About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 851476, 7 pages
http://dx.doi.org/10.1155/2013/851476
Research Article

On the Cauchy Problem for a Class of Weakly Dissipative One-Dimensional Shallow Water Equations

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 21 June 2013; Accepted 22 August 2013

Academic Editor: Sining Zheng

Copyright © 2013 Jingjing Xu and Zaihong Jiang. 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. Z. Jiang and S. Hakkaev, “Wave breaking and propagation speed for a class of one-dimensional shallow water equations,” Abstract and Applied Analysis, vol. 2011, Article ID 647368, 15 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. 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
  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. B. Fuchssteiner and A. S. Fokas, “Symplectic structures, their Bäcklund transformations and hereditary symmetries,” Physica D, vol. 4, no. 1, pp. 47–66, 1981-1982. View at Publisher · View at Google Scholar · View at MathSciNet
  5. 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
  6. Y. A. Li and P. J. Olver, “Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,” Journal of Differential Equations, vol. 162, no. 1, pp. 27–63, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H. P. McKean, “Breakdown of a shallow water equation,” The Asian Journal of Mathematics, vol. 2, no. 4, pp. 867–874, 1998. View at Zentralblatt MATH · View at MathSciNet
  8. Z. Jiang, L. Ni, and Y. Zhou, “Wave breaking of the Camassa-Holm equation,” Journal of Nonlinear Science, vol. 22, no. 2, pp. 235–245, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Constantin and W. A. Strauss, “Stability of peakons,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp. 603–610, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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
  11. 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
  12. A. Degasperis and M. Procesi, “Asymptotic integrability,” in Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta, Eds., pp. 23–37, World Scientific, Singapore, 1999. View at Zentralblatt MATH · View at MathSciNet
  13. D. D. Holm and M. F. Staley, “Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE,” Physics Letters. A, vol. 308, no. 5-6, pp. 437–444, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Zhou, “Blow-up phenomenon for the integrable Degasperis-Procesi equation,” Physics Letters A, vol. 328, no. 2-3, pp. 157–162, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y. Zhou, “On solutions to the Holm-Staley b-family of equations,” Nonlinearity, vol. 23, no. 2, pp. 369–381, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  16. 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, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Z. Guo, “Some properties of solutions to the weakly dissipative Degasperis-Procesi equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4332–4344, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. W. Niu and S. Zhang, “Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation,” Journal of Mathematical Analysis and Applications, vol. 374, no. 1, pp. 166–177, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  19. M. Zhu and Z. Jiang, “Some properties of solutions to the weakly dissipative b-family equation,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 158–167, 2012. View at Publisher · View at Google Scholar · View at MathSciNet