Lee, Mi Jin

  • Received : 2016.04.05
  • Accepted : 2016.04.27
  • Published : 2016.05.31


Global attractor is a basic concept to study the long-time behavior of solutions of the various equations. This paper is investigated with the existence of a global attractor for the beam equation $$u_{tt}+{\Delta}^2u-{\nabla}{\cdot}\{{\sigma}({\mid}{\nabla}u{\mid}^2){\nabla}u\}+f(u)+a(x)g(u_t)=h,$$ using multipliers technique and Nakao's Lemma.


Beam equation;Global attractor;Long-time behavior


  1. J. M. Ball, Stability theory for an extensible beam, J. Differential Equations 14 (1973),399-418.
  2. J. M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61-90.
  3. A. V. Babin, M. I. Vishik, Attrators of evolution equations, Nauka, Moscow, 1989(1992) (English translation, North-Holland 1992).
  4. R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl. 29 (1970) 443-454.
  5. R. W. Dickey, Dynamic stability of equilibrium states of the extensible beam, J. Proc. Amer. Math. Soc. 41 (1973) 94-102.
  6. C. M. Dafermos, Asymptotic behavior of solutions of evolution equations in nonlinear evolution equations(M. G. Crandall, ed.), Funkcial. Ekvac. 38 (1995) 545-568.
  7. J. K. Hale, Asymptotic behavior of dissipative systems, AMS, Providence, RI, 1988.
  8. A. Kh. Khanmamedov, Existence of a global attractor for the plate equation with the critical exponent in an unbounded domain, Appl. Math. Letter 18 (2005) 827-832.
  9. A. Kh. Khanmamedov, Global attractors for plate equation with a localized damping and critical exponent in an unbounded domain, J. Differential Equations 225 (2006) 528-548.
  10. To Fu Ma, V. Narciso, Global attractor for a model of extensible beam with nonlinear damping and source terms, Nonlinear Anal. 73 (2010) 3402-3412.
  11. M. Nakao, Global attractors for wave equaitons with nonlinear dissipative terms, J. Differential Equations 227 (2006) 204-229.
  12. M. Nakao, Global attractors for some quasi-linear wave equations with a strong with a strong dissipation, Advances in Math. Sciences Appl. 17 (2007), no.1, 89-105.
  13. M. Nakao and C. Chen, On global attractors for a nonlinear parabolic equation of m-laplacian type in RN, Funkcialaj Ekvacioj 50 (2007) 449-458.
  14. M. Nakao and N. Aris, On global attractor for nonlinear parabolic equations of m-laplacian type, J. Math. Anal. Appl. 331 (2007) 793-809.
  15. R. Temam, Infinite dimensional dynamic system in meachanics and physics, Springer, New York, 1997.
  16. Y. Xie, C. Zhong, Asymptotic behavior of a class of nonlinear evolution equations, Nonlinear Anal. 71 (2009) 5095-5105.
  17. L. Yang, C. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Anal. 69 (2008) 3802-3810.
  18. G. Yue, C. Zhong, Golbal attractors for plate equations with critical exponent in locallay uniform spaces, Nonlinear Anal. 71 (2009) 4105-4114.
  19. Y. Zhijian, Longtime behavior for a nonlinear wave equation arising in elasto-plastic flow, Math. Meth. Appl. Sci. 32 (2009) 1082-1104.