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Persistence properties for the generalized Camassa-Holm equation
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 |
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.
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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