About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 163070, 8 pages
http://dx.doi.org/10.1155/2013/163070
Research Article

Persistence Property and Asymptotic Description for DGH Equation with Strong Dissipation

Department of Basic, Zhejiang Dongfang Vocational Technical College, Wenzhou, Zhejiang 325011, China

Received 9 February 2013; Accepted 10 March 2013

Academic Editor: Yonghui Xia

Copyright © 2013 Ke-chuang 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. R. Dullin, G. A. Gottwald, and D. D. Holm, “An integrable shallow water equation with linear and nonlinear dispersion,” Physical Review Letters, vol. 87, no. 19, Article ID 194501, 4 pages, 2001. View at Scopus
  2. J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II. The KdV equation,” Geometric and Functional Analysis, vol. 3, no. 3, pp. 209–262, 1993.
  3. R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Constantin, “Finite propagation speed for the Camassa-Holm equation,” Journal of Mathematical Physics, vol. 46, no. 2, Article ID 023506, 4 pages, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,” Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475–504, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. A. Himonas, G. Misiołek, G. Ponce, and Y. Zhou, “Persistence properties and unique continuation of solutions of the Camassa-Holm equation,” Communications in Mathematical Physics, vol. 271, no. 2, pp. 511–522, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. P. McKean, “Breakdown of the Camassa-Holm equation,” Communications on Pure and Applied Mathematics, vol. 57, no. 3, pp. 416–418, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Z. Xin and P. Zhang, “On the weak solutions to a shallow water equation,” Communications on Pure and Applied Mathematics, vol. 53, no. 11, pp. 1411–1433, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Y. Zhou, “Wave breaking for a shallow water equation,” Nonlinear Analysis: Theory, Methods & Application, vol. 57, no. 1, pp. 137–152, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Zhou, “Wave breaking for a periodic shallow water equation,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 591–604, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Y. Zhou, “Stability of solitary waves for a rod equation,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 977–981, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. Zhou, “Local well-posedness and blow-up criteria of solutions for a rod equation,” Mathematische Nachrichten, vol. 278, no. 14, pp. 1726–1739, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. L. Ni and Y. Zhou, “Wave breaking and propagation speed for a class of nonlocal dispersive θ-equations,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 592–600, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  15. L. Tian, G. Gui, and Y. Liu, “On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation,” Communications in Mathematical Physics, vol. 257, no. 3, pp. 667–701, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations: Proceedings of the Symposium Held at Dundee, Scotland, 1–19 July 1974, vol. 448 of Lecture Notes in Mathetics, pp. 25–70, Springer, Berlin, Germany, 1975. View at Zentralblatt MATH · View at MathSciNet
  17. Y. Zhou, “Blow-up of solutions to the DGH equation,” Journal of Functional Analysis, vol. 250, no. 1, pp. 227–248, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y. Zhou and Z. Guo, “Blow up and propagation speed of solutions to the DGH equation,” Discrete and Continuous Dynamical Systems B, vol. 12, no. 3, pp. 657–670, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J.-M. Ghidaglia, “Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time,” Journal of Differential Equations, vol. 74, no. 2, pp. 369–390, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Z. Guo, “Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa-Holm equation,” Journal of Mathematical Physics, vol. 49, no. 3, Article ID 033516, 9 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Z. Guo and L. Ni, “Wave breaking for the periodic weakly dissipative Dullin-Gottwald-Holm equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 965–973, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Wu and Z. Yin, “Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4309–4321, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. A. V. Busuioc and T. S. Ratiu, “The second grade fluid and averaged Euler equations with Navier-slip boundary conditions,” Nonlinearity, vol. 16, no. 3, pp. 1119–1149, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet