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March  2017, 37(3): 1733-1748. doi: 10.3934/dcds.2017072

Conserved quantities, global existence and blow-up for a generalized CH equation

1. 

Department of Mathematics, Hangzhou Dianzi University, Zhejiang 310018, China

2. 

Department of Mathematics, The University of Texas Rio Grande Valley, Edinburg, TX 78541, USA

3. 

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

Received  June 2016 Revised  August 2016 Published  December 2016

In this paper, we study conserved quantities, blow-up criterions, and global existence of solutions for a generalized CH equation. We investigate the classification of self-adjointness, conserved quantities for this equation from the viewpoint of Lie symmetry analysis. Then, based on these conserved quantities, blow-up criterions and global existence of solutions are presented.

Citation: Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072
References:
[1]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. Google Scholar

[2]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl.(Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[3]

V. Busuioc, On second grade fluids with vanishing viscosity, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1241-1246. doi: 10.1016/S0764-4442(99)80447-9. Google Scholar

[4]

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

[5]

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

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757. Google Scholar

[7]

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

[8]

A. Constantin and J. Escher, Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Super. Pisa Cl. Sci., 26 (1998), 303-328. Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[11]

A. ConstantinR. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012. Google Scholar

[12]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. Google Scholar

[13]

A. Constantin and H. 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

[14]

H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. Google Scholar

[15]

A. DegasperisD. Holm and A. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. Google Scholar

[16]

A. DegasperisD. Holm and A. Hone, Integral and non-integrable equations with peakons, in Nonlinear Physics: Theory and Experiment, Ⅱ (Gallipoli, 2002), World Sci. Publ., River Edge, NJ, (2003), 37-43. doi: 10.1142/9789812704467_0005. Google Scholar

[17]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Scientific Publ., River Edge, NJ, (1999), 23-37. Google Scholar

[18]

H. DullinG. Gottwald and D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D., 190 (2004), 1-14. doi: 10.1016/j.physd.2003.11.004. Google Scholar

[19]

H. Dullin, G. Gottwald and D. Holm, An integrable shallow water equation with linear and nonlinear dispersion Phys. Rev. Lett. 87 (2001), 194501. doi: 10.1103/PhysRevLett.87.194501. Google Scholar

[20]

J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the $b$-equation, J. Reine Angew. Math., 624 (2008), 51-80. doi: 10.1515/CRELLE.2008.080. Google Scholar

[21]

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

[22]

M. Gandarias, Weak self-adjoint differential equations, J. Phys. A: Math. Theor. 44 (2011), 262001.Google Scholar

[23]

M. GandariasM. Redondo and M. Bruzon, Some weak self-adjoint Hamilton Jacobi Bellman equations arising in financial mathematics, Nonlinear Anal.: RWA, 13 (2012), 340-347. doi: 10.1016/j.nonrwa.2011.07.041. Google Scholar

[24]

K. Grayshan and A. Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232. Google Scholar

[25]

G. GuiY. Liu and T. Tian, Global existence and blow-up phenomena for the peakon $b$-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234. doi: 10.1512/iumj.2008.57.3213. Google Scholar

[26]

A. Himonas and C. Holliman, The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differ. Equations, 19 (2014), 161-200. Google Scholar

[27]

A. Himonas and C. Thompson, Persistence properties and unique continuation for a generalized Camassa-Holm equation J. Math. Phys. 55 (2014), 091503, 12pp. doi: 10.1063/1.4895572. Google Scholar

[28]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047. Google Scholar

[29]

D. Holm and M. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380. doi: 10.1137/S1111111102410943. Google Scholar

[30]

J. Holmes, Continuity properties of the data-to-solution map for the generalized Camassa-Holm equation, J. Math. Anal. Appl., 417 (2014), 635-642. doi: 10.1016/j.jmaa.2014.03.033. Google Scholar

[31]

A. HoneH. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation, Dyn. Partial Differential Eqns., 6 (2009), 253-289. doi: 10.4310/DPDE.2009.v6.n3.a3. Google Scholar

[32]

A. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity J. Phys. A 41 (2008), 372002, 10pp. doi: 10.1088/1751-8113/41/37/372002. Google Scholar

[33]

Y. HouP. ZhaoE. Fan and Z. Qiao, Algebro-geometric solutions for the Degasperis-Procesi hierarchy, SIAM J. Math. Anal., 45 (2013), 1216-1266. doi: 10.1137/12089689X. Google Scholar

[34]

N. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328. doi: 10.1016/j.jmaa.2006.10.078. Google Scholar

[35]

N. Ibragimov, Quasi-self-adjoint differential equations, Archives of ALGA., 4 (2007), 55-60. Google Scholar

[36]

N. Ibragimov, Nonlinear self-adjointness and conservation laws J. Phys. A: Math. Theor. 44 (2011), 432002, 8pp. doi: 10.1088/1751-8113/44/43/432002. Google Scholar

[37]

N. Ibragimov, M. Torrisi and R. Traciná, Self-adjointness and conservation laws of a generalized Burgers equation J. Phys. A: Math. Theor. 44 (2011), 145201, 5pp. doi: 10.1088/1751-8113/44/14/145201. Google Scholar

[38]

D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1. Google Scholar

[39]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. Google Scholar

[40]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111. doi: 10.1016/S0169-5983(03)00036-4. Google Scholar

[41]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations, (Proc. Sympos. , Dundee, 1974; dedicated to Konrad Jorgens), pp. 25-70. Lecture Notes in Math. , Vol. 448, Springer, Berlin, 1975. Google Scholar

[42]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3. Google Scholar

[43]

H. McKean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874. doi: 10.4310/AJM.1998.v2.n4.a10. Google Scholar

[44]

A. Mikhailov and V. Novikov, Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790. doi: 10.1088/0305-4470/35/22/309. Google Scholar

[45]

L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differential Equations, 250 (2011), 3002-3021. doi: 10.1016/j.jde.2011.01.030. Google Scholar

[46]

W. Niu and S. Zhang, Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation, J. Math. Anal. Appl., 374 (2011), 166-177. doi: 10.1016/j.jmaa.2010.08.002. Google Scholar

[47]

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

[48]

P. Olver, Applications of Lie Groups to Differential Equations New York: Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4350-2. Google Scholar

[49]

Z. Qiao, The Camassa-Holm hierarchy, $N$-dimensional integrable systems, and algebrogeometric solution on a symplectic submanifold, Commu. Math. Phys., 239 (2003), 309-341. doi: 10.1007/s00220-003-0880-y. Google Scholar

[50]

Z. Qiao, Integrable hierarchy, $3× 3$ constrained systems, and parametric and stationary solutions, Acta Appl. Math., 83 (2004), 199-220. doi: 10.1023/B:ACAP.0000038872.88367.dd. Google Scholar

[51]

L. Wei, Conservation laws for a modified lubrication equation, Nonlinear Analysis: RWA, 26 (2015), 44-55. doi: 10.1016/j.nonrwa.2015.04.005. Google Scholar

[52]

L. Wei and J. Zhang, Self-adjointness and conservation laws for Kadomtsev-Petviashvili-Burgers equation, Nonlinear Analysis: RWA, 23 (2015), 123-128. doi: 10.1016/j.nonrwa.2014.11.008. Google Scholar

[53]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Sup. Pisa CI. Sci., 11 (2012), 707-727. Google Scholar

[54]

Z. Xin and P. Zhang, On the weak solution to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar

[55]

W. YanY. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation, J. Differential Equations, 253 (2012), 298-318. doi: 10.1016/j.jde.2012.03.015. Google Scholar

[56]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. Google Scholar

[57]

S. Zhou and C. Mu, The properties of solutions for a generalized $b$-family equation with peakons, J. Nonlinear Sci., 23 (2013), 863-889. doi: 10.1007/s00332-013-9171-8. Google Scholar

[58]

S. Zhou, The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted spaces, Discrete Contin. Dyn. Syst., 34 (2014), 4967-4986. doi: 10.3934/dcds.2014.34.4967. Google Scholar

[59]

S. ZhouC. Mu and L. Wang, Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation, Discrete Contin. Dyn. Syst., 34 (2014), 843-867. doi: 10.3934/dcds.2014.34.843. Google Scholar

show all references

References:
[1]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. Google Scholar

[2]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl.(Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[3]

V. Busuioc, On second grade fluids with vanishing viscosity, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1241-1246. doi: 10.1016/S0764-4442(99)80447-9. Google Scholar

[4]

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

[5]

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

[6]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757. Google Scholar

[7]

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

[8]

A. Constantin and J. Escher, Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Super. Pisa Cl. Sci., 26 (1998), 303-328. Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[11]

A. ConstantinR. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012. Google Scholar

[12]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. Google Scholar

[13]

A. Constantin and H. 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

[14]

H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. Google Scholar

[15]

A. DegasperisD. Holm and A. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. Google Scholar

[16]

A. DegasperisD. Holm and A. Hone, Integral and non-integrable equations with peakons, in Nonlinear Physics: Theory and Experiment, Ⅱ (Gallipoli, 2002), World Sci. Publ., River Edge, NJ, (2003), 37-43. doi: 10.1142/9789812704467_0005. Google Scholar

[17]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Scientific Publ., River Edge, NJ, (1999), 23-37. Google Scholar

[18]

H. DullinG. Gottwald and D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D., 190 (2004), 1-14. doi: 10.1016/j.physd.2003.11.004. Google Scholar

[19]

H. Dullin, G. Gottwald and D. Holm, An integrable shallow water equation with linear and nonlinear dispersion Phys. Rev. Lett. 87 (2001), 194501. doi: 10.1103/PhysRevLett.87.194501. Google Scholar

[20]

J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the $b$-equation, J. Reine Angew. Math., 624 (2008), 51-80. doi: 10.1515/CRELLE.2008.080. Google Scholar

[21]

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

[22]

M. Gandarias, Weak self-adjoint differential equations, J. Phys. A: Math. Theor. 44 (2011), 262001.Google Scholar

[23]

M. GandariasM. Redondo and M. Bruzon, Some weak self-adjoint Hamilton Jacobi Bellman equations arising in financial mathematics, Nonlinear Anal.: RWA, 13 (2012), 340-347. doi: 10.1016/j.nonrwa.2011.07.041. Google Scholar

[24]

K. Grayshan and A. Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl., 8 (2013), 217-232. Google Scholar

[25]

G. GuiY. Liu and T. Tian, Global existence and blow-up phenomena for the peakon $b$-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234. doi: 10.1512/iumj.2008.57.3213. Google Scholar

[26]

A. Himonas and C. Holliman, The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differ. Equations, 19 (2014), 161-200. Google Scholar

[27]

A. Himonas and C. Thompson, Persistence properties and unique continuation for a generalized Camassa-Holm equation J. Math. Phys. 55 (2014), 091503, 12pp. doi: 10.1063/1.4895572. Google Scholar

[28]

H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Contin. Dyn. Syst., 24 (2009), 1047-1112. doi: 10.3934/dcds.2009.24.1047. Google Scholar

[29]

D. Holm and M. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380. doi: 10.1137/S1111111102410943. Google Scholar

[30]

J. Holmes, Continuity properties of the data-to-solution map for the generalized Camassa-Holm equation, J. Math. Anal. Appl., 417 (2014), 635-642. doi: 10.1016/j.jmaa.2014.03.033. Google Scholar

[31]

A. HoneH. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation, Dyn. Partial Differential Eqns., 6 (2009), 253-289. doi: 10.4310/DPDE.2009.v6.n3.a3. Google Scholar

[32]

A. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity J. Phys. A 41 (2008), 372002, 10pp. doi: 10.1088/1751-8113/41/37/372002. Google Scholar

[33]

Y. HouP. ZhaoE. Fan and Z. Qiao, Algebro-geometric solutions for the Degasperis-Procesi hierarchy, SIAM J. Math. Anal., 45 (2013), 1216-1266. doi: 10.1137/12089689X. Google Scholar

[34]

N. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311-328. doi: 10.1016/j.jmaa.2006.10.078. Google Scholar

[35]

N. Ibragimov, Quasi-self-adjoint differential equations, Archives of ALGA., 4 (2007), 55-60. Google Scholar

[36]

N. Ibragimov, Nonlinear self-adjointness and conservation laws J. Phys. A: Math. Theor. 44 (2011), 432002, 8pp. doi: 10.1088/1751-8113/44/43/432002. Google Scholar

[37]

N. Ibragimov, M. Torrisi and R. Traciná, Self-adjointness and conservation laws of a generalized Burgers equation J. Phys. A: Math. Theor. 44 (2011), 145201, 5pp. doi: 10.1088/1751-8113/44/14/145201. Google Scholar

[38]

D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312. doi: 10.2991/jnmp.2007.14.3.1. Google Scholar

[39]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224. Google Scholar

[40]

R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res., 33 (2003), 97-111. doi: 10.1016/S0169-5983(03)00036-4. Google Scholar

[41]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations, (Proc. Sympos. , Dundee, 1974; dedicated to Konrad Jorgens), pp. 25-70. Lecture Notes in Math. , Vol. 448, Springer, Berlin, 1975. Google Scholar

[42]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3. Google Scholar

[43]

H. McKean, Breakdown of a shallow water equation, Asian J. Math., 2 (1998), 867-874. doi: 10.4310/AJM.1998.v2.n4.a10. Google Scholar

[44]

A. Mikhailov and V. Novikov, Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790. doi: 10.1088/0305-4470/35/22/309. Google Scholar

[45]

L. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation, J. Differential Equations, 250 (2011), 3002-3021. doi: 10.1016/j.jde.2011.01.030. Google Scholar

[46]

W. Niu and S. Zhang, Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation, J. Math. Anal. Appl., 374 (2011), 166-177. doi: 10.1016/j.jmaa.2010.08.002. Google Scholar

[47]

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

[48]

P. Olver, Applications of Lie Groups to Differential Equations New York: Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4350-2. Google Scholar

[49]

Z. Qiao, The Camassa-Holm hierarchy, $N$-dimensional integrable systems, and algebrogeometric solution on a symplectic submanifold, Commu. Math. Phys., 239 (2003), 309-341. doi: 10.1007/s00220-003-0880-y. Google Scholar

[50]

Z. Qiao, Integrable hierarchy, $3× 3$ constrained systems, and parametric and stationary solutions, Acta Appl. Math., 83 (2004), 199-220. doi: 10.1023/B:ACAP.0000038872.88367.dd. Google Scholar

[51]

L. Wei, Conservation laws for a modified lubrication equation, Nonlinear Analysis: RWA, 26 (2015), 44-55. doi: 10.1016/j.nonrwa.2015.04.005. Google Scholar

[52]

L. Wei and J. Zhang, Self-adjointness and conservation laws for Kadomtsev-Petviashvili-Burgers equation, Nonlinear Analysis: RWA, 23 (2015), 123-128. doi: 10.1016/j.nonrwa.2014.11.008. Google Scholar

[53]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Sup. Pisa CI. Sci., 11 (2012), 707-727. Google Scholar

[54]

Z. Xin and P. Zhang, On the weak solution to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar

[55]

W. YanY. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation, J. Differential Equations, 253 (2012), 298-318. doi: 10.1016/j.jde.2012.03.015. Google Scholar

[56]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666. Google Scholar

[57]

S. Zhou and C. Mu, The properties of solutions for a generalized $b$-family equation with peakons, J. Nonlinear Sci., 23 (2013), 863-889. doi: 10.1007/s00332-013-9171-8. Google Scholar

[58]

S. Zhou, The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted spaces, Discrete Contin. Dyn. Syst., 34 (2014), 4967-4986. doi: 10.3934/dcds.2014.34.4967. Google Scholar

[59]

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