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Asymptotic Dynamics of Second Order Nonautonomous Systems on Infinite Lattices

Ahmed Y. Abdallah

Department of Mathematics, The University of Jordan, Amman 11942, Jordan

https://doi.org/10.1142/S0218127416500036Cited by:2 (Source: Crossref)

Abstract

We have introduced abstract sufficient conditions for the existence of a uniform exponential attractor for a special family of second order nonautonomous lattice dynamical systems with quasiperiodic symbols in a standard space of infinite sequences. Compared with the lattice dynamical system in [Zhou & Zhao, 2014], here a generalized nonlinear part and weaker assumptions have been presented, kindly see Remark Remark 2.1 for more details.

Keywords:

References

Abdallah, A. Y. [2006] “ Asymptotic behavior of the Klein–Gordon–Schrödinger lattice dynamical systems,” Commun. Pure Appl. Anal. 5, 55–69. Crossref, Web of Science, Google Scholar
Abdallah, A. Y. [2008] “ Exponential attractors for first order lattice dynamical systems,” J. Math. Anal. Appl. 339, 217–224. Crossref, Web of Science, Google Scholar
Abdallah, A. Y. [2009] “ Exponential attractors for second order lattice dynamical systems,” Commun. Pure Appl. Anal. 8, 803–813. Crossref, Web of Science, Google Scholar
Abdallah, A. Y. [2010] “ Uniform global attractors for first order non-autonomous lattice dynamical systems,” Proc. Amer. Math. Soc. 138, 3219–3228. Crossref, Web of Science, Google Scholar
Abdallah, A. Y. [2011] “ Uniform exponential attractors for first order non-autonomous lattice dynamical systems,” J. Diff. Eqs. 251, 1489–1504. Crossref, Web of Science, Google Scholar
Amigó, J. M., Giménez, Á., Morillas, F. & Valero, J. [2010] “ Attractors for a lattice dynamical system generated by non-Newtonian fluids modeling suspensions,” Int. J. Bifurcation and Chaos 20, 2681–2700. Link, Web of Science, Google Scholar
Bates, P. W., Lu, K. & Wang, B. [2001] “ Attractors for lattice dynamical systems,” Int. J. Bifurcation and Chaos 11, 143–153. Link, Web of Science, Google Scholar
Chate, H. & Courbage, M. (eds.) [1997] “ Lattice systems,” Physica D 103, 1–612. Crossref, Web of Science, Google Scholar
Chepyzhov, V. V. & Vishik, M. I. [1992] “Non-autonomous dynamical systems and their attractors,” Appendix in the book: M. I. Vishik, Asymptotic Behavior of Solutions of Evolutionary Equations (Cambridge University Press). Google Scholar
Chepyzhov, V. V. & Vishik, M. I. [1993] “ Non-autonomous evolution equations and their attractor,” Russ. J. Math. Phys. 1, 165–190. Google Scholar
Chepyzhov, V. V. & Vishik, M. I. [1994] “ Attractors of non-autonomous dynamical systems and their dimension,” J. Math. Pures Appl. 73, 279–333. Web of Science, Google Scholar
Chow, S. N. [2003] “ Lattice dynamical systems,” Dynamical System, Lecture Notes in Mathematics, Vol. 1822 (Springer, Berlin), pp. 1–102. Crossref, Google Scholar
Eden, A., Foias, C., Nicolaenko, B. & Temam, R. [1994] Exponential Attractors for Dissipative Evolution Equations, Research in Applied Mathematics, Vol. 37 (Masson/John Wiley co-publication, Paris). Google Scholar
Eden, A., Foias, C. & Kalantarov, V. [1998] “ A remark on two constructions of exponential attractors for $α$ $α$ -contractions,” J. Dyn. Diff. Eqs. 10, 37–45. Crossref, Google Scholar
Fabrie, P. & Miranville, A. [1998] “ Exponential attractors for nonautonomous first-order evolution equations,” Discr. Contin. Dyn. Syst. 4, 225–240. Crossref, Google Scholar
Fan, X. & Wang, Y. [2007] “ Attractors for a second order nonautonomous lattice dynamical system with nonlinear damping,” Phys. Lett. A 365, 17–27. Crossref, Web of Science, Google Scholar
Fan, X. [2008] “ Exponential attractor for a first-order dissipative lattice dynamical system,” J. Appl. Math. 2008, Art. ID 354652, 8 pp. Crossref, Web of Science, Google Scholar
Foias, C., Sell, G. & Temam, R. [1988] “ Inertial manifolds for nonlinear evolution equations,” J. Diff. Eqs. 73, 309–353. Crossref, Web of Science, Google Scholar
Hale, J. [1988] Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, Vol. 25 (Amer. Math. Soc., Providence, RI). Google Scholar
Han, X. [2011] “ Exponential attractors for lattice dynamical systems in weighted spaces,” Discr. Contin. Dyn. Syst. 31, 445–467. Crossref, Web of Science, Google Scholar
Huang, J., Han, X. & Zhou, S. [2009] “ Uniform attractors for non-autonomous Klein–Gordon–Schrödinger lattice systems,” Appl. Math. Mech. (English Ed.) 30, 1597–1607. Crossref, Web of Science, Google Scholar
Li, X., Wei, K. & Zhang, H. [2011] “ Exponential attractor for lattice dynamical systems in weighted spaces,” Acta Appl. Math. 114, 157–172. Crossref, Web of Science, Google Scholar
Oliveira, J. C., Pereira, J. M. & Perla Menzala, G. [2008] “ Attractors for second order periodic lattices with nonlinear damping,” J. Diff. Eqs. Appl. 14, 899–921. Crossref, Web of Science, Google Scholar
Wang, B. [2006] “ Dynamics of systems on infinite lattices,” J. Diff. Eqs. 221, 224–245. Crossref, Web of Science, Google Scholar
Wang, B. [2007] “ Asymptotic behavior of non-autonomous lattice systems,” J. Math. Anal. Appl. 331, 121–136. Crossref, Web of Science, Google Scholar
Zhou, S. [2002] “ Attractors for second order lattice dynamical systems,” J. Diff. Eqs. 179, 605–624. Crossref, Web of Science, Google Scholar
Zhou, S. [2003] “ Attractors for first order dissipative lattice dynamical systems,” Physica D 178, 51–61. Crossref, Web of Science, Google Scholar
Zhou, S. [2004] “ Attractors and approximations for lattice dynamical systems,” J. Diff. Eqs. 200, 342–368. Crossref, Web of Science, Google Scholar
Zhou, S., Zhao, C. & Liao, X. [2007] “ Compact uniform attractors for dissipative non-autonomous lattice dynamical systems,” Commun. Pure Appl. Anal. 6, 1087–1111. Crossref, Web of Science, Google Scholar
Zhou, S. [2013] “ Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces,” Nonlin. Anal. 78, 141–155. Crossref, Web of Science, Google Scholar
Zhou, S. & Zhao, M. [2014] “ Uniform exponential attractor for second order lattice system with quasi-periodic external forces in weighted space,” Int. J. Bifurcation and Chaos 24, 1450006-1–9. Link, Web of Science, Google Scholar