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May  2020, 25(5): 1623-1630. doi: 10.3934/dcdsb.2019243

Persistence properties for the generalized Camassa-Holm equation

Yongsheng Mi 1, , Boling Guo 2, and  Chunlai Mu 3,

1. 

College of Mathematics and Statistics, Yangtze Normal University, Fuling 408100, Chongqing, China
2. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
3. 

College of Mathematics and and Statistics, Chongqing University, Chongqing 401331, China

Received  February 2019 Published  May 2020 Early access  November 2019

Fund Project: The first author is supported by NSF of China (Grant No: 11671055), NSF of Chongqing (Grant No: cstc2018jcyj AX0273) and Key project of science and technology research program of Chongqing Education Commission (Grant No: KJZD-K20180140). The second author is supported by NSF of China (Grant No: 11771062). The third author is supported by NSF of China (Grant No: 11601053).

In present paper, we study the Cauchy problem for a generalized Camassa-Holm equation, which was discovered by Novikov. Our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted spaces.

Keywords: Persistence properties, Camassa-Holm equation, local well-posedness.
Mathematics Subject Classification: 35B30, 35G25.
Citation: Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1623-1630. doi: 10.3934/dcdsb.2019243
References:
[1]

A. Aldroubi and K. Grochenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., 43 (2001), 585-620.  doi: 10.1137/S0036144501386986.

[2]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not., IMRN, 2012 (2012), 5161-5181.  doi: 10.1093/imrn/rnr218.

[3]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[4]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.  doi: 10.1006/jfan.1997.3231.

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc., Lond. A, 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701.

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.

[7]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.

[8]

A. Constantin and J. Escher, Particle trajectores in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7.

[9]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017.

[10]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[11]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[12]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[13]

A. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522.  doi: 10.1007/s00220-006-0172-4.

[14]

J. L. Li and Z. Y. Yin, Well-posedness and global existence for a generalized Degasperis-Procesi equation, Nonlinear Anal. RWA., 28 (2016), 72-90.  doi: 10.1016/j.nonrwa.2015.09.003.

[15]

Y. S. Mi, Y. Liu, B. L. Guo and T. Luo, The Cauchy problem for a generalized Camassa-Holm equation, J. Differential Equations, 266 (2019), 6739-6770.  doi: 10.1016/j.jde.2018.11.019.

[16]

V. Novikov, Generalization of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14 pp. doi: 10.1088/1751-8113/42/34/342002.

[17]

S. M. Zhou, Persistence properties for a generalized Camassa-Holm equation in weighted $L^{p}$ spaces, J. Math. Anal. Appl., 410 (2014), 932-938.  doi: 10.1016/j.jmaa.2013.09.022.

show all references

References:
[1]

A. Aldroubi and K. Grochenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., 43 (2001), 585-620.  doi: 10.1137/S0036144501386986.

[2]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not., IMRN, 2012 (2012), 5161-5181.  doi: 10.1093/imrn/rnr218.

[3]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[4]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363.  doi: 10.1006/jfan.1997.3231.

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc., Lond. A, 457 (2001), 953-970.  doi: 10.1098/rspa.2000.0701.

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.

[7]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.

[8]

A. Constantin and J. Escher, Particle trajectores in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7.

[9]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.  doi: 10.1088/0266-5611/22/6/017.

[10]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.  doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[11]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.  doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[12]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.  doi: 10.1016/0167-2789(81)90004-X.

[13]

A. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522.  doi: 10.1007/s00220-006-0172-4.

[14]

J. L. Li and Z. Y. Yin, Well-posedness and global existence for a generalized Degasperis-Procesi equation, Nonlinear Anal. RWA., 28 (2016), 72-90.  doi: 10.1016/j.nonrwa.2015.09.003.

[15]

Y. S. Mi, Y. Liu, B. L. Guo and T. Luo, The Cauchy problem for a generalized Camassa-Holm equation, J. Differential Equations, 266 (2019), 6739-6770.  doi: 10.1016/j.jde.2018.11.019.

[16]

V. Novikov, Generalization of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14 pp. doi: 10.1088/1751-8113/42/34/342002.

[17]

S. M. Zhou, Persistence properties for a generalized Camassa-Holm equation in weighted $L^{p}$ spaces, J. Math. Anal. Appl., 410 (2014), 932-938.  doi: 10.1016/j.jmaa.2013.09.022.

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