Mathematical Physics, Analysis and Geometry

, Volume 14, Issue 3, pp 197–209

Blow-up, Global Existence and Persistence Properties for the Coupled Camassa–Holm equations



In this paper, we consider the coupled Camassa–Holm equations. First, we present some new criteria on blow-up. Then global existence and blow-up rate of the solution are also established. Finally, we discuss persistence properties of this system.


Coupled Camassa–Holm equations Blow-up Global existence Blow-up rate Persistence properties 

Mathematics Subject Classifications (2010)

37L05 35Q58 26A12 


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  1. 1.
    Constantin, A.: On the Cauchy problem for the periodic Camassa–Holm equation. J. Differ. Equ. 141, 218–235 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier 50, 321–362 (2000)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Constantin, A.: Finite propagation speed for the CamassaHolm equation. J. Math. Phys. 46, 023506 (2005)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math 166, 523–535 (2006)MathSciNetADSCrossRefMATHGoogle Scholar
  5. 5.
    Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equation. Acta Math. 181, 229–243 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Constantin, A., Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 75–91 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Math. Ann. 173, 559–568 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fu, Y., Liu, Y., Qu, C.: Well-possdness and blow-up solution for a modified two-component Camassa–Holm system with peakons. Math. Ann. 348, 415–448 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fu, Y., Qu, C.: Well-possdness and blow-up solution for a new coupled Camassa–Holm equations with peakons. J. Math. Phys. 50, 012906 (2009)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Guo, Z.: Blow up and global solutions to a new integrable model with two components. J. Math. Anal. Appl. 372, 316–327 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Guo, Z., Zhou, Y.: On solutions to a two-component generalized Camassa–Holm equation. Stud. Appl. Math. 124, 307–322 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Henry, D.: Compactly supported solutions of the Camassa–Holm equation. J. Nonlin. Math. Phys. 12, 342–347 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Henry, D.: Persistence properties for a family of nonlinear partial differential equations. Nonlinear Anal. 70, 1565–1573 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007)MathSciNetADSCrossRefMATHGoogle Scholar
  16. 16.
    Jin, L., Guo, Z.: On a two-component Degasperis-Procesi shallow water system. Nonlinear Anal. 11, 4164–4173 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jin, L., Liu, Y., Zhou, Y.: Blow-up of solutions to a periodic nonlinear dispersive rod equation. Doc. Math. 15, 267–283 (2010)MathSciNetMATHGoogle Scholar
  18. 18.
    Mckean, H.P.: Breakdown of a shallow water equation. Asian J. Math. 2, 767–774 (1998)MathSciNetGoogle Scholar
  19. 19.
    Ni, L.: The Cauchy problem for a two-component generalized θ-equations. Nonlinear Anal. 73, 1338–1349 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ni, L., Zhou, Y.: A new asymptotic behavior of solutions to the Camassa–Holm equation. Proc. Am. Math. Soc. (2011). doi:10.1007/s11040-011-9094-2
  21. 21.
    Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)MathSciNetMATHGoogle Scholar
  22. 22.
    Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57, 137–152 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zhou, Y.: Blow-up phenomenon for a periodic rod equation. Phys. Lett. A 353, 479–486 (2006)MathSciNetADSCrossRefMATHGoogle Scholar
  24. 24.
    Zhou, Y.: Blow-up of solutions to the DGH equation. J. Funct. Anal. 250(1), 227–248 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaPeople’s Republic of China

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