Skip to main content
SpringerLink
Log in
Book cover

Qualitative and Quantitative Analysis of Nonlinear Systems pp 125–158Cite as

Strongest Convergence Results for Weak Solutions of Differential-Operator Equations and Inclusions

  • Chapter
  • First Online:

  • 967 Accesses

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 111))

Abstract

In this chapter we establish strongest convergence results for weak solutions of differential-operator equations and inclusions. In Sect. 6.1 we consider first order differential-operator equations and inclusions . Section 6.2 devoted to convergence results for weak solutions of second order operator differential equations and inclusions. In Sect. 6.3 we consider the following examples of applications: nonlinear parabolic equations of divergent form; nonlinear problems on manifolds with and without boundary: a climate energy balance model; a model of conduction of electrical impulses in nerve axons; viscoelastic problems with nonlinear “reaction-displacement” law.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    I.e., \(V_i\) is a real reflexive separable Banach space continuously and densely embedded into a real Hilbert space H, H is identified with its topologically conjugated space \(H^*\), \(V_i^*\) is a dual space to \(V_i\). So, there is a chain of continuous and dense embeddings: \(V_i\subset H\equiv H^*\subset V_i^*\) (see, for example, Gajewski, Gröger, and Zacharias [12, Chap. I]).

  2. 2.

    We remark that operators A and B are continuous on V [12, Chap. III].

  3. 3.

    We remark that \(\sqrt{\langle Au, u\rangle _V}\) is equivalent norm on V, generated by inner product \(\langle A u, v\rangle _V\).

  4. 4.

    This section is based on results of [23] and references therein.

References

  1. Aubin, J.-P., Cellina, A.: Differential Inclusions. Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften, vol. 264. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  2. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Systems Control Foundation Application. Birkhauser, Boston (1990)

    Google Scholar 

  3. Babin, A.V., Vishik, M.I.: Maximal attractors of semigroups corresponding to evolution differential equations. Math. USSR-Sb. 54(2), 387–408 (1986)

    Article  MATH  Google Scholar 

  4. Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations [in Russian]. Nauka, Moscow (1989)

    MATH  Google Scholar 

  5. Chepyzhov, V.V., Vishik, M.I.: Trajectory attractors for evolution equations. C. R. Acad. Sci. Paris Ser. I Math. 321(10), 1309–1314 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Chepyzhov, V.V., Vishik, M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76(10), 913–964 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chepyzhov, V.V., Vishik, M.I.: Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete Contin. Dyn. Syst. 27(4), 1493–1509 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  9. Dubinskii, YuA: Higher-order nonlinear parabolic equations. J. Soviet Math. 56(4), 2557–2607 (1991)

    Article  MATH  Google Scholar 

  10. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)

    Book  MATH  Google Scholar 

  11. Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations. RAM Research in Applied Mathematics, vol. 37. Wiley, Chichester (1994)

    MATH  Google Scholar 

  12. Gajewski, H., Groger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferential-gleichungen. Akademie-Verlag, Berlin (1974)

    MATH  Google Scholar 

  13. Gorban, N.V., Kapustyan, O.V., Kasyanov, P.O.: Uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory’s nonlinearity. Nonlinear Anal. Theory Methods Appl. 98, 13–26 (2014). doi:10.1016/j.na.2013.12.004

    Article  MathSciNet  MATH  Google Scholar 

  14. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence (1988)

    MATH  Google Scholar 

  15. Kasyanov, P.O.: Multi-valued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity. Math. Notes 92(2), 57–70 (2012)

    MATH  Google Scholar 

  16. Kasyanov, P.O., Toscano, L., Zadoianchuk, N.V.: Long-time behavior of solutions for autonomous evolution hemivariational inequality with multidimensional “reaction-displacement” law. Abstr. Appl. Anal. (2012). doi:10.1155/2012/450984

  17. Kuttler, K.: Non-degenerate implicit evolution inclusions. Electron. J. Differ. Equ. 34, 1–20 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Ladyzhenskaya, O.A.: The Dynamical System That is Generated by the Navier–Stokes Equations [in Russian]. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov, vol. 27(6), pp. 91–115. Nauka, Leningrad (1972)

    Google Scholar 

  19. Ladyzhenskaya, O.A.: The Infinite-dimensionality of Bounded Invariant Sets for the Navier–Stokes System and Other Dissipative Systems [in Russian]. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov, vol. 115(6), pp. 137–155. Nauka, Leningrad (1982)

    Google Scholar 

  20. Ladyzhenskaya, O.: Attractors for Semigroups and Evolution Equations. Lezioni Lincee. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  21. Lions, J.-L.: Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  22. Migorski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Global Optim. 31(3), 505–533 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Migórski, S.: Dynamic hemivariational inequalities in contact mechanics. Nonlinear Anal. 63, 77–86 (2005). doi:10.1016/j.na.2005.01.012

    Article  MATH  Google Scholar 

  24. Melnik, V.S., Valero, J.: On attractors of multi-valued semi-flows and generalized differential equations. Set-Valued Anal. 6(1), 83–111 (1998)

    Article  MathSciNet  Google Scholar 

  25. Sell, G.R.: Global attractors for the three-dimensional Navier-Stokes equations. J. Dyn. Differ. Equ. 8(1), 1–33 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)

    MATH  Google Scholar 

  27. Vishik, M.I., Chepyzhov, V.V.: Trajectory and global attractors of the three-dimensional Navier-Stokes system. Mat. Zametki. 71(2), 194–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vishik, M.I., Zelik, S.V., Chepyzhov, V.V.: Strong trajectory attractor for a dissipative reaction-diffusion system. Dokl. Ross. Akad. Nauk. 435(2), 155–159 (2010)

    MATH  Google Scholar 

  29. Zadoyanchuk, N.V., Kas’yanov, P.O.: Faedo-Galerkin method for second-order evolution inclusions with \(W_\lambda \)-pseudomonotone mappings. Ukr. Math. J. 61(2), 236–258 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zgurovsky, M.Z., Mel’nik, V.S., Kasyanov, P.O.: Evolution Inclusions and Variation Inequalities for Earth Data Processing II. Springer, Berlin (2011)

    MATH  Google Scholar 

  31. Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V.: Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine

    Michael Z. Zgurovsky

  2. Institute for Applied System Analysis, Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine

    Pavlo O. Kasyanov

Authors
  1. Michael Z. Zgurovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pavlo O. Kasyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Z. Zgurovsky .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Cite this chapter

Zgurovsky, M.Z., Kasyanov, P.O. (2018). Strongest Convergence Results for Weak Solutions of Differential-Operator Equations and Inclusions. In: Qualitative and Quantitative Analysis of Nonlinear Systems. Studies in Systems, Decision and Control, vol 111. Springer, Cham. https://doi.org/10.1007/978-3-319-59840-6_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-59840-6_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59839-0

  • Online ISBN: 978-3-319-59840-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation