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Journal of Function Spaces
Volume 2017, Article ID 4896161, 10 pages
https://doi.org/10.1155/2017/4896161
Research Article

Global Attractors of the Extensible Plate Equations with Nonlinear Damping and Memory

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Correspondence should be addressed to Qiaozhen Ma; nc.ude.unwn@hzqam

Received 12 January 2017; Accepted 2 March 2017; Published 4 April 2017

Academic Editor: Hugo Leiva

Copyright © 2017 Xiaobin Yao and Qiaozhen Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Woinowsky-Krieger, “The effect of axial force on the vibration of hinged bars,” Journal of Applied Mechanics, vol. 17, pp. 35–36, 1950. View at Google Scholar
  2. H. M. Berger, “A new approach to the analysis of large deflections of plates,” Journal of Applied Mechanics, vol. 22, pp. 465–472, 1955. View at Google Scholar · View at MathSciNet
  3. J. E. Rivera and L. H. Fatori, “Smoothing effect and propagations of singularities for viscoelastic plates,” Journal of Mathematical Analysis and Applications, vol. 206, no. 2, pp. 397–427, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and T. Ma, “Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains,” Differential and Integral Equations, vol. 17, no. 5-6, pp. 495–510, 2004. View at Google Scholar · View at MathSciNet
  5. M. Conti and P. G. Geredeli, “Existence of smooth global attractors for nonlinear viscoelastic equations with memory,” Journal of Evolution Equations, vol. 15, no. 3, pp. 533–558, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. C. M. Dafermos, “Asymptotic stability in viscoelasticity,” Archive for Rational Mechanics and Analysis, vol. 37, no. 4, pp. 297–308, 1970. View at Publisher · View at Google Scholar · View at Scopus
  7. H. Xiao, “Asymptotic dynamics of plate equations with a critical exponent on unbounded domain,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 3, pp. 1288–1301, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. L. Yang and C.-K. Zhong, “Global attractor for plate equation with nonlinear damping,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 11, pp. 3802–3810, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. G. Yue and C. Zhong, “Global attractors for plate equations with critical exponent in locally uniform spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 9, pp. 4105–4114, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. L. Yang, “Uniform attractor for non-autonomous plate equation with a localized damping and a critical nonlinearity,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1243–1254, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. A. Kh. Khanmamedov, “Existence of a global attractor for the plate equation with a critical exponent in an unbounded domain,” Applied Mathematics Letters, vol. 18, no. 7, pp. 827–832, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. A. K. Khanmamedov, “Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain,” Journal of Differential Equations, vol. 225, no. 2, pp. 528–548, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. A. K. Khanmamedov, “A global attractor for the plate equation with displacement-dependent damping,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 5, pp. 1607–1615, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. W. Ma and Q. Ma, “Attractors for stochastic strongly damped plate equations with additive noise,” Electronic Journal of Differential Equations, no. 111, pp. 1–12, 2013. View at Google Scholar · View at MathSciNet
  15. H. Wu, “Long-time behavior for a nonlinear plate equation with thermal memory,” Journal of Mathematical Analysis and Applications, vol. 348, no. 2, pp. 650–670, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. A. R. A. Barbosa and T. F. Ma, “Long-time dynamics of an extensible plate equation with thermal memory,” Journal of Mathematical Analysis and Applications, vol. 416, no. 1, pp. 143–165, 2014. View at Publisher · View at Google Scholar · View at Scopus
  17. M. A. Jorge Silva and T. F. Ma, “Long-time dynamics for a class of Kirchhoff models with memory,” Journal of Mathematical Physics, vol. 54, no. 2, Article ID 021505, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. C. Giorgi, J. E. Rivera, and V. Pata, “Global attractors for a semilinear hyperbolic equation in viscoelasticity,” Journal of Mathematical Analysis and Applications, vol. 260, no. 1, pp. 83–99, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. C. Giorgi, V. Pata, and A. Marzocchi, “Asymptotic behavior of a semilinear problem in heat conduction with memory,” Nonlinear Differential Equations and Applications, vol. 5, no. 3, pp. 333–354, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  20. C. Giorgi, M. Grasselli, and V. Pata, “Well-posedness and longtime behavior of the phase-field model with memory in a history space setting,” Quarterly of Applied Mathematics, vol. 59, no. 4, pp. 701–736, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. X. Shen and Q. Ma, “The existence of random attractors for plate equations with memory and additive white noise,” Korean Journal of Mathematics, vol. 24, no. 3, pp. 447–467, 2016. View at Publisher · View at Google Scholar
  22. M. M. Cavalcanti, A. D. D. Cavalcanti, I. Lasiecka, and X. Wang, “Existence and sharp decay rate estimates for a von Karman system with long memory,” Nonlinear Analysis: Real World Applications, vol. 22, pp. 289–306, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  23. I. Lasiecka and X. Wang, “Intrinsic decay rate estimates for semilinear abstract second order equations with memory,” in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, vol. 10 of Springer INdAM Series, pp. 271–303, Springer International Publishing, Cham, Switzerland, 2014. View at Publisher · View at Google Scholar
  24. J. Zhou, “Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping,” Applied Mathematics and Computation, vol. 265, pp. 807–818, 2015. View at Publisher · View at Google Scholar · View at Scopus
  25. C. Giorgi, V. Pata, and E. Vuk, “On the extensible viscoelastic beam,” Nonlinearity, vol. 21, no. 4, pp. 713–733, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. C. Giorgi, M. G. Naso, V. Pata, and M. Potomkin, “Global attractors for the extensible thermoelastic beam system,” Journal of Differential Equations, vol. 246, no. 9, pp. 3496–3517, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. B. W. Feng, “Uniform attractors for a nonlinear non-autonomous extensible plate equation,” Chinese Annals of Mathematics. Series A, vol. 37, no. 1, pp. 15–30, 2016. View at Google Scholar · View at MathSciNet
  28. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, “Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation,” Communications in Contemporary Mathematics, vol. 6, no. 5, pp. 705–731, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  30. I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long-Time Dynamics, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  31. I. Chueshov and I. Lasiecka, “Long-time behavior of second order evolution equations with nonlinear damping,” Memoirs of the American Mathematical Society, vol. 195, no. 912, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  32. I. Chueshov and I. Lasiecka, “Long-time dynamics of a semilinear wave equation with nonlinear interior/boundary damping and sources of critical exponents,” in Control Methods in PDE-Dynamical Systems, vol. 426 of AMS Contemporary Mathematics, pp. 153–192, American Chemical Society, Providence, RI, USA, 2007. View at Publisher · View at Google Scholar · View at MathSciNet