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The regularity of local solutions for a generalized Camassa-Holm type equation

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Abstract

The regularity of the Cauchy problem for a generalized Camassa-Holm type equation is investigated. The pseudoparabolic regularization approach is employed to obtain some prior estimates under certain assumptions on the initial value of the equation. The local existence of its solution in Sobolev space H s(ℝ) with 1 < s ≤ 3/2 is derived.

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References

  1. Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71, 1661–1664 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Camassa, R., Holm, D., Hyman, J. M.: A new integral shallow water equation. Adv. Appl. Mech., 31, 1–33 (1994)

    Article  Google Scholar 

  3. Li, Y., Olver, P.: Well-posedness and blow solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eqs., 162, 27–63 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Comm. Pure Appl. Math., 53, 1411–1433 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Xin, X., Zhang, P.: On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Comm. Partial Differential Equations, 27(9–10), 1815–1844 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Wazwaz, A. M.: A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions. Appl. Math. Comput., 165, 485–501 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lim, W. K.: Global well-posedness for the viscous Camassa-Holm equation. J. Math. Anal. Appl., 326, 432–442 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Tian, L. X., Song X. Y.: New peaked solitary wave solutions of the generalized Camassa-Holm equation. Chaos, Solitons and Fractals, 19, 621–637 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Yin, Z. Y.: On the blow-up scenario for the generalized Camassa-Holm equation. Comm. Partial Differential Equations, 29, 867–877 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fu, Y. P., Guo, B. L.: Time periodic solution of the viscous Camassa-Holm equation. J. Math. Anal. Appl., 313, 311–321 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ma, L., Chen, D. Z., Yang, Y.: Some results on subelliptic equations. Acta Mathematica Sinica, English Series, 22, 1695–1704 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Miao, C. X., Zhu, Y. B.: Well-posedness in the energy space for non-linear system of wave equations with critical growth. Acta Mathematica Sinica, English Series, 23, 17–26 (2007)

    Article  MathSciNet  Google Scholar 

  13. Kokilashili, V., S. Samko, S.: Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent. Acta Mathematica Sinica, English Series, 24, 1775–1800 (2008)

    Article  MathSciNet  Google Scholar 

  14. Ji, R. H., Zheng, S. N.: Asymptotic estimates to non-global solutions of a multi-coupled parabolic system. Acta Mathematica Sinica, English Series, 24, 1713–1726 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bona, J., Smith, R.: The initial value problem for the Korteweg-de Vries equation. Phil. Trans. Roy. Soc. London Ser. A, 278, 555–601 (1975)

    Article  MathSciNet  Google Scholar 

  16. Kato, T., Ponce, G.: Commutator estimayes and the Euler and Navier-Stokes equation. Commun. Pure Appl. Math., 41, 891–907 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  17. Walter, W.: Differential and Integral Inequalities, Springer-Verlag, New York, 1970

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, P. R. China

    Shao Yong Lai

  2. Department of Mathematics, Sichuan Normal University, Chengdu, 610066, P. R. China

    Jian Zhang

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  1. Shao Yong Lai
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  2. Jian Zhang
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Correspondence to Shao Yong Lai.

Additional information

Supported by Key Project of Chinese Ministry of Education (Grant No. 109140) and the SWUFE’s third period construction item funds of the 211 project (Grant No. 211D3T06)

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Lai, S.Y., Zhang, J. The regularity of local solutions for a generalized Camassa-Holm type equation. Acta. Math. Sin.-English Ser. 25, 2065–2076 (2009). https://doi.org/10.1007/s10114-009-8104-9

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  • DOI: https://doi.org/10.1007/s10114-009-8104-9

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