google Scholar
  5. F. E. Browder: Nonlinear maximal monotone operators in Banach spaces, Math. Ann. 175 (1968) 89–113.https://doi.org/10.1007/BF01418765Google Scholar
  6. M. Cyrot, D. Pavuna: Introduction to Superconductivity and High-Tc Materials, World Scientific, Singapore (1992).https://doi.org/10.1142/1039Google Scholar
  7. T. Fukao, N. Kenmochi: Parabolic variational inequalities with weakly time-dependent constraints, Adv. Math. Sci. Appl. 23 (2013) 365–395.Google Scholar
  8. T. Fukao, N. Kenmochi: Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint, Discrete Contin. Dyn. Systems 6 (2015) 2523–2538.https://doi.org/10.3934/dcds.2015.35.2523Google Scholar
  9. M. Gokieli, N. Kenmochi, M. Niezgódka: Variational inequalities of Navier-Stokes type with time dependent constraints, J. Math. Analysis Appl. 449 (2017) 1229–1247.https://doi.org/10.1016/j.jmaa.2016.12.048Google Scholar
  10. M. Gokieli, N. Kenmochi, M. Niezgódka: A new compactness theorem for variational inequalities of parabolic type, Houston J. Math. 44 (2018) 319–350.Google Scholar
  11. M. Gokieli, N. Kenmochi, M. Niezgódka: Mathematical modelling of biofilm development, Nonlinear Analysis Real World Appl. 42 (2018) 422–447.https://doi.org/10.1016/j.nonrwa.2018.01.005Google Scholar
  12. M. Gokieli, N. Kenmochi, M. Niezgódka: Variational and quasi-variational inequalities of Navier-Stokes type with velocity constraints, Adv. Math. Sci. Appl. 27 (2018) 359–402.Google Scholar
  13. M. Hintermüller, C. Rautenberg: Parabolic variational inequalities with gradient-type constraints, SIAM J. Optimization 23 (2013) 2090–2123.https://doi.org/10.1137/120874308Google Scholar
  14. M. Hintermüller, C. Rautenberg: On the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities, Port. Math. 74 (2017) 1–35.https://doi.org/10.4171/pm/1991Google Scholar
  15. J.-L. Joly, U. Mosco: A propos de l’existence et de la régularité des solutions de certaines inéquations quasi-variationnelles, J. Funct. Analysis 34 (1979) 107–137.https://doi.org/10.1016/0022-1236(79)90028-4Google Scholar
  16. A. Kadoya, Y. Murase, N. Kenmochi: A class of nonlinear parabolic systems with environmental constraints, Adv. Math. Sci. Appl. 20 (2010) 281–313.Google Scholar
  17. R. Kano, N. Kenmochi, Y. Murase: Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publication 86 (2009) 175–194.https://doi.org/10.4064/bc86-0-11Google Scholar
  18. N. Kenmochi: Solvability of nonlinear evolution equations with time-dependent constraits and applications, Bull. Fac. Education, Chiba Univ. 30 (1981) 1–87.Google Scholar
  19. N. Kenmochi: Parabolic quasi-variational diffusion problems with gradient constraints, Discrete Contin. Dyn. Systems 6 (2013) 423–438.https://doi.org/10.3934/dcdss.2013.6.423Google Scholar
  20. N. Kenmochi, M. Niezgódka: Weak solvability for parabolic variational inclusions and applications to quasi-variational inequalities, Adv. Math. Sci. Appl. 25 (2016) 62–97.Google Scholar
  21. M. Kunze, J. F. Rodrigues: An elliptic quasi-variational inequality with gradient constraints and some of applications, Math. Methods Appl. Sci. 23 (2000) 897–908.https://doi.org/10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-HGoogle Scholar
  22. J. Leray, J. L. Lions: Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France 93 (1965) 97–107.https://doi.org/10.24033/bsmf.1617Google Scholar
  23. F. Mignot, J.-P. Puel: Inéquations d’évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi variationelles d’évolution, Arch. Rational Mech. Analysis 64 (1977) 493–519.https://doi.org/10.1007/BF00280179Google Scholar
  24. U. Mosco: On the continuity of the Young-Fenchel transform, J. Math. Analysis Appl. 35 (1971) 518–535.https://doi.org/10.1016/0022-247X(71)90200-9Google Scholar
  25. Y. Murase: Abstract quasi-variational inequalities of elliptic type and applications, Banach Center Publication 86 (2009) 235–246.https://doi.org/10.4064/bc86-0-15Google Scholar
  26. L. Prigozhin: On the Bean critical-state model in superconductivity, European J. Appl. Math. 7 (1996) 237–247.https://doi.org/10.1017/S0956792500002333Google Scholar
  27. J.-F. Rodrigues, L. Santos: A parabolic quasi-variational inequality arising in a superconductivity model, Ann. Scuola Norm. Sup. Pisa, Sci. 29 (2000) 153–169.Google Scholar
  28. U. Stefanelli: Nonlocal quasi-variational evolution equations, J. Diff. Equations 229 (2006) 204–228.https://doi.org/10.1016/j.jde.2006.05.004Google Scholar
  29. L. Tartar: Inéquations quasi variationnelles abstraites, C. R. Acad. Sci. Paris 278 (1974) 1193–1196.Google Scholar
  30. Y. Yamada: On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo Sect. IA 23 (1976) 491–515.Google Scholar

Suggested citation

M. Gokieli, N. Kenmochi, M. Niezgódka. “Parabolic Quasi-Variational Inequalities. I: Semimonotone Operator Approach.” Journal of Convex Analysis 29 (2022), No. 2, 531–558. https://doi.org/10.68381/jca29031

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2022. Rights now held by Banach Press LLC.