DOI QR코드

DOI QR Code

ATTRACTORS OF LOCAL SEMIFLOWS ON TOPOLOGICAL SPACES

  • Li, Desheng ;
  • Wang, Jintao ;
  • Xiong, Youbing
  • Received : 2016.04.01
  • Published : 2017.05.01

Abstract

In this paper we introduce a notion of an attractor for local semiflows on topological spaces, which in some cases seems to be more suitable than the existing ones in the literature. Based on this notion we develop a basic attractor theory on topological spaces under appropriate separation axioms. First, we discuss fundamental properties of attractors such as maximality and stability and establish some existence results. Then, we give a converse Lyapunov theorem. Finally, the Morse decomposition of attractors is also addressed.

Keywords

topological space;local semiflow;attractor;existence;stability;Lyapunov function;Morse decomposition

References

  1. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989. English translation, North-Holland, Amsterdam, 1992.
  2. N. P. Bhatia and O. Hajek, Local Semi-dynamical Systems, Lecture Notes in Mathematics, Vol. 90, Springer-Verlag, Berlin, 1969.
  3. D. N. Cheban, Global Attractors of Non-autonomous Dissipative Dynamical Systems, World Scientific Publishing Co. Pte. Ltd., Singapore, 2004.
  4. V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.
  5. F. Gazzola and M. Sardella, Attractors for families of processes in weak topologies of Banach spaces, Discrete Contin. Dyn. Syst. 4 (1998), no. 3, 455-466. https://doi.org/10.3934/dcds.1998.4.455
  6. A. Giraldo, M. A. Moron, F. R. Ruiz Del Portal, and J. M. R. Sanjurjo, Shape of global attractors in topological spaces, Nonlinear Anal. 60 (2005), no. 5, 837-847. https://doi.org/10.1016/j.na.2004.03.036
  7. W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Colloq. Publ. Vol. 36, Amer. Math. Soc., Providence, RI, 1955.
  8. J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys Monographs, Vol. 25, Amer. Math. Soc., Providence, RI, 1998.
  9. L. Kapitanski and I. Rodnianski, Shape and Morse theory of attractors, Commun. Pure Appl. Math. 53 (2000), no. 2, 218-242. https://doi.org/10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO;2-W
  10. O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Lizioni Lincei, Cambridge Univ. Press, Cambridge, New-York, 1991.
  11. D. S. Li, G. S. Shi, and X. F. Song, A linking theory of dynamical systems with applications to PDEs, http://arxiv.org/abs/1312.1868v3.
  12. D. S. Li and Y. Wang, Smooth Morse-Lyapunov functions of strong attractors for differential inclusions, SIAM J. Control Optim. 50 (2012), no. 1, 368-387. https://doi.org/10.1137/10081280X
  13. Q. F. Ma, S. H. Wang, and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J. 51 (2002), no. 6, 1541-1559. https://doi.org/10.1512/iumj.2002.51.2255
  14. L. Markus, The global theory of ordinary differential equations, Lecture Notes, Univ. Minnesota, Minneapolis, 1964.
  15. A. Marzocchi and S. Zandonella Necca, Attractors for dynamical systems in topological spaces, Discrete Contin. Dyn. Syst. 8 (2002), no. 3, 585-597. https://doi.org/10.3934/dcds.2002.8.585
  16. J. C. Robinson, Infinite-dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.
  17. K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 1987.
  18. G. R. Sell, Nonautonomous Differential Equations and Topological Dynamics I. The basic Theory, Trans. Amer. Math. Soc. 127 (1967), 241-262.
  19. G. R. Sell and Y. C. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.
  20. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed, Springer-Verlag, New York, 1997.
  21. M. I. Vishik, Asymptotic Behavior of Solutions of Evolutionary Equations, Cambridge Univ. Press, Cambridge, 1992.
  22. J. T. Wang, D. S. Li, and J. Q. Duan, On the shape conley index theory of semiflows on complete metric spaces, Discrete Contin. Dyn. Syst. 36 (2016), no. 3, 1629-1647.

Acknowledgement

Supported by : NSF of China