June  2017, 37(6): 3503-3519. doi: 10.3934/dcds.2017149

The Cauchy problem for a generalized Novikov equation

1. 

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

2. 

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China, and, Faculty of Information Technology, Macau University of Science and Technology, Macau, China

1 Corresponding author

Received  September 2016 Revised  December 2016 Published  February 2017

Fund Project: This work was partially supported by NNSFC (No.11671407 and No.11271382), FDCT (No. 098/2013/A3), Guangdong Special Support Program (No. 8-2015), and the key project of NSF of Guangdong province (No. 2016A030311004)

We establish the local well-posedness for a generalized Novikov equation in nonhomogeneous Besov spaces. Besides, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.

Citation: Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149
References:
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A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z. Google Scholar

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

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A. Boutet de MonvelA. KostenkoD. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588. doi: 10.1137/090748500. Google Scholar

[5]

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

[6]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91. doi: 10.1016/j.jfa.2005.07.008. Google Scholar

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A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701. Google Scholar

[8]

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

[9]

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

[10]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar

[11]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup 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

[12]

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

[13]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar

[14]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2001), 559-568. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[15]

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

[16]

A. ConstantinR. I. 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

[17]

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

[18]

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

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A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. Google Scholar

[20]

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

[21]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. Google Scholar

[22]

A. DegasperisD. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. Google Scholar

[23]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (Rome, 1998), World Sci. Publ., River Edge, NJ, 133 (1999), 23-37. Google Scholar

[24]

H. R. DullinG. A. Gottwald and D. 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

[25]

J. EscherY. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022. Google Scholar

[26]

J. EscherY. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117. doi: 10.1512/iumj.2007.56.3040. Google Scholar

[27]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[28]

G. Gui and Y. Liu, On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math., 69 (2011), 445-464. doi: 10.1090/S0033-569X-2011-01216-5. Google Scholar

[29]

G. GuiY. LiuP. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759. doi: 10.1007/s00220-012-1566-0. Google Scholar

[30]

A. N. W. 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

[31]

H. He and Z. Yin, On a generalized Camassa-Holm equation with the flow generated by velocity and its gradient, Appl. Anal., 96 (2017), 679-701. doi: 10.1080/00036811.2016.1151498. Google Scholar

[32]

J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82. doi: 10.1016/j.jmaa.2004.11.038. Google Scholar

[33]

J. Li and Z. Yin, Well-posedness and global existence for a generalized Degasperis-Procesi equation, Nonlinear Anal. Real World Appl., 28 (2016), 72-90. doi: 10.1016/j.nonrwa.2015.09.003. Google Scholar

[34]

J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143. doi: 10.1016/j.jde.2016.08.031. Google Scholar

[35]

J. Li and Z. Yin, Well-posedness and analytic solutions of the two-component Euler-Poincaré system, Monatsh Math, (2016), 1-29. doi: 10.1007/s00605-016-0927-8. Google Scholar

[36]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820. doi: 10.1007/s00220-006-0082-5. Google Scholar

[37]

Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Procesi equation Int. Math. Res. Not. IMRN (2007), Art. ID rnm117, 22 pp. doi: 10.1093/imrn/rnm117. Google Scholar

[38]

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

[39]

W. Luo and Z. Yin, Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space, Nonlinear Anal., 122 (2015), 1-22. doi: 10.1016/j.na.2015.03.022. Google Scholar

[40]

T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107. doi: 10.3934/dcds.2014.34.3095. Google Scholar

[41]

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

[42]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. Google Scholar

[43]

X. Tu and Z. Yin, Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation in the critical Besov space, Nonlinear Anal., 128 (2015), 1-19. doi: 10.1016/j.na.2015.07.017. Google Scholar

[44]

V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals, 20 (2004), 1059-1073. doi: 10.1016/j.chaos.2003.09.043. Google Scholar

[45]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation J. Phys. A 44 (2011), 055202, 17pp. doi: 10.1088/1751-8113/44/5/055202. Google Scholar

[46]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707-727. Google Scholar

[47]

X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137. doi: 10.1080/00036811.2011.649735. Google Scholar

[48]

Z. Xin and P. Zhang, On the weak solutions 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

[49]

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

[50]

W. YanY. Li and Y. Zhang, The Cauchy problem for the Novikov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1157-1169. doi: 10.1007/s00030-012-0202-1. Google Scholar

[51]

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

[52]

Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139. doi: 10.1016/S0022-247X(03)00250-6. Google Scholar

[53]

Z. Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209. doi: 10.1512/iumj.2004.53.2479. Google Scholar

[54]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. doi: 10.1016/j.jfa.2003.07.010. Google Scholar

show all references

References:
[1]

H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations Springer-Verlag Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7. Google Scholar

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. 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., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[4]

A. Boutet de MonvelA. KostenkoD. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588. doi: 10.1137/090748500. Google Scholar

[5]

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

[6]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91. doi: 10.1016/j.jfa.2005.07.008. Google Scholar

[7]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701. Google Scholar

[8]

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

[9]

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

[10]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033. Google Scholar

[11]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup 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

[12]

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

[13]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7. Google Scholar

[14]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2001), 559-568. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[15]

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

[16]

A. ConstantinR. I. 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

[17]

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

[18]

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

[19]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. Google Scholar

[20]

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

[21]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. Google Scholar

[22]

A. DegasperisD. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. Google Scholar

[23]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (Rome, 1998), World Sci. Publ., River Edge, NJ, 133 (1999), 23-37. Google Scholar

[24]

H. R. DullinG. A. Gottwald and D. 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

[25]

J. EscherY. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022. Google Scholar

[26]

J. EscherY. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117. doi: 10.1512/iumj.2007.56.3040. Google Scholar

[27]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[28]

G. Gui and Y. Liu, On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math., 69 (2011), 445-464. doi: 10.1090/S0033-569X-2011-01216-5. Google Scholar

[29]

G. GuiY. LiuP. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759. doi: 10.1007/s00220-012-1566-0. Google Scholar

[30]

A. N. W. 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

[31]

H. He and Z. Yin, On a generalized Camassa-Holm equation with the flow generated by velocity and its gradient, Appl. Anal., 96 (2017), 679-701. doi: 10.1080/00036811.2016.1151498. Google Scholar

[32]

J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82. doi: 10.1016/j.jmaa.2004.11.038. Google Scholar

[33]

J. Li and Z. Yin, Well-posedness and global existence for a generalized Degasperis-Procesi equation, Nonlinear Anal. Real World Appl., 28 (2016), 72-90. doi: 10.1016/j.nonrwa.2015.09.003. Google Scholar

[34]

J. Li and Z. Yin, Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143. doi: 10.1016/j.jde.2016.08.031. Google Scholar

[35]

J. Li and Z. Yin, Well-posedness and analytic solutions of the two-component Euler-Poincaré system, Monatsh Math, (2016), 1-29. doi: 10.1007/s00605-016-0927-8. Google Scholar

[36]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820. doi: 10.1007/s00220-006-0082-5. Google Scholar

[37]

Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Procesi equation Int. Math. Res. Not. IMRN (2007), Art. ID rnm117, 22 pp. doi: 10.1093/imrn/rnm117. Google Scholar

[38]

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

[39]

W. Luo and Z. Yin, Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space, Nonlinear Anal., 122 (2015), 1-22. doi: 10.1016/j.na.2015.03.022. Google Scholar

[40]

T. Lyons, Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107. doi: 10.3934/dcds.2014.34.3095. Google Scholar

[41]

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

[42]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. Google Scholar

[43]

X. Tu and Z. Yin, Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation in the critical Besov space, Nonlinear Anal., 128 (2015), 1-19. doi: 10.1016/j.na.2015.07.017. Google Scholar

[44]

V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals, 20 (2004), 1059-1073. doi: 10.1016/j.chaos.2003.09.043. Google Scholar

[45]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation J. Phys. A 44 (2011), 055202, 17pp. doi: 10.1088/1751-8113/44/5/055202. Google Scholar

[46]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707-727. Google Scholar

[47]

X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137. doi: 10.1080/00036811.2011.649735. Google Scholar

[48]

Z. Xin and P. Zhang, On the weak solutions 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

[49]

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

[50]

W. YanY. Li and Y. Zhang, The Cauchy problem for the Novikov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1157-1169. doi: 10.1007/s00030-012-0202-1. Google Scholar

[51]

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

[52]

Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139. doi: 10.1016/S0022-247X(03)00250-6. Google Scholar

[53]

Z. Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209. doi: 10.1512/iumj.2004.53.2479. Google Scholar

[54]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. doi: 10.1016/j.jfa.2003.07.010. Google Scholar

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