A new asymptotic behavior of solutions to the Camassa-Holm equation
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- by Lidiao Ni and Yong Zhou PDF
- Proc. Amer. Math. Soc. 140 (2012), 607-614 Request permission
Abstract:
The present work is mainly concerned with an algebraic decay rate of the strong solution to the Camassa-Holm equation in $L^{\infty }$-space. In particular, it is proved that the solution decays algebraically with the same exponent as that of the initial datum.References
- Alberto Bressan and Adrian Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal. 183 (2007), no. 2, 215–239. MR 2278406, DOI 10.1007/s00205-006-0010-z
- Adrian Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys. 46 (2005), no. 2, 023506, 4. MR 2121730, DOI 10.1063/1.1845603
- Adrian Constantin and Joachim Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998), no. 5, 475–504. MR 1604278, DOI 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
- Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664. MR 1234453, DOI 10.1103/PhysRevLett.71.1661
- Raphaël Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations 14 (2001), no. 8, 953–988. MR 1827098
- B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981/82), no. 1, 47–66. MR 636470, DOI 10.1016/0167-2789(81)90004-X
- H. P. McKean, Breakdown of a shallow water equation, Asian J. Math. 2 (1998), no. 4, 867–874. Mikio Sato: a great Japanese mathematician of the twentieth century. MR 1734131, DOI 10.4310/AJM.1998.v2.n4.a10
- A. Alexandrou Himonas, Gerard Misiołek, Gustavo Ponce, and Yong Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys. 271 (2007), no. 2, 511–522. MR 2287915, DOI 10.1007/s00220-006-0172-4
- Z. Jiang, L. Ni and Y. Zhou, Wave breaking for the Camassa-Holm equation. Preprint (2010).
- L. Ni and Y. Zhou, Wave breaking and propagation speed for a class of nonlocal dispersive $\theta$-equations. Nonlinear Anal.: Real World Appl. 12 (2011), no. 1, 592-600.
- Guillermo Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal. 46 (2001), no. 3, Ser. A: Theory Methods, 309–327. MR 1851854, DOI 10.1016/S0362-546X(01)00791-X
- Zhouping Xin and Ping Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math. 53 (2000), no. 11, 1411–1433. MR 1773414, DOI 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.3.CO;2-X
- Yong Zhou, Wave breaking for a shallow water equation, Nonlinear Anal. 57 (2004), no. 1, 137–152. MR 2055991, DOI 10.1016/j.na.2004.02.004
- Yong Zhou, Wave breaking for a periodic shallow water equation, J. Math. Anal. Appl. 290 (2004), no. 2, 591–604. MR 2033045, DOI 10.1016/j.jmaa.2003.10.017
- Yong Zhou, On solutions to the Holm-Staley $b$-family of equations, Nonlinearity 23 (2010), no. 2, 369–381. MR 2578483, DOI 10.1088/0951-7715/23/2/008
- Yong Zhou and Zhengguang Guo, Blow up and propagation speed of solutions to the DGH equation, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), no. 3, 657–670. MR 2525163, DOI 10.3934/dcdsb.2009.12.657
Additional Information
- Lidiao Ni
- Affiliation: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, People’s Republic of China
- Email: ni.lidiao@gmail.com
- Yong Zhou
- Affiliation: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, People’s Republic of China
- Email: yzhoumath@zjnu.edu.cn
- Received by editor(s): April 21, 2010
- Received by editor(s) in revised form: November 28, 2010
- Published electronically: May 12, 2011
- Additional Notes: The second author is the corresponding author and is partially supported by the Zhejiang Innovation Project (Grant No. T200905), ZJNSF (Grant No. R6090109) and NSFC (Grant No. 10971197)
- Communicated by: Walter Craig
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 607-614
- MSC (2010): Primary 37L05; Secondary 35Q58, 26A12
- DOI: https://doi.org/10.1090/S0002-9939-2011-10922-5
- MathSciNet review: 2846329