Abstract
We investigate the dynamics of solutions of an autonomous wave equation in ℝn with continuous nonlinearity. A priori estimates are obtained. We substantiate the existence of an invariant global attractor for an m-semiflow.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations [in Russian], Nauka, Moscow (1989).
O. V. Kapustyan and G. Iovane, “Global attractor for a nonautonomous wave equation without uniqueness of solution,” Syst. Doslid. Inform. Tekhn., No. 2, 107–120 (2006).
O. V. Kapustyan, “Kneser property for a nonautonomous wave equation without uniqueness of solution,” Nauk. Visti “KPI” Nauk. Tekhn. Univ. Ukr., No. 2, 137–141 (2007).
J. M. Ball, “Global attractors for damped semilinear wave equations,” Discrete Contin. Dynam. Syst., 10, 31–52 (2004).
V. Belleri and V. Pata, “Attractors for semilinear strongly damped wave equations on ℝ3,” Discrete Contin. Dynam. Syst., 7, No. 4, 719–735 (2001).
V. V. Chepyzhov and M. I. Vishik, “Attractors on non-autonomous dynamical systems and their dimension,” J. Math. Pures Appl., 73, No. 3, 279–333 (1994).
V. V. Chepyzhov and M. I. Vishik, “Evolution equations and their trajectory attractors,” J. Math. Pures Appl., 76, No. 10, 913–964 (1997).
O. V. Kapustyan, “The global attractors of multi-valued semiflows, which are generated by some evolutionary equations,” Nelin. Granich. Zad., No. 11, 65–70 (2001).
A. V. Kapustyan, V. S. Melnik, and J. Valero, “Attractors of multivalued dynamical processes generated by phase-field equations,” Int. Bifurcat. Chaos, 13, 1969–1983 (2003).
A. V. Kapustyan, V. S. Melnik, and J. Valero, “A weak attractor and properties of solutions for the three-dimensional Bénard problem,” Discrete Contin. Dynam. Syst., 18, 449–481 (2007).
V. S. Melnik, Multivalued Dynamics of Nonlinear Infinite-Dimensional Systems, Preprint No. 94-17, Institute of Cybernetics, Ukrainian Academy of Sciences, Kyiv (1994).
V. S. Melnik, “Estimates of the fractal and Hausdorff dimensions of sets invariant under multimappings,” Math. Notes, 63, 190–196 (1998).
V. S. Melnik, V. Slastikov, and S. I. Vasilkevich, “On global attractors of multivalued semi-processes,” Dokl. Akad. Nauk Ukr., No. 7, 12–17 (1999).
V. S. Melnik and J. Valero, “On attractors of multivalued semi-flows and differential inclusions,” Set-Valued Analysis, 6, 83–111 (1998).
F. Morillas and J. Valero, “Attractors for reaction-diffusion equation in ℝn with continuous nonlinearity,” Asympt. Analysis, 44, 111–130 (2005).
A. Rodriguez-Bernal and B. Wang, “Attractors for partly dissipative reaction-diffusion systems in ℝn,” J. Math. Anal. Appl., 252, 790–803 (2000).
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin (1988).
B. Wang, “Attractors for reaction-diffusion equations in unbounded domains,” Physica D, 128, 41–52 (1999).
S. V. Zelik, “The attractor for nonlinear hyperbolic equation in the unbounded domain,” Discrete Contin. Dynam. Syst., 7, No. 3, 593–641 (2001).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 260–267, February, 2008.
Rights and permissions
About this article
Cite this article
Stanzhyts’kyi, O.M., Horban’, N.V. Global attractor for the autonomous wave equation in ℝn with continuous nonlinearity. Ukr Math J 60, 299–309 (2008). https://doi.org/10.1007/s11253-008-0059-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-008-0059-y