Advertisement

Acta Mathematicae Applicatae Sinica

, Volume 20, Issue 4, pp 547–556 | Cite as

The Inflated Attractors of Non–autonomous Strongly Damped Wave Equations

  • Xiao-ming FanEmail author
  • Sheng-fan Zhou
Original Papers

Abstract

We investigate the existence and the continuity of the inflated attractors for a class of non–autonomous strongly damped wave equations through differential inclusion.

Keywords

ε–inflated system global attractor setvalued process 

2000 MR Subject Classification

35B41 35L05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cheban, D.N., Kloeden, P.E., Schmalfuß, B. Pullback attractors in dissipative nonau–tonomous differential equations under discretization. DANSE–Preprint 5/98, FU Berlin, 1998Google Scholar
  2. 2.
    Chepyzhov, V.V., Vishik, M.I. A Hausdorff dimension estimate for kernel sections of nonautonomous evolution equations. Indiana Univ. Math. Journal, 140: 193–206 (1991)Google Scholar
  3. 3.
    Chepyzhov, V.V., Vishik, M.I. Attractors of nonautonomous systems and their dimensions. J. Math. Pures Appl., 73: 279–333, 365–393 (1994)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Ghidaghlia, J.M.,Temam, R. Attractors for damped nonlinear hyperbolic equations. J. Math. Pures Appl., 66: 273–319 (1987)MathSciNetGoogle Scholar
  5. 5.
    Ghidaglia, J.M., Marzocchi, A. Longtime behaviour of strongly damped wave equations, global attractors and their dimensions. SIAM J. Math. Anal., 22: 879–895 (1991)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Hale, J. Asymptotic behavior of dissipative dynamical systems. Amer. Math. Soc., Providence, RI, 1988Google Scholar
  7. 7.
    Kloeden, P.E., Lorenz, J. Stable attracting sets in dynamical systems and their one-step discretizations. SIAM J. Numer. Anal., 23: 986–995 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kloeden, P.E., B. SchmalfußLyapunov functions and attractors under variable time–step discretization. Discrete and Conts. Dynamical Systems, 2: 163–172 (1996)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Kloeden, P.E., Kozyakin, V.S. The inflation of attractors and their discretization: the autonomous case. Nonlinear Analysis TMA. Special Issue in Honour of V. Laksh- mikantham, 2000Google Scholar
  10. 10.
    Massatt, P. Longtime behaviour of strongly damped nonlinear wave equations. J. Differential Equations, 48: 334–349 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Pazy, A. Semigroups of linear operators and applications to partial equations. Appl. Math. Sci., 44, Springer–Verlag, New York, 1983Google Scholar
  12. 12.
    Sell, G.R. Nonautonomous differential equations and topological dynamics I, II. Amer. Math. Soc., 127: 241–289 (1967)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Stuart, A.M., Humphries, A.R. Numerical analysis and dynamical systems. Cambridge University Press, Cambridge, 1996Google Scholar
  14. 14.
    Temam, R. Infinite dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York, Second Edition, 1997Google Scholar
  15. 15.
    Webb, G.F. Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Can. J. Math., 32: 631–643 (1980)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Zhou, S. Global attractor for strongly damped nonlinear wave equations. Functional Differential Equations, 6: 451–470 (1999)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Zhou, S., Fan, X. Kernel sections for non-autonomous strongly damped wave equations. J. Math. Anal. Appl., 275: 850–869 (2002)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.School of Applied MathematicsUniversity of Electronic Science and TechnologyChengduChina
  2. 2.Department of MathematicsShanghai UniversityShanghaiChina

Personalised recommendations