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Existence and Invariance of Global Attractors for Impulsive Parabolic System Without Uniqueness

  • Sergey Dashkovskiy
  • Petro Feketa
  • Oleksiy V. KapustyanEmail author
  • Iryna V. Romaniuk
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

In this paper, we apply the abstract theory of global attractors for multi-valued impulsive dynamical systems to weakly-nonlinear impulsively perturbed parabolic system without uniqueness of a solution to the Cauchy problem. We prove that for a sufficiently wide class of impulsive perturbations (including multi-valued ones) the global attractor of the corresponding multi-valued impulsive dynamical system has an invariant non-impulsive part.

Notes

Acknowledgements

This work was partially supported by the German Academic Exchange Service (DAAD). Oleksiy Kapustyan was partially supported by the State Fund For Fundamental Research, Grant of President of Ukraine.

References

  1. 1.
    Akhmet, M.: Principles of Discontinuous Dynamical Systems. Springer, Berlin (2010)CrossRefGoogle Scholar
  2. 2.
    Ball, J.M.: Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations. J. Nonlinear Sci. 7(5), 475–502 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bonotto, E.M.: Flows of characteristic 0+ in impulsive semidynamical systems. J. Math. Anal. Appl. 332, 81–96 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonotto, E.M., Demuner, D.P.: Attractors of impulsive dissipative semidynamical systems. Bull. Sci. Math. 137, 617–642 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bonotto, E.M., Bortolan, M.C., Carvalho, A.N., Czaja, R.: Global attractors for impulsive dynamical systems – a precompact approach. J. Differ. Equ. 259, 2602–2625 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bonotto, E.M., Bortolan, M.C., Collegari, R., Czaja, R.: Semicontinuity of attractors for impulsive dynamical systems. J. Differ. Equ. 261, 4338–4367 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chepyzhov, V.V., Vishik, M.I.: Attractors of Equations of Mathematical Physics. Colloquium Publications, vol. 49. American Mathematical Society, Providence (2002)Google Scholar
  8. 8.
    Ciesielski, K.: On stability in impulsive dynamical systems. Bull. Pol. Acad. Sci. Math. 52, 81–91 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dashkovskiy, S., Feketa, P.: Input-to-state stability of impulsive systems and their interconnections. Nonlinear Anal. Hybrid Syst. 26, 190–200 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dashkovskiy, S., Mironchenko, A.: Input-to-state stability of nonlinear impulsive systems. SIAM J. Control Optim. 51(3), 1962–1987 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dashkovskiy, S., Kapustyan, O., Romanjuk, I.: Global attractors of impulsive parabolic inclusions. Discrete Contin. Dyn. Syst. Ser. B 22(5), 1875–1886 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dashkovskiy, S., Feketa, P., Kapustyan, O., Romaniuk, I.: Invariance and stability of global attractors for multi-valued impulsive dynamical systems. J. Math. Anal. Appl. 458(1), 193–218 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Feketa, P., Bajcinca, N.: Stability of nonlinear impulsive differential equations with non-fixed moments of jumps. In: Proceedings of 17th European Control Conference, Limassol, Cyprus, 900–905 (2018)Google Scholar
  14. 14.
    Feketa, P., Perestyuk, Yu.: Perturbation theorems for a multifrequency system with pulses. J. Math. Sci. (N.Y.) 217(4), 515–524 (2016)Google Scholar
  15. 15.
    Gorban, N.V., Kapustyan, O.V., Kasyanov, P.O.: Uniform trajectory attractor for non-autonomous reactiondiffusion equations with Caratheodorys nonlinearity. Nonlinear Anal. 98, 13–26 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Iovane, G., Kapustyan, O.V., Valero, J.: Asymptotic behaviour of reaction-diffusion equations with non-damped impulsive effects. Nonlinear Anal. 68, 2516–2530 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kapustyan, A.V.: Global attractors of non-autonomous reaction-diffusion equation. Diff. Equ. 38, 1467–1471 (2002)CrossRefGoogle Scholar
  18. 18.
    Kapustyan, A.V., Melnik, V.S.: On global attractors of multivalued semidynamical systems and their approximations. Dokl. Akad. Nauk. 366(4), 445–448 (1999)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kapustyan, O.V., Shkundin, D.V.: Global attractor of one nonlinear parabolic equation. Ukr. Math. J. 55(4), 446–455 (2003)CrossRefGoogle Scholar
  20. 20.
    Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Pullback attractors for some class of extremal solutions of 3D Navier-Stokes system. J. Math. Anal. Appl. 373, 535–547 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kapustyan, O.V., Kasyanov, P.O., Valero, J.: Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Commun. Pure Appl. Anal. 13, 1891–1906 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kapustyan, O., Perestyuk, M., Romaniuk, I.: Global attractor of weakly nonlinear parabolic system with discontinuous trajectories. Mem. Differ. Equ. Math. Phys. 72, 59–70 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kasyanov, P.O.: Multivalued dynamics of solutions of autonomous differential-operator inclusion with pseudomonotone nonlinearity. Cybern. Syst. Anal. 47(5), 800–811 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kaul, S.K.: On impulsive semidynamical systems. J. Math. Anal. Appl. 150(1), 120–128 (1990)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kaul, S.K.: Stability and asymptotic stability in impulsive semidynamical systems. J. Appl. Math. Stoch. Anal. 7(4), 509–523 (1994)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)CrossRefGoogle Scholar
  27. 27.
    Melnik, V.S.: Families of multi-valued semiflows and their attractors. Dokl. Math. 55, 195–196 (1997)Google Scholar
  28. 28.
    Melnik, V.S., Valero, J.: On attractors of multi-valued semi-flows and differential inclusions. Set-Valued Var. Anal. 6, 83–111 (1998)CrossRefGoogle Scholar
  29. 29.
    Perestyuk, M.O., Feketa, P.V.: Invariant manifolds of one class of systems of impulsive differential equations. Nonlinear Oscil. 13(2), 260–273 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Perestyuk, M., Feketa, P.: Invariant sets of impulsive differential equations with particularities in ω-limit set. Abstr. Appl. Anal. 2011, ID 970469, 14 pp. (2011)Google Scholar
  31. 31.
    Perestyuk, M.O., Kapustyan, O.V.: Long-time behavior of evolution inclusion with non-damped impulsive effects. Mem. Differ. Equ. Math. Phys. 56, 89–113 (2012)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Perestyuk, M.O., Kapustyan, O.V.: Global attractors of impulsive infinite-dimensional systems. Ukr. Math. J. 68(4), 517–528 (2016)MathSciNetGoogle Scholar
  33. 33.
    Pichkur, V.V., Sasonkina, M.S.: Maximum set of initial conditions for the problem of weak practical stability of a discrete inclusion. J. Math. Sci. 194, 414–425 (2013)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rozko, V.: Stability in terms of Lyapunov of discontinuous dynamic systems. Differ. Uravn. 11(6), 1005–1012 (1975)MathSciNetGoogle Scholar
  35. 35.
    Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)CrossRefGoogle Scholar
  36. 36.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin (1988)CrossRefGoogle Scholar
  37. 37.
    Zgurovsky, M.Z., Kasyanov, P.O., Kapustyan, O.V., Valero, J., Zadoianchuk, N.V.: Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Long-Time Behavior of Evolution Inclusions Solutions in Earth Data Analysis. Springer, Berlin, 330 pp. (2012)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Sergey Dashkovskiy
    • 1
  • Petro Feketa
    • 2
  • Oleksiy V. Kapustyan
    • 3
    Email author
  • Iryna V. Romaniuk
    • 3
  1. 1.University of WürzburgWürzburgGermany
  2. 2.University of KaiserslauternKaiserslauternGermany
  3. 3.Taras Shevchenko National University of KyivKyivUkraine

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