Global Periodic Attractors for a Class of Infinite Dimensional Dissipative Dynamical Systems

ABSTRACT

In this paper we consider the existence of a global periodic attractor for a class of infinite dimensional dissipative equations under homogeneous Dirichlet boundary conditions. It is proved that in a certain parameter, for an arbitrary timeperiodic driving force, the system has a unique periodic solution attracting any bounded set exponentially in the phase space, which implies that the system behaves exactly as a one-dimensional system. We mention, in particular, that the obtained result can be used to prove the existence of the global periodic attractor for abstract parabolic problems.

Cite this paper

H. Li, "Global Periodic Attractors for a Class of Infinite Dimensional Dissipative Dynamical Systems,"*Advances in Pure Mathematics*, Vol. 3 No. 5, 2013, pp. 472-474. doi: 10.4236/apm.2013.35067.

H. Li, "Global Periodic Attractors for a Class of Infinite Dimensional Dissipative Dynamical Systems,"

References

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[11] X. Fu and X. Liu, “Existence of Periodic Solutions for Abstract Neutral Non-Autonomous Equations with Infinite Delay,” Journal of Mathematical Analysis and Applications, Vol. 325, No. 1, 2007, pp. 249-267. doi:10.1016/j.jmaa.2006.01.048

[1] J. K. Cholewa and T. Dlokto, “Global Attractors in Abstract Parabolic Problem,” Cambridge University Press, Cambridge, 2000. doi:10.1017/CBO9780511526404

[2] J. K. Hale, “Asymptotic Behavior of Dissipative Systems,” AMS, 1988.

[3] R. Temam, “Infinite-Dimensional Dynamical Systems in Mechanics and Physics,” Springer-Verlag, New York, 1997.

[4] M. A. Hussain, “On a Nonlinear Integrodifferential Equation in Banach Space,” Indian Journal of Pure and Applied Mathematics, Vol. 19, 1988, pp. 516-529.

[5] S. M. Rankin III, “Existence and Asymptotic Behavior of a Functional Differential Equation in Banach Space,” Journal of Mathematical Analysis and Applications, Vol. 88, No. 2, 1982, pp. 531-542. doi:10.1016/0022-247X(82)90211-6

[6] W. E. Fitzgibbon, “Semilinear Functional Equations in Banach Space,” Journal of Differential Equations, Vol. 29, No. 1, 1978, pp. 1-14. doi:10.1016/0022-0396(78)90037-2

[7] C. Zhong, C. Sun and M. Niu, “On the Existence of Global Attractor for a Class of Infinite Dimensional Dissipative Nonlinear Dynamical Systems,” Chinese Annals of Mathematics, Vol. 26, No. 3, 2005, pp. 393-400. doi:10.1142/S0252959905000312

[8] H. R. Henriquez, “Periodic Solutions of Quasi-Linear Partial Functional Differential Equations with Unbounded Delay,” Funkcialaj Ekvacioj, Vol. 37, No. 2, 1994, pp. 329-343.

[9] E. Hernandez and H. R. Henriquez, “Existence Results for Partial Neutral Functional Differential Equations with Unbounded Delay,” Journal of Mathematical Analysis and Applications, Vol. 221, No. 2, 1998, pp. 452-475. doi:10.1006/jmaa.1997.5875

[10] E. Hernandez and H. R. Henriquez, “Existence of Periodic Solutions of Partial Neutral Functional Differential Equations with Unbounded Delay,” Journal of Mathematical Analysis and Applications, Vol. 221, No. 2, 1998, pp. 499-522. doi:10.1006/jmaa.1997.5899

[11] X. Fu and X. Liu, “Existence of Periodic Solutions for Abstract Neutral Non-Autonomous Equations with Infinite Delay,” Journal of Mathematical Analysis and Applications, Vol. 325, No. 1, 2007, pp. 249-267. doi:10.1016/j.jmaa.2006.01.048