Abstract and Applied Analysis

Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation

Panpan Zhai, Zhengguang Guo, and Weiming Wang

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Abstract

This paper is concerned with blow-up phenomena and persistence properties for an integrable two-component Dullin-Gottwald-Holm shallow water system. We give sufficient conditions on the initial data which guarantee blow-up phenomena of solutions in finite time for both periodic and nonperiodic cases, respectively. Furthermore, the persistence properties of solutions to the system are investigated.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 750315, 13 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449401

Digital Object Identifier
doi:10.1155/2013/750315

Mathematical Reviews number (MathSciNet)
MR3067061

Zentralblatt MATH identifier
07095324

Citation

Zhai, Panpan; Guo, Zhengguang; Wang, Weiming. Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 750315, 13 pages. doi:10.1155/2013/750315. https://projecteuclid.org/euclid.aaa/1393449401


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References

  • 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.
  • J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations,” Geometric and Functional Analysis, vol. 3, no. 3, pp. 209–262, 1993.
  • C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,” Communications on Pure and Applied Mathematics, vol. 46, no. 4, pp. 527–620, 1993.
  • P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, Mass, USA, 1989.
  • 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.
  • A. A. Himonas and G. Misiołek, “The Cauchy problem for an integrable shallow-water equation,” Differential and Integral Equations, vol. 14, no. 7, pp. 821–831, 2001.
  • 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.
  • G. Misiołek, “Classical solutions of the periodic Camassa-Holm equation,” Geometric and Functional Analysis, vol. 12, no. 5, pp. 1080–1104, 2002.
  • S. Shkoller, “Geometry and curvature of diffeomorphism groups with ${H}^{1}$ metric and mean hydrodynamics,” Journal of Functional Analysis, vol. 160, no. 1, pp. 337–365, 1998.
  • A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.
  • 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.
  • 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.
  • H. P. McKean, “Breakdown of a shallow water equation,” The Asian Journal of Mathematics, vol. 2, no. 4, pp. 867–874, 1998.
  • Y. Zhou, “Wave breaking for a periodic shallow water equation,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 591–604, 2004.
  • Y. Zhou, “Wave breaking for a shallow water equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 57, no. 1, pp. 137–152, 2004.
  • 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.
  • 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.
  • F. Cooper and H. Shepard, “Solitons in the Camassa-Holm shallow water equation,” Physics Letters A, vol. 194, no. 4, pp. 246–250, 1994.
  • L. Tian and X. Song, “New peaked solitary wave solutions of the generalized Camassa-Holm equation,” Chaos, Solitons and Fractals, vol. 19, no. 3, pp. 621–637, 2004.
  • L. Tian and J. Yin, “New compacton solutions and solitary wave solutions of fully nonlinear generalized Camassa-Holm equations,” Chaos, Solitons and Fractals, vol. 20, no. 2, pp. 289–299, 2004.
  • 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.
  • Y. Liu, “Global existence and blow-up solutions for a nonlinear shallow water equation,” Mathematische Annalen, vol. 335, no. 3, pp. 717–735, 2006.
  • Y. Zhou, “Blow-up of solutions to the DGH equation,” Journal of Functional Analysis, vol. 250, no. 1, pp. 227–248, 2007.
  • 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.
  • A. Constantin and R. I. Ivanov, “On an integrable two-component Camassa-Holm shallow water system,” Physics Letters A, vol. 372, no. 48, pp. 7129–7132, 2008.
  • J. Escher, O. Lechtenfeld, and Z. Yin, “Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,” Discrete and Continuous Dynamical Systems A, vol. 19, no. 3, pp. 493–513, 2007.
  • Y. Fu, Y. Liu, and C. Qu, “Well-posedness and blow-up solution for a modified two-component periodic Camassa-Holm system with peakons,” Mathematische Annalen, vol. 348, no. 2, pp. 415–448, 2010.
  • C. Guan and Z. Yin, “Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,” Journal of Differential Equations, vol. 248, no. 8, pp. 2003–2014, 2010.
  • G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010.
  • Z. Guo, “Blow-up and global solutions to a new integrable model with two components,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 316–327, 2010.
  • Z. Guo and Y. Zhou, “On solutions to a two-component generalized Camassa-Holm equation,” Studies in Applied Mathematics, vol. 124, no. 3, pp. 307–322, 2010.
  • Z. Guo and M. Zhu, “Wave breaking for a modified two-component Camassa-Holm system,” Journal of Differential Equations, vol. 252, no. 3, pp. 2759–2770, 2012.
  • F. Guo, H. Gao, and Y. Liu, “On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system,” Journal of the London Mathematical Society, vol. 86, no. 3, pp. 810–834, 2012.
  • M. Zhu and J. Xu, “On the wave-breaking phenomena for the periodic two-component Dullin-Gottwald-Holm system,” Journal of Mathematical Analysis and Applications, vol. 391, no. 2, pp. 415–428, 2012.
  • T. Kato, “On the Cauchy problem for the (generalized) Korteweg-de Vries equation,” in Studies in Applied Mathematics, vol. 8, pp. 93–128, Academic Press, New York, NY, USA, 1983.
  • P. Zhang and Y. Liu, “Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system,” International Mathematics Research Notices, vol. 2010, no. 11, pp. 1981–2021, 2010.
  • Y. Zhou, “Blow-up of solutions to a nonlinear dispersive rod equation,” Calculus of Variations and Partial Differential Equations, vol. 25, no. 1, pp. 63–77, 2006.
  • Z. Guo and Y. Zhou, “Wave breaking and persistence properties for the dispersive rod equation,” SIAM Journal on Mathematical Analysis, vol. 40, no. 6, pp. 2567–2580, 2009.