Advertisement

Wuhan University Journal of Natural Sciences

, Volume 10, Issue 6, pp 961–965 | Cite as

Blow-up of the solutions of the initial boundary value problem of Camassa-Holm equation

  • Zhu Xu-sheng
  • Wang Wei-ke
Article

Abstract

Any classical non-null solution to the initial boundary value problem of Camassa-Holm equation on finite interval with homogeneous boundary condition must blow up in finite time. An initial boundary value problem of Camassa-Holm equation on half axis is also investigated in this paper. When the initial potential is nonnegative, then the classical solution exists globally; if the derivative of initial data on zero point is nonpositire, then the life span of nonzero solution must be finite.

Key words

Camassa-Holm equation initial boundary value problem blow-up 

CLC number

O 175. 24 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Camassa R, Holm D. An Integrable Shallow Water Equation with Peaked Solutions.Phys Rev Lett, 1993,71:1661–1664.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Camassa R, Holm D, Hyman J. A New Integrable Shallow Water Equation.Adv Appl Mech, 1994,31:1–33.CrossRefGoogle Scholar
  3. [3]
    Constantin A, Escher J. Global Existence and Blow-up for a Shallow Water Equation.Ann Scuola Norm Sup Pissa Cl Sci, 1998,26(4):303–328.MATHMathSciNetGoogle Scholar
  4. [4]
    Constantin A, Escher J. On the Cauchy Problem for a Family of Quasilinear Hyperbolic Equation.Comm Partial Diff Eqns, 1998,23:1449–1458.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Constantin A, Escher J. Wave Breaking for Nonlinear Nonlocal Shallow Water Equations.Acta Math, 1998,181:229–243.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Rodriguez-Blanco G. On the Cauchy Problem for the Camassa-Holm Equation.Nonlinear Anal, 2001,46:309–327.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Constantin A, Escher J. Global Weak Solutions for a Shallow Water Equation.Indiana Univ Math J, 1998,47: 1527–1545.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Constantin A, Molinet L. Global Weak Solutions for a Shallow Water Equation.Comm Math Phys, 2000,211:45–61.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Xin Z, Zhang P. On the Weak Solutions to a Shallow Water Equation.Comm Pure Appl Math, 2000,53:1411–1433.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Xin Z, Zhang P. On the Uniqueness and Large Time Behavior of the Weak Solutions to a Shallow Water Equation.Comm Partial Diff Eqns, 2002,27:1815–1844.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Constantin A. On the Cauchy Problem for the Periodic Camassa-Holm Equation.J Diff Eqns, 1997,141:218–235.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Constantin A, Escher J. Well-posedness, Global Existence, and Blow-up Phenomena for a Periodic Quasi-linear Hyperbolic Equation.J Comm Pure Appl Math, 1998,51:475–504.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Kwek K, Gao H, Zhang W,et al. An Initial Boundary Value Problem of Camassa-Holm Equation.J Math Phys, 2000,41:8279–8285.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Courant R, Hilbert D.Methods of Mathematical Physics, Vol. 1. New York: Interscience, 1953.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Zhu Xu-sheng
    • 1
    • 2
  • Wang Wei-ke
    • 1
  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhan, HubeiChina
  2. 2.Department of MathematicsEast China Jiaotong UniversityNanchang, JiangxiChina

Personalised recommendations