Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 353757, 8 pages
http://dx.doi.org/10.1155/2013/353757
Research Article

Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

Received 17 December 2012; Revised 15 May 2013; Accepted 30 May 2013

Academic Editor: Roberto Natalini

Copyright © 2013 Yu-Zhu Wang. 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. H. T. Banks, D. S. Gilliam, and V. I. Shubov, “Global solvability for damped abstract nonlinear hyperbolic systems,” Differential and Integral Equations, vol. 10, no. 2, pp. 309–332, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. G. Chen and F. Da, “Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 358–372, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. Song and Z. Yang, “Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation,” Mathematical Methods in the Applied Sciences, vol. 33, no. 5, pp. 563–575, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. G. Chen, “Initial boundary value problem for a damped nonlinear hyperbolic equation,” Journal of Partial Differential Equations, vol. 16, no. 1, pp. 49–61, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. G. Chen, R. Song, and S. Wang, “Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation,” Journal of Mathematical Analysis and Applications, vol. 368, no. 1, pp. 19–31, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y.-Z. Wang, “Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 465–482, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Y.-Z. Wang, F. G. Liu, and Y. Z. Zhang, “Global existence and asymptotic behavior of solutions for a semi-linear wave equation,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 836–853, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Wang and H. Xu, “On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term,” Journal of Differential Equations, vol. 252, no. 7, pp. 4243–4258, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. V. Pata and S. Zelik, “Smooth attractors for strongly damped wave equations,” Nonlinearity, vol. 19, no. 7, pp. 1495–1506, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques and Applications, Springer, Berlin, Germany, 1997. View at MathSciNet
  11. C. D. Sogge, Lecures on Nonlinear Wave Equations, vol. 2 of Monographs in Analysis, International Press, 1995. View at MathSciNet
  12. W.-R. Dai and D.-X. Kong, “Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields,” Journal of Differential Equations, vol. 235, no. 1, pp. 127–165, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D.-X. Kong and T. Yang, “Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems,” Communications in Partial Differential Equations, vol. 28, no. 5-6, pp. 1203–1220, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y.-Z. Wang, “Global existence of classical solutions to the minimal surface equation in two space dimensions with slow decay initial value,” Journal of Mathematical Physics, vol. 50, no. 10, Article ID 103506, 14 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y.-Z. Wang and Y.-X. Wang, “Global existence of classical solutions to the minimal surface equation with slow decay initial value,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 576–583, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Liu and W. Wang, “The pointwise estimates of solutions for dissipative wave equation in multi-dimensions,” Discrete and Continuous Dynamical Systems A, vol. 20, no. 4, pp. 1013–1028, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. M. Nakao and K. Ono, “Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,” Mathematische Zeitschrift, vol. 214, no. 2, pp. 325–342, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. K. Nishihara, “Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application,” Mathematische Zeitschrift, vol. 244, no. 3, pp. 631–649, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. K. Ono, “Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations,” Discrete and Continuous Dynamical Systems A, vol. 9, no. 3, pp. 651–662, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. W. Wang and W. Wang, “The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions,” Journal of Mathematical Analysis and Applications, vol. 366, no. 1, pp. 226–241, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. Z. Yang, “Longtime behavior of the Kirchhoff type equation with strong damping on n,” Journal of Differential Equations, vol. 242, no. 2, pp. 269–286, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Y. Liu and S. Kawashima, “Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,” Discrete and Continuous Dynamical Systems A, vol. 29, no. 3, pp. 1113–1139, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. Sugitani and S. Kawashima, “Decay estimates of solutions to a semilinear dissipative plate equation,” Journal of Hyperbolic Differential Equations, vol. 7, no. 3, pp. 471–501, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  24. T. T. Li and Y. M. Chen, Nonlinear Evolution Equations, Scientific Press, 1989 (Chinese).
  25. S. M. Zheng, Nonlinear Evolution Equations, vol. 133 of Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, 2004. View at Publisher · View at Google Scholar · View at MathSciNet