Almost periodic solutions of damping wave equation with impulsive action
Abstract
UDC 517.9
We obtain sufficient conditions for the existence of piecewise continuous almost periodic solutions to a strongly damped semilinear wave equation with impulsive action.
References
A. N. Carvalho, J. W. Cholewa, T. Dlotko, Strongly damped wave problems: bootstrapping and regularity of solutions, J. Different. Equat., 244, № 9, 2310–2333 (2008). DOI: https://doi.org/10.1016/j.jde.2008.02.011
A. N. Carvalho, J. W. Cholewa, Strongly damped wave equations in $W^{1,p}_0(Ω)× L^p(Ω)$, Discrete and Contin. Dyn. Syst., 2007, 230–239 (2007).
T. Diagana, Almost periodic solutions to some second-order nonautonomous differential equations, Proc. Amer. Math. Soc., 140, № 1, 279–289 (2012). DOI: https://doi.org/10.1090/S0002-9939-2011-10970-5
E. Hernandez, K. Balachandran, N. Annapoorani, Existence results for a damped second order abstract functional differential equation with impulses, Math. Comput. Model., 50, № 11–12, 1583–1594 (2009). DOI: https://doi.org/10.1016/j.mcm.2009.09.007
P. Massatt, Limiting behavior for strongly damped nonlinear wave equations, J. Different. Equat., 48, № 3, 334–349 (1983). DOI: https://doi.org/10.1016/0022-0396(83)90098-0
P. Massatt, Asymptotic behavior for a strongly damped nonlinear wave equation, Nonlinear Phenomena in Mathematical Sciences, Acad. Press (1982), p. 663–670 . DOI: https://doi.org/10.1016/B978-0-12-434170-8.50083-2
G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32, № 3, 631–643 (1980). DOI: https://doi.org/10.4153/CJM-1980-049-5
Q. Zhang, Global existence of $varepsilon$-regular solutions for the strongly damped wave equation, Electron. J. Qual. Theory Different. Equat., 62, 1–11 (2013). DOI: https://doi.org/10.14232/ejqtde.2013.1.62
A. Халанай, Д. Векслер, Качественная теория импульсных систем, Мир, Москва (1971).
A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Sci. Publ., Singapore (1995). DOI: https://doi.org/10.1142/2892
A. V. Dvornyk, V. I. Tkachenko, Almost periodic solutions for systems with delay and nonfixed times of impulsive actions, Ukrainian Math. J., 68, № 11, 1673–1693 (2017). DOI: https://doi.org/10.1007/s11253-017-1320-z
A. V. Dvornyk, O. O. Struk, V. I. Tkachenko, Almost periodic solutions of Lotka–Volterra systems with diffusion and impulse action, Ukrainian Math. J., 70, № 2, 197–216 (2018). DOI: https://doi.org/10.1007/s11253-018-1495-y
R. Hakl, M. Pinto, V. Tkachenko, S. Trofimchuk, Almost periodic evolution systems with impulse action at state-dependent moments, J. Math. Anal. and Appl., 446, № 1, 1030–1045 (2017). DOI: https://doi.org/10.1016/j.jmaa.2016.09.024
A. M. Samoilenko, S. I. Trofimchuk, Almost periodic impulsive systems, Different. Equat., 29, № 4, 684–691 (1993).
A. M. Samoilenko, S. I. Trofimchuk, Unbounded functions with almost periodic differences, Ukrainian Math. J., 43, № 10, 1306–1309 (1991). DOI: https://doi.org/10.1007/BF01061818
G. T. Stamov, Almost periodic solutions of impulsive differential equations, Lect. Notes Math., 2047, Springer, Heidelberg (2012). DOI: https://doi.org/10.1007/978-3-642-27546-3
V. Tkachenko, Almost periodic solutions of evolution differential equations with impulsive action, Mathematical Modelling and Applications in Nonlinear Dynamics, Springer, Cham (2016), p. 161–205. DOI: https://doi.org/10.1007/978-3-319-26630-5_7
A. V. Dvornyk, V. I. Tkachenko, On the stability of solutions of evolutionary equations with nonfixed times of pulse actions, J. Math. Sci., 220, № 4, 425–439 (2017). DOI: https://doi.org/10.1007/s10958-016-3193-3
D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math., 840, Springer, Berlin, Heidelberg (1981). DOI: https://doi.org/10.1007/BFb0089647
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