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

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

Received  February 2019 Published  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.

Citation: Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

A. ConstantinV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. A. HimonasG. MisiolekG. 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.  Google Scholar

[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.  Google Scholar

[15]

Y. S. MiY. LiuB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

A. ConstantinV. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

A. A. HimonasG. MisiolekG. 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.  Google Scholar

[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.  Google Scholar

[15]

Y. S. MiY. LiuB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[2]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[3]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[4]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[5]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[6]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049

[7]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269

[8]

Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386

[9]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[10]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

[11]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[12]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[13]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[14]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[15]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[16]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[17]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[18]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[19]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[20]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (205)
  • HTML views (208)
  • Cited by (0)

Other articles
by authors

[Back to Top]