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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[35]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[43]

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

[44]

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

[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.

[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.

[47]

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

[48]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[56]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[35]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[43]

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

[44]

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

[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.

[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.

[47]

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

[48]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[56]

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

[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.

[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.

[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.

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