Uniform Global Attractors for Nonautonomous Evolution Inclusions

  • Mikhail Z. Zgurovsky
  • Pavlo O. KasyanovEmail author
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 69)


In this note, we prove the existence and provide basic structure properties of compact (in the natural phase space) uniform global attractor for all global weak solutions of the general classes of nonautonomous evolution equations and inclusions that satisfy standard sign and polynomial growth conditions. The obtained results allow to reduce the problem of the complete qualitative investigation of various nonlinear systems into the “small” (compact) part of the natural phase space.


Weak Solution Global Attractor Stochastic Partial Differential Equation Global Weak Solution Nonempty Closed Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was partially supported by the Ukrainian State Fund for Fundamental Researches under grant GP/F66/14921, and by the National Academy of Sciences of Ukraine under grant 2284.


  1. 1.
    Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie-Verlag, Berlin (1978)zbMATHGoogle Scholar
  2. 2.
    Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  3. 3.
    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/ MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Zgurovsky, M.Z., Kasyanov, P.O.: Uniform trajectory attractors for nonautonomous dissipative dynamical systems. Continuous and Distributed Systems II. Studies in Systems, Decision and Control Volume 30, pp. 221–232. Springer, New York (2015)CrossRefGoogle Scholar
  5. 5.
    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)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chepyzhov, V.V., Vishik, M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76, 913–964 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Melnik, V.S., Valero, J.: On attractors of multivalued semi-flows and generalized differential equations. Set-Valued Anal. 6(1), 83–111 (1998)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Balibrea, F., Caraballo, T., Kloeden, P.E., Valero, J.: Recent developments in dynamical systems: three perspectives. Int. J. Bifurc. Chaos (2010). doi: 10.1142/S0218127410027246
  9. 9.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Volume II: Applications. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Series in Mathematical Analysis and Applications 9. Chapman & Hall/CRC, Boca Raton (2005)CrossRefzbMATHGoogle Scholar
  11. 11.
    Kasyanov, P.O.: Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity. Math. Notes 92, 205–218 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kasyanov, P.O., Toscano, L., Zadoianchuk, N.V.: Regularity of Weak Solutions and Their Attractors for a Parabolic Feedback Control Problem. Set-Valued Var. Anal. (2013). doi: 10.1007/s11228-013-0233-8
  13. 13.
    Mel’nik, V.S., Valero, J.: On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions. Set-Valued Anal. doi: 10.1023/A:1026514727329
  14. 14.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)CrossRefzbMATHGoogle Scholar
  15. 15.
    Zgurovsky, M.Z., Kasyanov, P.O.: Evolution inclusions in nonsmooth systems with applications for earth data processing. Advances in Global Optimization. Springer Proceedings in Mathematics & Statistics, vol. 95 (2014). doi: 10.1007/978-3-319-08377-3_29
  16. 16.
    Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Nauka, Moscow (1989). [in Russian]zbMATHGoogle Scholar
  17. 17.
    Chepyzhov, V.V., Vishik, M.I.: Trajectory attractors for evolution equations. C. R. Acad. Sci. Paris. Ser. I 321, 1309–1314 (1995)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Chepyzhov, V.V., Vishik, M.I.: Trajectory and global attractors for 3D Navier-Stokes system. Mat. Zametki. (2002). doi: 10.1023/A:1014190629738 zbMATHGoogle Scholar
  19. 19.
    Chepyzhov, V.V., Vishik, M.I.: Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete Contin. Dyn. Syst. 27(4), 1498–1509 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  21. 21.
    Hale, J.K.: Asymptotic behavior of dissipative systems. AMS, Providence (1988)zbMATHGoogle Scholar
  22. 22.
    Kasyanov, P.O.: Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity. Cybern. Syst. Anal. 47, 800–811 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes equations. J. Math. Anal. Appl. (2011). doi: 10.1016/j.jmaa.2010.07.040 MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar
  25. 25.
    Migórski, S., Ochal, A.: Optimal control of parabolic hemivariational inequalities. J. Glob Optim. 17, 285–300 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Migórski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Glob. Optim. 31(3), 505–533 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions. Birkhauser, Basel (1985)CrossRefzbMATHGoogle Scholar
  28. 28.
    Sell, G.R.: Global attractors for the three-dimensional Navier-Stokes equations. J. Dyn. Differ. Equ. 8(12), 1–33 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)zbMATHGoogle Scholar
  30. 30.
    Zgurovsky, M.Z., Mel’nik, V.S., Kasyanov, P.O.: Evolution Inclusions and Variation Inequalities for Earth Data Processing II. Springer, Berlin (2011)zbMATHGoogle Scholar
  31. 31.
    Zgurovsky, M.Z., Kasyanov, P.O., Zadoianchuk (Zadoyanchuk), N.V.: Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem. Appl. Math. Lett. 25(10), 1569–1574 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.National Technical University of Ukraine “Kyiv Polytechnic Institute”KyivUkraine
  2. 2.Institute for Applied System AnalysisNational Technical University of Ukraine “Kyiv Polytechnic Institute”KyivUkraine

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