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Uniform Trajectory Attractors for Nonautonomous Dissipative Dynamical Systems

  • Mikhail Z. Zgurovsky
  • Pavlo O. Kasyanov
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 30)

Abstract

For all global weak solutions of the general classes of nonautonomous evolution equations and inclusions that satisfy standard sign and polynomial growth conditions, the multivalued dynamics as time \(t\rightarrow +\infty \) is studied. The existence of a compact uniform trajectory attractor is justified. The obtained results allow to investigate long-time behavior of distributions of state functions for various mathematical models in geophysics, mechanics, biology, medicine etc.

Keywords

Evolution inclusion Pseudomonotone map Uniform trajectory attractor Feedback control 

References

  1. 1.
    Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Nauka, Moscow (1989) [in Russian]Google Scholar
  2. 2.
    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
  3. 3.
    Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Volume II: Applications. Kluwer, Dordrecht (2000)Google Scholar
  4. 4.
    Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie-Verlag, Berlin (1978)Google Scholar
  5. 5.
    Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Series in Mathematical Analysis and Applications, vol. 9. Chapman & Hall/CRC, Boca Raton (2005)Google Scholar
  6. 6.
    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 zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chepyzhov, V.V., Vishik, M.I.: Trajectory attractors for evolution equations. C. R. Acad. Sci. Paris. Ser. I 321, 1309–1314 (1995)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Chepyzhov, V.V., Vishik, M.I.: Evolution equations and their trajectory attractors. J. Math. Pures Appl. 76, 913–964 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chepyzhov, V.V., Vishik, M.I.: Trajectory and global attractors for 3D Navier-Stokes system. Mat. Zametki. (2002). doi: 10.1023/A:1014190629738 Google Scholar
  10. 10.
    Chepyzhov, V.V., Vishik, M.I.: Attractors for Equations of Mathematical Physics. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  11. 11.
    Chepyzhov, V.V., Vishik, M.I.: Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discret. Contin. Dyn. Syst. 27(4), 1498–1509 (2010)MathSciNetGoogle Scholar
  12. 12.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  13. 13.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston (2003)CrossRefGoogle Scholar
  14. 14.
    Hale, J.K.: Asymptotic Behavior of Dissipative Systems. AMS, Providence (1988)zbMATHGoogle Scholar
  15. 15.
    Kasyanov, P.O.: Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity. Math. Notes 92, 205–218 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Kasyanov, P.O.: Multivalued dynamics of solutions of an autonomous differential-operator inclusion with pseudomonotone nonlinearity. Cybern. Syst. Anal. 47, 800–811 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    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
  18. 18.
    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
  19. 19.
    Ladyzhenskaya, O.A.: Attractors for Semigroups and Evolution Equations. Cambridge University Press, Cambridge (1991)zbMATHCrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    Mel’nik, V.S., Valero, J.: On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions. Set-Valued Anal. (2000). doi: 10.1023/A:1026514727329
  22. 22.
    Migórski, S., Ochal, A.: Optimal control of parabolic hemivariational inequalities. J. Glob. Optim. 17, 285–300 (2000)zbMATHCrossRefGoogle Scholar
  23. 23.
    Migórski, S.: Boundary hemivariational inequalities of hyperbolic type and applications. J. Glob. Optim. 31(3), 505–533 (2005)zbMATHCrossRefGoogle Scholar
  24. 24.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications Convex and Nonconvex Energy Functions. Birkhauser, Basel (1985)zbMATHCrossRefGoogle Scholar
  25. 25.
    Sell, G.R.: Global attractors for the three-dimensional Navier-Stokes equations. J. Dyn. Differ. Equ. 8(12), 1–33 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988)zbMATHGoogle Scholar
  27. 27.
    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
  28. 28.
    Zgurovsky, M.Z., Kasyanov, P.O.: Evolution Inclusions in Nonsmooth Systems with Applications for Earth Data Processing, Advances in Global Optimization. In: Proceedings in Mathematics & Statistics, vol. 95. Springer (2014). doi: 10.1007/978-3-319-08377-3_29
  29. 29.
    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)zbMATHCrossRefGoogle Scholar
  30. 30.
    Zgurovsky, M.Z., Kasyanov, P.O.: Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem. Appl. Math. Lett. 25(10), 1569–1574 (2012)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

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

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