February  2014, 34(2): 803-820. doi: 10.3934/dcds.2014.34.803

Local Well-posedness and Persistence Property for the Generalized Novikov Equation

1. 

Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China, China

2. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China

Received  December 2012 Revised  February 2013 Published  August 2013

In this paper, we study the generalized Novikov equation which describes the motion of shallow water waves. By using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the generalized Novikov equation is locally well-posed in Besov space $B_{p,r}^{s}$ with $1\leq p, r\leq +\infty$ and $s>{\rm max}\{1+\frac{1}{p},\frac{3}{2}\}$. We also show the persistence property of the strong solutions which implies that the solution decays at infinity in the spatial variable provided that the initial function does.
Citation: Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803
References:
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G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonl. Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation,, J. Diff. Eqns., 141 (1997), 218. doi: 10.1006/jdeq.1997.3333. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603. Google Scholar

[10]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303. Google Scholar

[11]

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. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar

[12]

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

[13]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar

[14]

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

[15]

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

[16]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonl. Sci., 12 (2002), 415. doi: 10.1007/s00332-002-0517-x. Google Scholar

[17]

R. Danchin, A few remarks on the Camassa-Holm equation,, Diff. Integ. Eqns., 14 (2001), 953. Google Scholar

[18]

R. Danchin, "Fourier Analysis Method for PDEs,", Lecture Notes, (2005). Google Scholar

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R. Danchin, On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework,, J. Diff. Eqns., 248 (2010), 2130. doi: 10.1016/j.jde.2009.09.007. Google Scholar

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A. Degasperis, D. D. Holm and A. N. I. Hone, A new integral equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422. Google Scholar

[21]

A. Degasperis and M. Procesi, Asymptotic integrability,, in, (1999), 23. Google Scholar

[22]

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

[23]

H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid. Dyn. Res., 33 (2003), 73. doi: 10.1016/S0169-5983(03)00046-7. Google Scholar

[24]

J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457. doi: 10.1016/j.jfa.2006.03.022. Google Scholar

[25]

J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation,, Indiana Univ. Math. J., 56 (2007), 87. doi: 10.1512/iumj.2007.56.3040. Google Scholar

[26]

J. Escher and Z. Yin, On the initial boundary value problems for the Degasperis-Procesi equation,, Phys. Lett. A, 368 (2007), 69. doi: 10.1016/j.physleta.2007.03.073. Google Scholar

[27]

A. Fokas, B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditray symmetries,, Physica D., 4 (): 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[28]

X. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlinearity,, Nonlinearity, 22 (2009), 1847. doi: 10.1088/0951-7715/22/8/004. Google Scholar

[29]

D. Henry, Infinite propagation speed for the Degasperis-Procesi equation,, J. Math. Anal. Appl., 311 (2005), 755. doi: 10.1016/j.jmaa.2005.03.001. Google Scholar

[30]

D. Henry, Compactly supported solutions of the Camassa-Holm equation,, J. Nonlinear Math. Phys., 12 (2005), 342. doi: 10.2991/jnmp.2005.12.3.3. Google Scholar

[31]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonl. Anal., 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104. Google Scholar

[32]

D. Henry, Persistence properties for the Degasperis-Procesi equation,, J. Hyper. Diff. Eq., 5 (2008), 99. doi: 10.1142/S0219891608001404. Google Scholar

[33]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discr. Contin. Dyn. Syst. Ser. B., 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar

[34]

A. A. Himonas and C. Holliman, On well-posedness of the Degasperis-Procesi equation,, Discr. Contin. Dyn. Syst., 31 (2011), 469. doi: 10.3934/dcds.2011.31.469. Google Scholar

[35]

A. A. Himonas and G. Misio lek, The Cauchy problem for an integrable shallow water equation,, Diff. Int. Eqns., 14 (2001), 821. Google Scholar

[36]

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

[37]

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

[38]

A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation,, Dyn. Partial Diff. Eqns., 6 (2009), 253. Google Scholar

[39]

A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129. doi: 10.1088/0266-5611/19/1/307. Google Scholar

[40]

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

[41]

R. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities,, Z. Naturforsch. A, 61 (2006), 133. Google Scholar

[42]

Z. H. Jiang and L. D. Ni, Blow-up phenomenon for the integrable Novikov equation,, J. Math. Appl. Anal., 385 (2012), 551. doi: 10.1016/j.jmaa.2011.06.067. Google Scholar

[43]

S. Y. Lai and Y. H. Wu, The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,, J. Diff. Eqns., 248 (2010), 2038. doi: 10.1016/j.jde.2010.01.008. Google Scholar

[44]

J. Lenells, Conservation laws of the Camassa-Holm equation,, J. Phys. A, 38 (2005), 869. doi: 10.1088/0305-4470/38/4/007. Google Scholar

[45]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Diff. Eqns., 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar

[46]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801. doi: 10.1007/s00220-006-0082-5. Google Scholar

[47]

H. P. McKean, Breakdown of a shallow water equation,, Asian J. Math., 2 (1998), 867. Google Scholar

[48]

Y. S. Mi and C. L. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions,, J. Diff. Eqns., 254 (2013), 961. doi: 10.1016/j.jde.2012.09.016. Google Scholar

[49]

O. Mustafa, A note on the Degasperis-Procesi equation,, J. Nonl. Math. Phys., 12 (2005), 10. doi: 10.2991/jnmp.2005.12.1.2. Google Scholar

[50]

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

[51]

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

[52]

F. Tiǧlay, The periodic Cauchy problem for Novikov's equation,, Int. Math. Res. Notices IMRN, (2011), 4633. doi: 10.1093/imrn/rnq267. Google Scholar

[53]

M. Vishik, Hydrodynamics in Besov spaces,, Arch. Rat. Mech. Anal., 145 (1998), 197. doi: 10.1007/s002050050128. Google Scholar

[54]

W. Walter, "Differential and Integral Inequalities,", Ergebnisse der Mathematik und ihrer Grenzgebiete, (1970). Google Scholar

[55]

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

[56]

K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system,, J. Diff. Eqns., 252 (2012), 2131. doi: 10.1016/j.jde.2011.08.003. Google Scholar

[57]

W. Yan, Y. S. Li and Y. M. Zhang, The Cauchy problem for the integrable Novikov equation,, J. Diff. Eqns., 253 (2012), 298. doi: 10.1016/j.jde.2012.03.015. Google Scholar

[58]

W. Yan, Y. S. Li and Y. M. Zhang, The Cauchy problem for the Novikov equation,, Nonlinear Differ. Equ. Appl., 20 (2013), 1157. doi: 10.1007/s00030-012-0202-1. Google Scholar

[59]

W. Yan, Y. S. Li and Y. M. Zhang, Global existence and blow-up phenomena for the weakly dissipative Novikov equation,, Nonl. Anal., 75 (2012), 2464. doi: 10.1016/j.na.2011.10.044. Google Scholar

[60]

W. Yan, Y. S. Li and Y. M. Zhang, The Cauchy problem for the generalized Camassa-Holm equation,, to appear., (). Google Scholar

[61]

Z. Yin, Well-posedness, blowup, and global existence for an integrable shallow water equation,, Discr. Contin. Dyn. Syst. Ser., 11 (2004), 393. doi: 10.3934/dcds.2004.11.393. Google Scholar

[62]

Z. Yin, Global existence for a new periodic integrable equation,, J. Math. Anal. Appl., 283 (2003), 129. doi: 10.1016/S0022-247X(03)00250-6. Google Scholar

[63]

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

[64]

Z. Yin, Global solutions to a new integrable equation with peakons,, Indiana Univ. Math. J., 53 (2004), 1189. doi: 10.1512/iumj.2004.53.2479. Google Scholar

[65]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions,, J. Funct. Anal., 212 (2004), 182. doi: 10.1016/j.jfa.2003.07.010. Google Scholar

show all references

References:
[1]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonl. Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation,, J. Diff. Eqns., 141 (1997), 218. doi: 10.1006/jdeq.1997.3333. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603. Google Scholar

[10]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303. Google Scholar

[11]

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. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. Google Scholar

[12]

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

[13]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar

[14]

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

[15]

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

[16]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonl. Sci., 12 (2002), 415. doi: 10.1007/s00332-002-0517-x. Google Scholar

[17]

R. Danchin, A few remarks on the Camassa-Holm equation,, Diff. Integ. Eqns., 14 (2001), 953. Google Scholar

[18]

R. Danchin, "Fourier Analysis Method for PDEs,", Lecture Notes, (2005). Google Scholar

[19]

R. Danchin, On the well-posedness of the incompressible density-dependent Euler equations in the $L^p$ framework,, J. Diff. Eqns., 248 (2010), 2130. doi: 10.1016/j.jde.2009.09.007. Google Scholar

[20]

A. Degasperis, D. D. Holm and A. N. I. Hone, A new integral equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422. Google Scholar

[21]

A. Degasperis and M. Procesi, Asymptotic integrability,, in, (1999), 23. Google Scholar

[22]

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

[23]

H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid. Dyn. Res., 33 (2003), 73. doi: 10.1016/S0169-5983(03)00046-7. Google Scholar

[24]

J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457. doi: 10.1016/j.jfa.2006.03.022. Google Scholar

[25]

J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation,, Indiana Univ. Math. J., 56 (2007), 87. doi: 10.1512/iumj.2007.56.3040. Google Scholar

[26]

J. Escher and Z. Yin, On the initial boundary value problems for the Degasperis-Procesi equation,, Phys. Lett. A, 368 (2007), 69. doi: 10.1016/j.physleta.2007.03.073. Google Scholar

[27]

A. Fokas, B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditray symmetries,, Physica D., 4 (): 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[28]

X. Geng and B. Xue, An extension of integrable peakon equations with cubic nonlinearity,, Nonlinearity, 22 (2009), 1847. doi: 10.1088/0951-7715/22/8/004. Google Scholar

[29]

D. Henry, Infinite propagation speed for the Degasperis-Procesi equation,, J. Math. Anal. Appl., 311 (2005), 755. doi: 10.1016/j.jmaa.2005.03.001. Google Scholar

[30]

D. Henry, Compactly supported solutions of the Camassa-Holm equation,, J. Nonlinear Math. Phys., 12 (2005), 342. doi: 10.2991/jnmp.2005.12.3.3. Google Scholar

[31]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonl. Anal., 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104. Google Scholar

[32]

D. Henry, Persistence properties for the Degasperis-Procesi equation,, J. Hyper. Diff. Eq., 5 (2008), 99. doi: 10.1142/S0219891608001404. Google Scholar

[33]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation,, Discr. Contin. Dyn. Syst. Ser. B., 12 (2009), 597. doi: 10.3934/dcdsb.2009.12.597. Google Scholar

[34]

A. A. Himonas and C. Holliman, On well-posedness of the Degasperis-Procesi equation,, Discr. Contin. Dyn. Syst., 31 (2011), 469. doi: 10.3934/dcds.2011.31.469. Google Scholar

[35]

A. A. Himonas and G. Misio lek, The Cauchy problem for an integrable shallow water equation,, Diff. Int. Eqns., 14 (2001), 821. Google Scholar

[36]

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

[37]

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

[38]

A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation,, Dyn. Partial Diff. Eqns., 6 (2009), 253. Google Scholar

[39]

A. N. W. Hone and J. P. Wang, Prolongation algebras and Hamiltonian operators for peakon equations,, Inverse Problems, 19 (2003), 129. doi: 10.1088/0266-5611/19/1/307. Google Scholar

[40]

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

[41]

R. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities,, Z. Naturforsch. A, 61 (2006), 133. Google Scholar

[42]

Z. H. Jiang and L. D. Ni, Blow-up phenomenon for the integrable Novikov equation,, J. Math. Appl. Anal., 385 (2012), 551. doi: 10.1016/j.jmaa.2011.06.067. Google Scholar

[43]

S. Y. Lai and Y. H. Wu, The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,, J. Diff. Eqns., 248 (2010), 2038. doi: 10.1016/j.jde.2010.01.008. Google Scholar

[44]

J. Lenells, Conservation laws of the Camassa-Holm equation,, J. Phys. A, 38 (2005), 869. doi: 10.1088/0305-4470/38/4/007. Google Scholar

[45]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Diff. Eqns., 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar

[46]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801. doi: 10.1007/s00220-006-0082-5. Google Scholar

[47]

H. P. McKean, Breakdown of a shallow water equation,, Asian J. Math., 2 (1998), 867. Google Scholar

[48]

Y. S. Mi and C. L. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions,, J. Diff. Eqns., 254 (2013), 961. doi: 10.1016/j.jde.2012.09.016. Google Scholar

[49]

O. Mustafa, A note on the Degasperis-Procesi equation,, J. Nonl. Math. Phys., 12 (2005), 10. doi: 10.2991/jnmp.2005.12.1.2. Google Scholar

[50]

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

[51]

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

[52]

F. Tiǧlay, The periodic Cauchy problem for Novikov's equation,, Int. Math. Res. Notices IMRN, (2011), 4633. doi: 10.1093/imrn/rnq267. Google Scholar

[53]

M. Vishik, Hydrodynamics in Besov spaces,, Arch. Rat. Mech. Anal., 145 (1998), 197. doi: 10.1007/s002050050128. Google Scholar

[54]

W. Walter, "Differential and Integral Inequalities,", Ergebnisse der Mathematik und ihrer Grenzgebiete, (1970). Google Scholar

[55]

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

[56]

K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system,, J. Diff. Eqns., 252 (2012), 2131. doi: 10.1016/j.jde.2011.08.003. Google Scholar

[57]

W. Yan, Y. S. Li and Y. M. Zhang, The Cauchy problem for the integrable Novikov equation,, J. Diff. Eqns., 253 (2012), 298. doi: 10.1016/j.jde.2012.03.015. Google Scholar

[58]

W. Yan, Y. S. Li and Y. M. Zhang, The Cauchy problem for the Novikov equation,, Nonlinear Differ. Equ. Appl., 20 (2013), 1157. doi: 10.1007/s00030-012-0202-1. Google Scholar

[59]

W. Yan, Y. S. Li and Y. M. Zhang, Global existence and blow-up phenomena for the weakly dissipative Novikov equation,, Nonl. Anal., 75 (2012), 2464. doi: 10.1016/j.na.2011.10.044. Google Scholar

[60]

W. Yan, Y. S. Li and Y. M. Zhang, The Cauchy problem for the generalized Camassa-Holm equation,, to appear., (). Google Scholar

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