Skip to main content
Log in

Almost Periodic Solutions of the Wave Equation with Damping and Impulsive Action

  • Published:
Ukrainian Mathematical Journal Aims and scope

We obtain sufficient conditions for the existence of piecewise continuous almost periodic solutions to the damped wave equation with impulsive action.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. N. Carvalho, J. W. Cholewa, and T. Dlotko, “Strongly damped wave problems: bootstrapping and regularity of solutions,” J. Different. Equat., 244, No. 9, 2310–2333 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. N. Carvalho and J. W. Cholewa, “Strongly damped wave equations in W01,p (Ω) × Lp(Ω),Discrete Contin. Dyn. Syst., 2007, 230–239 (2007).

  3. T. Diagana, “Almost periodic solutions to some second-order nonautonomous differential equations,” Proc. Amer. Math. Soc., 140, No. 1, 279–289 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Hernandez, K. Balachandran, and N. Annapoorani, “Existence results for a damped second order abstract functional differential equation with impulses,” Math. Comput. Model., 50, No. 11–12, 1583–1594 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. Massatt, “Limiting behavior for strongly damped nonlinear wave equations,” J. Different. Equat., 48, No. 3, 334–349 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Massatt, “Asymptotic behavior for a strongly damped nonlinear wave equation,” in: Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York (1982), pp. 663–670.

  7. G. F.Webb, “Existence and asymptotic behavior for a strongly damped nonlinear wave equation,” Canad. J. Math., 32, No. 3, 631–643 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  8. Q. Zhang, “Global existence of -regular solutions for the strongly damped wave equation,” Electron. J. Qual. Theory Different. Equat., 62, 1–11 (2013).

    MathSciNet  Google Scholar 

  9. A. Halanay and D. Wexler, Teoria Calitativă a Sistemelor cu Impulsuri, Editura Academiei Republicii Socialiste România, Bucure¸sti (1968).

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

    Book  MATH  Google Scholar 

  11. A. V. Dvornyk and V. I. Tkachenko, “Almost periodic solutions for systems with delay and nonfixed times of impulsive actions,” Ukr. Mat. Zh., 68, No. 11, 1450–1466 (2016); English translation: 68, No. 11, 1673–1693 (2017).

  12. A. V. Dvornyk, O. O. Struk, and V. I. Tkachenko, “Almost periodic solutions of Lotka–Volterra systems with diffusion and pulsed action,” Ukr. Mat. Zh., 70, No. 2, 177–192 (2018); English translation: Ukr. Math. J., 70, No. 2, 197–216 (2018).

  13. R. Hakl, M. Pinto, V. Tkachenko, and S. Trofimchuk, “Almost periodic evolution systems with impulse action at state-dependent moments,” J. Math. Anal. Appl., 446, No. 1, 1030–1045 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. M. Samoilenko and S. I. Trofimchuk, “Almost periodic impulsive systems,” Different. Equat., 29, No. 4, 684–691 (1993).

    MATH  Google Scholar 

  15. A. M. Samoilenko and S. I. Trofimchuk, “Unbounded functions with almost periodic differences,” Ukr. Mat. Zh., 43, No. 10, 1409–1413 (1991); English translation: Ukr. Math. J., 43, No. 10, 1306–1309 (1991).

  16. G. T. Stamov, Almost Periodic Solutions of Impulsive Differential Equations, Lect. Notes Math., Vol. 2047, Springer, Heidelberg (2012).

  17. V. Tkachenko, “Almost periodic solutions of evolution differential equations with impulsive action,” in: Mathematical Modelling and Applications in Nonlinear Dynamics, Springer, Cham (2016), pp. 161–205.

  18. A. V. Dvornyk and V. I. Tkachenko, “On the stability of solutions of evolutionary equations with nonfixed times of pulse actions,” Nelin. Kolyv., 18, No. 4, 475–488 (2015); English translation: J. Math. Sci., 220, No. 4, 425–439 (2017).

  19. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., Vol. 840, Springer, Berlin–Heidelberg (1981).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. V. Dvornyk.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 1, pp. 62–71, January, 2023. Ukrainian DOI: 10.37863/umzh.v75i1.7400.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dvornyk, A.V., Tkachenko, V.I. Almost Periodic Solutions of the Wave Equation with Damping and Impulsive Action. Ukr Math J 75, 68–79 (2023). https://doi.org/10.1007/s11253-023-02186-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-023-02186-7

Navigation