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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 102594, 14 pages
http://dx.doi.org/10.1155/2014/102594
Research Article

Boundary Stabilization of a Nonlinear Viscoelastic Equation with Interior Time-Varying Delay and Nonlinear Dissipative Boundary Feedback

1College of Science, National University of Defense Technology, Changsha, Hunan 410083, China
2School of Mathematics, Hunan Institute of Science and Technology, Yueyang, Hunan 414006, China
3School of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

Received 22 January 2014; Revised 2 May 2014; Accepted 16 May 2014; Published 9 June 2014

Academic Editor: Chuangxia Huang

Copyright © 2014 Zaiyun Zhang et al. 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. K. Ammari, S. Nicaise, and C. Pignotti, “Feedback boundary stabilization of wave equations with interior delay,” Systems & Control Letters, vol. 59, no. 10, pp. 623–628, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Z.-Y. Zhang, Z.-H. Liu, and X.-Y. Gan, “Global existence and general decay for a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density,” Applicable Analysis, vol. 92, no. 10, pp. 2021–2048, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Z.-Y. Zhang and X.-J. Miao, “Global existence and uniform decay for wave equation with dissipative term and boundary damping,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 1003–1018, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao, and Y.-Z. Chen, “Global existence and uniform stabilization of a generalized dissipative Klein-Gordon equation type with boundary damping,” Journal of Mathematical Physics, vol. 52, no. 2, Article ID 023502, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  5. Z.-Y. Zhang, Z.-H. Liu, and X.-J. Miao, “Estimate on the dimension of global attractor for nonlinear dissipative Kirchhoff equation,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 271–282, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, and J. A. Soriano, “Asymptotic stability of the wave equation on compact surfaces and locally distributed damping–-a sharp result,” Transactions of the American Mathematical Society, vol. 361, no. 9, pp. 4561–4580, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, and J. A. Soriano, “Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result,” Archive for Rational Mechanics and Analysis, vol. 197, no. 3, pp. 925–964, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, “Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping,” Electronic Journal of Differential Equations, vol. 44, pp. 1–14, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Berrimi and S. A. Messaoudi, “Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping,” Electronic Journal of Differential Equations, vol. 88, pp. 1–10, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. M. Cavalcanti and H. P. Oquendo, “Frictional versus viscoelastic damping in a semilinear wave equation,” SIAM Journal on Control and Optimization, vol. 42, no. 4, pp. 1310–1324, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao, and Y.-Z. Chen, “A note on decay properties for the solutions of a class of partial differential equation with memory,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 85–102, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  12. W. Liu, “General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms,” Journal of Mathematical Physics, vol. 50, no. 11, Article ID 113506, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. W. Liu, “Uniform decay of solutions for a quasilinear system of viscoelastic equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2257–2267, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W. Liu, “General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 6, pp. 1890–1904, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. W. Liu, “General decay of solutions to a viscoelastic wave equation with nonlinear localized damping,” Annales Academiæ Scientiarum Fennicæ. Mathematica, vol. 34, no. 1, pp. 291–302, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. W. Liu, “General decay of solutions of a nonlinear system of viscoelastic equations,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 153–165, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. W. Liu, “Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping,” Journal of Applied Mathematics and Computing, vol. 32, no. 1, pp. 59–68, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. X. Han and M. Wang, “Global existence and uniform decay for a nonlinear viscoelastic equation with damping,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3090–3098, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. X. Han and M. Wang, “General decay of energy for a viscoelastic equation with nonlinear damping,” Journal of the Franklin Institute, vol. 347, no. 5, pp. 806–817, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. X. Han and M. Wang, “Global existence and asymptotic behavior for a coupled hyperbolic system with localized damping,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 2, pp. 965–986, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. X. Han and M. Wang, “Energy decay rate for a coupled hyperbolic system with nonlinear damping,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3264–3272, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. X. Han and M. Wang, “General decay of energy for a viscoelastic equation with nonlinear damping,” Mathematical Methods in the Applied Sciences, vol. 32, no. 3, pp. 346–358, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, “Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping,” Differential and Integral Equations, vol. 14, no. 1, pp. 85–116, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. S. A. Messaoudi and N.-E. Tatar, “Exponential and polynomial decay for a quasilinear viscoelastic equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 4, pp. 785–793, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. N.-E. Tatar, “Exponential decay for a viscoelastic problem with a singular kernel,” Zeitschrift für Angewandte Mathematik und Physik, vol. 60, no. 4, pp. 640–650, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. S. Berrimi and S. A. Messaoudi, “Existence and decay of solutions of a viscoelastic equation with a nonlinear source,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 10, pp. 2314–2331, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. S. A. Messaoudi, “General decay of the solution energy in a viscoelastic equation with a nonlinear source,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2589–2598, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. W. Liu, “Asymptotic behavior of solutions of time-delayed Burgers' equation,” Discrete and Continuous Dynamical Systems B, vol. 2, no. 1, pp. 47–56, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Y. F. Shang, G. Q. Xu, and Y. L. Chen, “Stability analysis of Euler-Bernoulli beam with input delay in the boundary control,” Asian Journal of Control, vol. 14, no. 1, pp. 186–196, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. R. Datko, “Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks,” SIAM Journal on Control and Optimization, vol. 26, no. 3, pp. 697–713, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. R. Datko, J. Lagnese, and M. P. Polis, “An example on the effect of time delays in boundary feedback stabilization of wave equations,” SIAM Journal on Control and Optimization, vol. 24, no. 1, pp. 152–156, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. S. Nicaise and C. Pignotti, “Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks,” SIAM Journal on Control and Optimization, vol. 45, no. 5, pp. 1561–1585, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. Nicaise and C. Pignotti, “Stabilization of the wave equation with boundary or internal distributed delay,” Differential and Integral Equations, vol. 21, no. 9-10, pp. 935–958, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. G. Q. Xu, S. P. Yung, and L. K. Li, “Stabilization of wave systems with input delay in the boundary control,” ESAIM. Control, Optimisation and Calculus of Variations, vol. 12, no. 4, pp. 770–785, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. E. M. Ait Benhassi, K. Ammari, S. Boulite, and L. Maniar, “Feedback stabilization of a class of evolution equations with delay,” Journal of Evolution Equations, vol. 9, no. 1, pp. 103–121, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. S. Nicaise, C. Pignotti, and J. Valein, “Exponential stability of the wave equation with boundary time-varying delay,” Discrete and Continuous Dynamical Systems S, vol. 4, no. 3, pp. 693–722, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. M. Kirane and B. Said-Houari, “Existence and asymptotic stability of a viscoelastic wave equation with a delay,” Zeitschrift für Angewandte Mathematik und Physik, vol. 62, no. 6, pp. 1065–1082, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. T. Caraballo, J. Real, and L. Shaikhet, “Method of Lyapunov functionals construction in stability of delay evolution equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1130–1145, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. Z.-Y. Zhang, Z.-H. Liu, X.-J. Miao, and Y.-Z. Chen, “Stability analysis of heat flow with boundary time-varying delay effect,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 6, pp. 1878–1889, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. S. Nicaise and C. Pignotti, “Interior feedback stabilization of wave equations with time dependent delay,” Electronic Journal of Differential Equations, vol. 41, pp. 1–20, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. S. Nicaise, J. Valein, and E. Fridman, “Stability of the heat and of the wave equations with boundary time-varying delays,” Discrete and Continuous Dynamical Systems S, vol. 2, no. 3, pp. 559–581, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. S. Nicaise and J. Valein, “Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks,” Networks and Heterogeneous Media, vol. 2, no. 3, pp. 425–479, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. E. Fridman, S. Nicaise, and J. Valein, “Stabilization of second order evolution equations with unbounded feedback with time-dependent delay,” SIAM Journal on Control and Optimization, vol. 48, no. 8, pp. 5028–5052, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. S. Nicaise and J. Valein, “Stabilization of second order evolution equations with unbounded feedback with delay,” ESAIM. Control, Optimisation and Calculus of Variations, vol. 16, no. 2, pp. 420–456, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, France, 1969. View at MathSciNet
  46. S. Zheng, Nonlinear Evolution Equations, CRC Press, Boca Raton, Fla, USA, 2004. View at Publisher · View at Google Scholar · View at MathSciNet