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:
[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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

[35]

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

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[44]

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

[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.

[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.

[47]

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

[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.

[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.

[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.

[51]

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

[52]

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

[53]

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

[54]

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

[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.

[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.

[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.

[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.

[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.

[60]

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

[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.

[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.

[63]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

[35]

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

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

[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.

[44]

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

[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.

[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.

[47]

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

[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.

[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.

[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.

[51]

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

[52]

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

[53]

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

[54]

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

[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.

[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.

[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.

[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.

[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.

[60]

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

[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.

[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.

[63]

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

[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.

[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.

[1]

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