Abstract and Applied Analysis

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

Jingjing Xu and Zaihong Jiang

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Abstract

We investigate a more general family of one-dimensional shallow water equations with a weakly dissipative term. First, we establish blow-up criteria for this family of equations. Then, global existence of the solution is also proved. Finally, we discuss the infinite propagation speed of this family of equations.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 851476, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512068

Digital Object Identifier
doi:10.1155/2013/851476

Mathematical Reviews number (MathSciNet)
MR3108660

Zentralblatt MATH identifier
07095430

Citation

Xu, Jingjing; Jiang, Zaihong. On the Cauchy Problem for a Class of Weakly Dissipative One-Dimensional Shallow Water Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 851476, 7 pages. doi:10.1155/2013/851476. https://projecteuclid.org/euclid.aaa/1393512068


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References

  • 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.
  • L. Ni and Y. Zhou, “Wave breaking and propagation speed for a class of nonlocal dispersive $\theta $-equations,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 592–600, 2011.
  • 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.
  • 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.
  • 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.
  • 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.
  • H. P. McKean, “Breakdown of a shallow water equation,” The Asian Journal of Mathematics, vol. 2, no. 4, pp. 867–874, 1998.
  • 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.
  • A. Constantin and W. A. Strauss, “Stability of peakons,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp. 603–610, 2000.
  • Y. Zhou, “Stability of solitary waves for a rod equation,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 977–981, 2004.
  • 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.
  • 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.
  • 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.
  • Y. Zhou, “Blow-up phenomenon for the integrable Degasperis-Procesi equation,” Physics Letters A, vol. 328, no. 2-3, pp. 157–162, 2004.
  • Y. Zhou, “On solutions to the Holm-Staley $b$-family of equations,” Nonlinearity, vol. 23, no. 2, pp. 369–381, 2010.
  • 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.
  • 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.
  • 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.
  • 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.