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October  2019, 39(10): 6023-6037. doi: 10.3934/dcds.2019263

Local-in-space blow-up criteria for two-component nonlinear dispersive wave system

1. 

Department of Mathematics, Istanbul Technical University, Turkey

2. 

Department of Mathematics, Gebze Technical University, Turkey

3. 

Department of Aeronautical Engineering, Istanbul Technical University, Turkey

* Corresponding author: Emil Novruzov

Received  January 2019 Revised  May 2019 Published  July 2019

We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial profiles. Presented result extends and specifies the earlier blow-up criteria for such type systems.

Citation: Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263
References:
[1]

H. Aratyn, J. F. Gomes and A. H. Zimerman, On a negative flow of the AKNS hierarchy and its relation to a two-Component Camassa-Holm equation, Symmetry, Integrability and Geom. Methods Appl., 2 (2006), Paper 070, 12 pp. doi: 10.3842/SIGMA.2006.070. Google Scholar

[2]

R. BealsD. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg de Vries hierarchy, Adv. Math., 140 (1998), 190-206. doi: 10.1006/aima.1998.1768. Google Scholar

[3]

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

[4]

L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 256 (2014), 3981–3998. doi: 10.1016/j.jde.2014.03.008. Google Scholar

[5]

L. Brandolese and M. F. Cortez, On permanent and breaking waves in hyperelastic rods and rings, Journal of Functional Analysis, 266 (2014), 6954-6987. doi: 10.1016/j.jfa.2014.02.039. Google Scholar

[6]

L. Brandolese, Local-in-space criteria for blowup in shallow water and dispersive rod equations,, Comm.Math.Phys., 330 (2014), 401-414. doi: 10.1007/s00220-014-1958-4. Google Scholar

[7]

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

[8]

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

[9]

R. Camassa and 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

[10]

R. CamassaD. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. Google Scholar

[11]

R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa–Holm system, Int. Math. Res. Not., 6 (2011), 1381–1416. doi: 10.1093/imrn/rnq118. Google Scholar

[12]

R. M. Chen, Y. Liu and Z. Qiao, Stability of solitary waves and global existence of a generalized two-component Camassa–Holm system, Comm. Partial Differ. Equ., 36 (2011), 2162–2188. doi: 10.1080/03605302.2011.556695. Google Scholar

[13]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4pp. doi: 10.1063/1.1845603. Google Scholar

[23]

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

[24]

H.-H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser A. Math. Phys. Eng. Sci., 456 (2000), 331–363. doi: 10.1098/rspa.2000.0520. Google Scholar

[25]

J. EscherO. Lechtenfeld and Z. Y. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513. doi: 10.3934/dcds.2007.19.493. Google Scholar

[26]

J. EscherD. HenryB. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271. doi: 10.1007/s10231-014-0461-z. Google Scholar

[27]

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

[28]

C. X. Guan and Z. Y. Yin, Global weak solutions for a two-component Camassa–Holm shallow water systems, J. Funct. Anal., 260 (2011), 1132–1154. doi: 10.1016/j.jfa.2010.11.015. Google Scholar

[29]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2. Google Scholar

[30]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa–Holm system, J. Funct. Anal., 258 (2011), 4251-4278. doi: 10.1016/j.jfa.2010.02.008. Google Scholar

[31]

D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math.Phys., 12 (2005), 342–347. doi: 10.2991/jnmp.2005.12.3.3. Google Scholar

[32]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, DiscreteContin.Dyn.Syst.Ser.B, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597. Google Scholar

[33]

D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565–1573. doi: 10.1016/j.na.2008.02.104. Google Scholar

[34]

A. Himonas, G, Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511–522. doi: 10.1007/s00220-006-0172-4. Google Scholar

[35]

A. N. W. Hone, V. Novikov and J. P. Wang, Two-component generalizations of the Camassa–Holm equation equation, Nonlinearity, 30 (2017), 622–658. doi: 10.1088/1361-6544/aa5490. Google Scholar

[36]

R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389–396. doi: 10.1016/j.wavemoti.2009.06.012. Google Scholar

[37]

Z. Jiang, L. Ni and Y. Zhou, Wave breaking of the Camassa-Holm equation, J. Nonlinear Sci., 22 (2012), 235–245. doi: 10.1007/s00332-011-9115-0. Google Scholar

[38]

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

[39]

T. Kato, Quasi-linear equation of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations", Lecture Notes in Math., 448 (1975), 25-70. Google Scholar

[40]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. Google Scholar

[41]

Y. Liu and P. Zhang, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not., 2010 (2010), 1981-2021. doi: 10.1093/imrn/rnp211. Google Scholar

[42]

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

[43]

T. Lyons, The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209-218. doi: 10.1007/s00021-016-0249-6. Google Scholar

[44]

T. Lyons, The pressure distribution in extreme Stokes waves, Nonlinear Anal. Real World Appl., 31 (2016), 77-87. doi: 10.1016/j.nonrwa.2016.01.008. Google Scholar

[45]

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

[46]

E.Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation, J. Math. Phys., 54 (2013), 092703, 8pp. doi: 10.1063/1.4820786. Google Scholar

[47]

E. Novruzov and A. Hagverdiyev, On the behavior of the solution of the dissipative Camassa-Holm equation with the arbitrary dispersion coefficient, J. Differential Equations, 257 (2014), 4525-4541. doi: 10.1016/j.jde.2014.08.016. Google Scholar

[48]

E. Novruzov, Local-in-space blow-up criteria for a class of nonlinear dispersive wave equations, J. Differ. Equ., 263 (2017), 5773-5786. doi: 10.1016/j.jde.2017.06.031. Google Scholar

[49]

P. Olver and P. Rosenau, Tri-hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. Google Scholar

[50]

C. TianW. Yan and H. Zhang, The Cauchy problem for the generalized hyperelastic rod equation, Math. Nachr., 287 (2014), 2116-2137. doi: 10.1002/mana.201200243. Google Scholar

[51]

S. Yang and T. Xu, Local-in-space blow-up and symmetric waves for a generalized two-component Camassa–Holm system, Applied Mathematics and Computation, 347 (2019), 514-521. doi: 10.1016/j.amc.2018.10.032. Google Scholar

[52]

Z. Yin, On the Blow-Up Scenario for the Generalized Camassa–HolmEquation, Communications in Partial Differential Equations, 29 (2004), 867-877. doi: 10.1081/PDE-120037334. Google Scholar

[53]

Z. Yin, Well-posedness, global solutions and blow-up phenomena for a nonlinearly dispersive wave equation, J. Evo. Eqns., 4 (2004), 391-419. doi: 10.1007/s00028-004-0166-7. Google Scholar

[54]

Y. Zhou and H. Chen, Wave breaking and propagation speed for the Camassa–Holm equation with $k\neq 0$, Nonlinear Anal.: Real World Appl., 12 (2011), 1875-1882. doi: 10.1016/j.nonrwa.2010.12.005. Google Scholar

[55]

Y. Zhou, Local well-posedness and blow-up criteria of solutions for a rod equation, Math. Nachr., 278 (2005), 1726-1739. doi: 10.1002/mana.200310337. Google Scholar

[56]

M. Zhu and J. Xu, On the wave-breaking phenomena for the periodic two-component Dullin–Gottwald–Holm system, J. Math. Anal. Appl., 391 (2012), 415-428. doi: 10.1016/j.jmaa.2012.02.058. Google Scholar

show all references

References:
[1]

H. Aratyn, J. F. Gomes and A. H. Zimerman, On a negative flow of the AKNS hierarchy and its relation to a two-Component Camassa-Holm equation, Symmetry, Integrability and Geom. Methods Appl., 2 (2006), Paper 070, 12 pp. doi: 10.3842/SIGMA.2006.070. Google Scholar

[2]

R. BealsD. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg de Vries hierarchy, Adv. Math., 140 (1998), 190-206. doi: 10.1006/aima.1998.1768. Google Scholar

[3]

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

[4]

L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 256 (2014), 3981–3998. doi: 10.1016/j.jde.2014.03.008. Google Scholar

[5]

L. Brandolese and M. F. Cortez, On permanent and breaking waves in hyperelastic rods and rings, Journal of Functional Analysis, 266 (2014), 6954-6987. doi: 10.1016/j.jfa.2014.02.039. Google Scholar

[6]

L. Brandolese, Local-in-space criteria for blowup in shallow water and dispersive rod equations,, Comm.Math.Phys., 330 (2014), 401-414. doi: 10.1007/s00220-014-1958-4. Google Scholar

[7]

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

[8]

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

[9]

R. Camassa and 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

[10]

R. CamassaD. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. Google Scholar

[11]

R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa–Holm system, Int. Math. Res. Not., 6 (2011), 1381–1416. doi: 10.1093/imrn/rnq118. Google Scholar

[12]

R. M. Chen, Y. Liu and Z. Qiao, Stability of solitary waves and global existence of a generalized two-component Camassa–Holm system, Comm. Partial Differ. Equ., 36 (2011), 2162–2188. doi: 10.1080/03605302.2011.556695. Google Scholar

[13]

A. Constantin and R. I. Ivanov, On an integrable two-component Camassa–Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4pp. doi: 10.1063/1.1845603. Google Scholar

[23]

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

[24]

H.-H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. Lond. Proc. Ser A. Math. Phys. Eng. Sci., 456 (2000), 331–363. doi: 10.1098/rspa.2000.0520. Google Scholar

[25]

J. EscherO. Lechtenfeld and Z. Y. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513. doi: 10.3934/dcds.2007.19.493. Google Scholar

[26]

J. EscherD. HenryB. Kolev and T. Lyons, Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271. doi: 10.1007/s10231-014-0461-z. Google Scholar

[27]

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

[28]

C. X. Guan and Z. Y. Yin, Global weak solutions for a two-component Camassa–Holm shallow water systems, J. Funct. Anal., 260 (2011), 1132–1154. doi: 10.1016/j.jfa.2010.11.015. Google Scholar

[29]

G. Gui and Y. Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z., 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2. Google Scholar

[30]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa–Holm system, J. Funct. Anal., 258 (2011), 4251-4278. doi: 10.1016/j.jfa.2010.02.008. Google Scholar

[31]

D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math.Phys., 12 (2005), 342–347. doi: 10.2991/jnmp.2005.12.3.3. Google Scholar

[32]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, DiscreteContin.Dyn.Syst.Ser.B, 12 (2009), 597-606. doi: 10.3934/dcdsb.2009.12.597. Google Scholar

[33]

D. Henry, Persistence properties for a family of nonlinear partial differential equations, Nonlinear Anal., 70 (2009), 1565–1573. doi: 10.1016/j.na.2008.02.104. Google Scholar

[34]

A. Himonas, G, Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511–522. doi: 10.1007/s00220-006-0172-4. Google Scholar

[35]

A. N. W. Hone, V. Novikov and J. P. Wang, Two-component generalizations of the Camassa–Holm equation equation, Nonlinearity, 30 (2017), 622–658. doi: 10.1088/1361-6544/aa5490. Google Scholar

[36]

R. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389–396. doi: 10.1016/j.wavemoti.2009.06.012. Google Scholar

[37]

Z. Jiang, L. Ni and Y. Zhou, Wave breaking of the Camassa-Holm equation, J. Nonlinear Sci., 22 (2012), 235–245. doi: 10.1007/s00332-011-9115-0. Google Scholar

[38]

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

[39]

T. Kato, Quasi-linear equation of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations", Lecture Notes in Math., 448 (1975), 25-70. Google Scholar

[40]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. Google Scholar

[41]

Y. Liu and P. Zhang, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not., 2010 (2010), 1981-2021. doi: 10.1093/imrn/rnp211. Google Scholar

[42]

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

[43]

T. Lyons, The pressure in a deep-water Stokes wave of greatest height, J. Math. Fluid Mech., 18 (2016), 209-218. doi: 10.1007/s00021-016-0249-6. Google Scholar

[44]

T. Lyons, The pressure distribution in extreme Stokes waves, Nonlinear Anal. Real World Appl., 31 (2016), 77-87. doi: 10.1016/j.nonrwa.2016.01.008. Google Scholar

[45]

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

[46]

E.Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation, J. Math. Phys., 54 (2013), 092703, 8pp. doi: 10.1063/1.4820786. Google Scholar

[47]

E. Novruzov and A. Hagverdiyev, On the behavior of the solution of the dissipative Camassa-Holm equation with the arbitrary dispersion coefficient, J. Differential Equations, 257 (2014), 4525-4541. doi: 10.1016/j.jde.2014.08.016. Google Scholar

[48]

E. Novruzov, Local-in-space blow-up criteria for a class of nonlinear dispersive wave equations, J. Differ. Equ., 263 (2017), 5773-5786. doi: 10.1016/j.jde.2017.06.031. Google Scholar

[49]

P. Olver and P. Rosenau, Tri-hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. Google Scholar

[50]

C. TianW. Yan and H. Zhang, The Cauchy problem for the generalized hyperelastic rod equation, Math. Nachr., 287 (2014), 2116-2137. doi: 10.1002/mana.201200243. Google Scholar

[51]

S. Yang and T. Xu, Local-in-space blow-up and symmetric waves for a generalized two-component Camassa–Holm system, Applied Mathematics and Computation, 347 (2019), 514-521. doi: 10.1016/j.amc.2018.10.032. Google Scholar

[52]

Z. Yin, On the Blow-Up Scenario for the Generalized Camassa–HolmEquation, Communications in Partial Differential Equations, 29 (2004), 867-877. doi: 10.1081/PDE-120037334. Google Scholar

[53]

Z. Yin, Well-posedness, global solutions and blow-up phenomena for a nonlinearly dispersive wave equation, J. Evo. Eqns., 4 (2004), 391-419. doi: 10.1007/s00028-004-0166-7. Google Scholar

[54]

Y. Zhou and H. Chen, Wave breaking and propagation speed for the Camassa–Holm equation with $k\neq 0$, Nonlinear Anal.: Real World Appl., 12 (2011), 1875-1882. doi: 10.1016/j.nonrwa.2010.12.005. Google Scholar

[55]

Y. Zhou, Local well-posedness and blow-up criteria of solutions for a rod equation, Math. Nachr., 278 (2005), 1726-1739. doi: 10.1002/mana.200310337. Google Scholar

[56]

M. Zhu and J. Xu, On the wave-breaking phenomena for the periodic two-component Dullin–Gottwald–Holm system, J. Math. Anal. Appl., 391 (2012), 415-428. doi: 10.1016/j.jmaa.2012.02.058. Google Scholar

[1]

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