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Uniform Attractor for an N-Dimensional Parabolic System with Impulsive Perturbation

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We consider a weakly nonlinear N-dimensional parabolic system whose solutions are subjected to impulsive perturbations upon attainment of a certain fixed subset in the phase space. For broad classes of impulsive perturbations, it is proved that the system generates an impulsive semiflow with the minimal compact uniformly attracting set (uniform attractor) in the phase space. The invariance and stability of the nonimpulsive part of the uniform attractor are established under additional restrictions imposed on impulsive parameters.

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References

  1. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).

    Google Scholar 

  2. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989).

    Book  Google Scholar 

  3. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific , Singapore (1995).

    Book  Google Scholar 

  4. N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Impulsive Differential Equations with Set-Valued and Discontinuous Right-Hand Sides [in Russian], Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev (2007).

    Google Scholar 

  5. M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer, New York (2010).

    Book  Google Scholar 

  6. N. A. Perestyuk, “Invariant sets of a class of discontinuous dynamical systems,” Ukr. Mat. Zh., 36, No. 1, 63–68 (1984); English translation: Ukr. Math. J., 36, No. 1, 58–62 (1984).

  7. S. K. Kaul, “Stability and asymptotic stability in impulsive semidynamical systems,” J. Appl. Math. Stochast. Anal., 7, No. 4, 509–523 (1994).

    Article  MathSciNet  Google Scholar 

  8. T. Pavlidis, “Stability of a class of discontinuous dynamical systems,” Inf. Contr., 9, 298–322 (1996).

    Article  MathSciNet  Google Scholar 

  9. K. Ciesielski, “On stability in impulsive dynamical systems,” Bull. Pol. Acad. Sci. Math., 52, 81–91 (2004).

    Article  MathSciNet  Google Scholar 

  10. E. M. Bonotto, “Flows of characteristic 0+ in impulsive semidynamical systems,” J. Math. Anal. Appl., 332, 81–96 (2007).

    Article  MathSciNet  Google Scholar 

  11. Yu. M. Perestyuk, “Discontinuous oscillations in one impulsive system,” Nelin. Kolyv., 15, No. 4, 494–503 (2012); English translation: J. Math. Sci., 194, No. 4, 404–413 (2013).

  12. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York (1988).

    Book  Google Scholar 

  13. O. V. Kapustyan, P. O. Kasyanov, and J. Valero, “Regular solutions and global attractors for reaction-diffusion systems without uniqueness,” Comm. Pure Appl. Anal., 13, No. 5, 1891–1906 (2014).

    Article  MathSciNet  Google Scholar 

  14. N. V. Gorban, O. V. Kapustyan, P. O. Kasyanov, and L. S. Paliichuk, “On global attractors for autonomous damped wave equation with discontinuous nonlinearity,” Solid Mech. Appl., 211, 221–237 (2014).

    MathSciNet  MATH  Google Scholar 

  15. N. V. Gorban, A. V. Kapustyan, E. A. Kapustyan, and O. V. Khomenko, “Strong global attractor for the three-dimensional Navier–Stokes system of equations in unbounded domain of channel type,” J. Autom. Inform. Sci., 47, Issue 11, 48–59 (2015).

    Article  Google Scholar 

  16. O. V. Kapustyan, A. V. Pankov, and J. Valero, “Global attractors of multivalued semiflows, generated by continuous solutions of 3D Bénard system,” Set-Valued Var. Anal., 20, No. 3, 445–465 (2012).

    Article  MathSciNet  Google Scholar 

  17. M. O. Perestyuk and O. V. Kapustyan, “Long-time behavior of evolution inclusion with non-damped impulsive effects,” Mem. Differ. Equat. Math. Phys., 56, 89–113 (2012).

    MathSciNet  MATH  Google Scholar 

  18. E. M. Bonotto, M. C. Bortolan, A. N. Carvalho, and R. Czaja, “Global attractors for impulsive dynamical systems—a precompact approach,” J. Different. Equat., 259, 2602–2625 (2015).

    Article  MathSciNet  Google Scholar 

  19. O. V. Kapustyan and M. O. Perestyuk, “Global attractors of impulsive infinite-dimensional systems,” Ukr. Mat. Zh., 68, No. 4, 517–528 (2016); English translation: Ukr. Math. J., 68, No. 4, 583–597 (2016).

  20. S. Dashkovskiy, O. V. Kapustyan, and I. V. Romaniuk, “Global attractors of impulsive parabolic inclusions,” Discrete Contin. Dynam. Syst., Ser. B, 22, No. 5, 1875–1886 (2017).

    Article  MathSciNet  Google Scholar 

  21. O. Kapustyan, M. O. Perestyuk, and I. V. Romaniuk, “Global attractor of weakly nonlinear parabolic system with discontinuous trajectories,” Mem. Differ. Equat. Math. Phys., 72, 59–70 (2017).

    MathSciNet  MATH  Google Scholar 

  22. S. Dashkovskiy, P. Feketa, O. Kapustyan, and I. Romaniuk, “Invariance and stability of global attractors for multi-valued impulsive dynamical systems,” J. Math. Anal. Appl., 458, 193–218 (2018).

    Article  MathSciNet  Google Scholar 

  23. O. V. Kapustyan, M. O. Perestyuk, and I. V. Romanyuk, “Stability of global attractors of impulsive infinite-dimensional systems,” Ukr. Mat. Zh., 70, No. 1, 29–39 (2018); English translation: Ukr. Math. J., 70, No. 1, 30–41 (2018).

  24. N. P. Bhatia and G. P. Szeg Ro; Stability Theory of Dynamical Systems, Springer, New York (2002).

    Google Scholar 

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Authors and Affiliations

  1. T. Shevchenko Kyiv National University, Volodymyrs’ka Str., 64, Kyiv, 01601, Ukraine

    O. V. Kapustyan & F. A. Asrorov

  2. L. Ukrainka East-European National University, Volya Ave., 13, Lutsk, 43025, Ukraine

    V. V. Sobchuk

Authors
  1. O. V. Kapustyan
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  2. F. A. Asrorov
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  3. V. V. Sobchuk
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Correspondence to O. V. Kapustyan.

Additional information

Translated from Neliniini Kolyvannya, Vol. 22, No. 4, pp. 474–481, October–December, 2019.

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Kapustyan, O.V., Asrorov, F.A. & Sobchuk, V.V. Uniform Attractor for an N-Dimensional Parabolic System with Impulsive Perturbation. J Math Sci 254, 219–228 (2021). https://doi.org/10.1007/s10958-021-05299-1

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  • DOI: https://doi.org/10.1007/s10958-021-05299-1

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