Journal of Applied Mathematics

Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation

Yongsheng Mi and Chunlai Mu

Full-text: Open access

Abstract

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 547261, 11 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808127

Digital Object Identifier
doi:10.1155/2013/547261

Mathematical Reviews number (MathSciNet)
MR3100837

Zentralblatt MATH identifier
06950742

Citation

Mi, Yongsheng; Mu, Chunlai. Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation. J. Appl. Math. 2013 (2013), Article ID 547261, 11 pages. doi:10.1155/2013/547261. https://projecteuclid.org/euclid.jam/1394808127


Export citation

References

  • 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.
  • M. Chen, S.-Q. Liu, and Y. Zhang, “A two-component generalization of the Camassa-Holm equation and its solutions,” Letters in Mathematical Physics, vol. 75, no. 1, pp. 1–15, 2006.
  • Z. Popowicz, “A 2-component or $N=2$ supersymmetric Camassa-Holm equation,” Physics Letters A, vol. 354, no. 1-2, pp. 110–114, 2006.
  • H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation–-a Lagrangian point of view,” Communications in Partial Differential Equations, vol. 32, no. 10-12, pp. 1511–1549, 2007.
  • H. Holden and X. Raynaud, “Dissipative solutions for the Camassa-Holm equation,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 4, pp. 1047–1112, 2009.
  • Y. Zhou, “Stability of solitary waves for a rod equation,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 977–981, 2004.
  • 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.
  • Y. Zhou and H. Chen, “Wave breaking and propagation speed for the Camassa-Holm equation with $\kappa {\,\neq\,}0$,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 12, no. 3, pp. 1875–1882, 2011.
  • 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.
  • G. Falqui, “On a Camassa-Holm type equation with two dependent variables,” Journal of Physics A, vol. 39, no. 2, pp. 327–342, 2006.
  • 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.
  • 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.
  • C. Guan and Z. Yin, “Global weak solutions for a two-component Camassa-Holm shallow water system,” Journal of Functional Analysis, vol. 260, no. 4, pp. 1132–1154, 2011.
  • D. Henry, “Infinite propagation speed for a two component Camassa-Holm equation,” Discrete and Continuous Dynamical Systems B, vol. 12, no. 3, pp. 597–606, 2009.
  • G. Gui and Y. Liu, “On the Cauchy problem for the two-component Camassa-Holm system,” Mathematische Zeitschrift, vol. 268, no. 1-2, pp. 45–66, 2011.
  • O. G. Mustafa, “On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system,” Wave Motion, vol. 46, no. 6, pp. 397–402, 2009.
  • Q. Hu and Z. Yin, “Well-posedness and blow-up phenomena for a periodic two-component Camassa-Holm equation,” Proceedings of the Royal Society of Edinburgh A, vol. 141, no. 1, pp. 93–107, 2011.
  • Q. Hu and Z. Yin, “Global existence and blow-up phenomena for a periodic 2-component Camassa-Holm equation,” Monatshefte für Mathematik, vol. 165, no. 2, pp. 217–235, 2012.
  • D. Holm, L. Ó. Náraigh, and C. Tronci, “Singular solutions of a modified two-component Camassa-Holm equation,” Physical Review E, vol. 79, Article ID 016601, 2009.
  • G. Lv and M. Wang, “Some remarks for a modified periodic Camassa-Holm system,” Discrete and Continuous Dynamical Systems A, vol. 30, no. 4, pp. 1161–1180, 2011.
  • 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.
  • S. Yu, “Well-posedness and blow-up for a modified two-component Camassa-Holm equation,” Applicable Analysis, vol. 91, no. 7, pp. 1321–1337, 2012.
  • C. Guan, K. Karlsen, and Z. Yin, “Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation, Nonlinear partial differential equations and hyperbolic wave phenomena,” in Contemporary Mathematics, vol. 526, pp. 199–220, American Mathematical Society, Providence, RI, USA, 2010.
  • W. Tan and Z. Yin, “Global dissipative solutions of a modified two-component Camassa-Holm shallow water system,” Journal of Mathematical Physics, vol. 52, no. 3, Article ID 033507, 2011.
  • Z. Guo and L. Ni, “Persistence properties and unique continuation of solutions to a two-component Camassa-Holm equation,” Mathematical Physics, Analysis and Geometry, vol. 14, no. 2, pp. 101–114, 2011.
  • W. Tan and Z. Yin, “Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation,” Journal of Functional Analysis, vol. 261, no. 5, pp. 1204–1226, 2011.
  • Z. Guo, M. Zhu, and L. Ni, “Blow-up criteria of solutions to a modified two-component Camassa-Holm system,” Nonlinear Analysis. Real World Applications, vol. 12, no. 6, pp. 3531–3540, 2011.
  • 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.
  • L. Jin and Z. Guo, “A note on a modified two-component Camassa-Holm system,” Nonlinear Analysis. Real World Applications, vol. 13, no. 2, pp. 887–892, 2012.
  • W. Rui and Y. Long, “Integral bifurcation method together with a translation-dilation transformation for solving an integrable 2-component Camassa-Holm shallow water system,” Journal of Applied Mathematics, vol. 2012, Article ID 736765, 21 pages, 2012.
  • J. B. Li and Y. S. Li, “Bifurcations of travelling wave solutions for a two-component Camassa-Holm equation,” Acta Mathematica Sinica, vol. 24, no. 8, pp. 1319–1330, 2008.
  • 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.
  • Q. Hu and Z. Yin, “Blowup and blowup rate of solutions to a weakly dissipative periodic rod equation,” Journal of Mathematical Physics, vol. 50, no. 8, Article ID 083503, 2009.
  • Q. Hu, “Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa-Holm system,” Applicable Analysis, vol. 92, no. 2, pp. 398–410, 2013.
  • Q. Hu, “Global existence and blow-up phenomena for a weakly dissipative periodic 2-component Camassa-Holm system,” Journal of Mathematical Physics, vol. 52, no. 10, Article ID 103701, 2011.
  • T. Kato, “Quasi-linear equations of evolution with application to partial differential equations,” in Spectral Theory and Differential Equations, pp. 25–70, Springer, Berlin, Germany, 1975.
  • A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.
  • L. Ni and Y. Zhou, “A new asymptotic behavior of solutions to the Camassa-Holm equation,” Proceedings of the American Mathematical Society, vol. 140, no. 2, pp. 607–614, 2012.
  • 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.
  • Z. Guo, “Asymptotic profiles of solutions to the two-component Camassa-Holm system,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 75, no. 1, pp. 1–6, 2012.