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Blow-up phenomena and the local well-posedness and ill-posedness of the generalized Camassa–Holm equation in critical Besov spaces

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Abstract

In this paper, we first establish the local well-posednesss for the Cauchy problem of a generalized Camassa–Holm (gCH) equation in Besov spaces \(B^{1+\frac{1}{p}}_{p,1}\) with \(1\le p<+\infty .\) Then we gain two blow-up criterions, and present two new blow-up results. Finally, we prove the ill-posedness of the gCH equation in critical Besov spaces \(B^{\frac{3}{2}}_{2,r},~r\in (1,+\infty ].\)

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References

  1. Anco, S. C., Recio, E., Gandarias, M. L., Bruzon, M.: A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Dyn. Syst. Differ. Equ. Appl. 29–37 (2015)

  2. Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Heidelberg (2011)

    Book  MATH  Google Scholar 

  3. Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. 5(1), 1–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183(2), 215–239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byers, P.: Existence time for the Camassa–Holm equation and the critical Sobolev index. Indiana Univ. Math. J. 55(3), 941–954 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin, A.: On the scattering problem for the Camassa–Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2008), 953–970 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26(2), 303–328 (1998)

    MathSciNet  MATH  Google Scholar 

  9. Constantin, A., Escher, J.: Well-posedness, global existence, and blow up phenomena for a periodic quasilinear hyperbolic equation. Commun. Pure Appl. Math. 51(5), 475–504 (1998)

    Article  MATH  Google Scholar 

  10. Constantin, A., Gerdjikov, V.S., Ivanov, R.I.: Inverse scattering transform for the Camassa–Holm equation. Inverse Probl. 22(6), 2197–2207 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Constantin, A., McKean, H.P.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52(8), 949–982 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211(1), 45–61 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Danchin, R.: A few remarks on the Camassa–Holm equation. Differ. Integral Equ. 14(8), 953–988 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Danchin, R.: A note on well-posedness for Camassa–Holm equation. J. Differ. Equ. 192(2), 429–444 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4(1), 47–66 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, Z., Liu, X., Molinet, L., Yin, Z.: Ill-posedness of the Camassa–Holm and related equations in the critical space. J. Differ. Equ. 266(2–3), 1698–1707 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hakkaev, S., Kirchev, K.: Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation. Commun. Partial Differ. Equ. 30(4–6), 761–781 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, H., Yin, Z.: On a generalized Camassa–Holm equation with the flow generated by velocity and its gradient. Appl. Anal. 96(4), 679–701 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Himonas, A.A., Holliman, C.: The Cauchy problem for the Novikov equation. Nonlinearity 25(2), 449–479 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, J., Yin, Z.: Remarks on the well-posedness of Camassa–Holm type equations in Besov spaces. J. Differ. Equ. 261(11), 6125–6143 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, J., Yu, Y., Zhu, W.: Ill-posedness for the Camassa–Holm and related equations in Besov spaces. J. Differ. Equ. 306, 403–417 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, Y., Mu, C., Zhou, S., Tu, X.: The global conservative solutions for the generalized Camassa–Holm equation. Electron. Res. Arch. 27, 37–67 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mi, Y., Mu, C.: Well-posedness and analyticity for the Cauchy problem for the generalized Camassa–Holm equation. J. Math. Anal. Appl. 405(1), 173–182 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mi, Y., Mu, C.: On the Cauchy problem for the generalized Camassa–Holm equation. Monatsh. Math. 176(3), 423–457 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Commun. Pure Appl. Math. 53(11), 1411–1433 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ye, W., Yin, Z., Guo, Y.: A new result for the local well-posedness of the Camassa–Holm type equations in critial Besov spaces \(B^{1+\frac{1}{p}}_{p,1}, 1\le p<\infty \). arXiv:2101.00803

  27. Zhou, S.: Persistence properties for a generalized Camassa–Holm equation in weighted \(L^p\) spaces. J. Math. Anal. Appl. 410(2), 932–938 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was partially supported by NNSFC (Grant Nos. 12171493 and 11671407), FDCT (Grant No. 0091/2018/A3), the Guangdong Special Support Program (Grant No. 8-2015), and the key project of NSF of Guangdong province (Grant No. 2016A03031104). The authors thank the referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Zhiying Meng & Zhaoyang Yin

  2. Faculty of Information Technology, Macau University of Science and Technology, Macau, China

    Zhaoyang Yin

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  1. Zhiying Meng
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  2. Zhaoyang Yin
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Correspondence to Zhaoyang Yin.

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Communicated by Adrian Constantin.

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Meng, Z., Yin, Z. Blow-up phenomena and the local well-posedness and ill-posedness of the generalized Camassa–Holm equation in critical Besov spaces. Monatsh Math 200, 933–954 (2023). https://doi.org/10.1007/s00605-022-01719-9

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  • DOI: https://doi.org/10.1007/s00605-022-01719-9

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