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July  2020, 19(7): 3769-3784. doi: 10.3934/cpaa.2020166

Blow-up for two-component Camassa-Holm equation with generalized weak dissipation

1. 

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013, China

2. 

School of Mathematical Science, Nanjing Normal University, Nanjing, Jiangsu, 210023, China

*Corresponding author

Received  October 2019 Revised  January 2020 Published  April 2020

Fund Project: The first author is supported by by the National Natural Science Foundation of China(No. 11731014), and the Nature Science Foundation of Jiangsu Province(No. BK20171294)

This paper is concerned with blow-up solution for the Cauchy problem of two-component Camassa-Holm equation with generalized weak dissipation. By Kato's theorem and monotonicity, we investigate the local well-posedness of Cauchy problem and establish the blow-up criteria and the blow-up rate. Moreover, the property of blow-up points set is characterized.

Citation: Wenxia Chen, Jingyi Liu, Danping Ding, Lixin Tian. Blow-up for two-component Camassa-Holm equation with generalized weak dissipation. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3769-3784. doi: 10.3934/cpaa.2020166
References:
[1]

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.

[2]

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.

[3]

M. ChenS. Liu and Y. Zhang, A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.  doi: 10.1007/s11005-005-0041-7.

[4]

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

[5]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation, Ann. Inst. Fourier, 50 (2000), 321-362. 

[6]

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

[7]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Commun. Pure Appl. Math., 51 (1998), 475-504.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[8]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91.  doi: 10.1007/PL00004793.

[9]

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

[10]

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.

[11]

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

[12]

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

[13]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278.  doi: 10.1016/j.jfa.2010.02.008.

[14]

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.

[15]

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

[16]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Vol. 448, (1975), 25–70.

[17]

T. Kato, On the Korteweg-de Vries equation, Manuscr. Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.

[18]

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

[19]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer Science Business Media, 2012. doi: 10.1007/978-1-4612-5561-1.

[20]

G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1980.

[21]

S. Wu and Z. Yin, Blow-up, blow-up rate and decay of the weakly dissipative Camassa-Holm equation, J. Math. Phys., 47 (2006), Art. 013504. doi: 10.1063/1.2158437.

[22]

Z. Yin, Well-posedness, blowup and global existence for an integrable shallow water equation, Discrete Contin. Dyn. Syst., 11 (2004), 393-411.  doi: 10.3934/dcds.2004.11.393.

[23]

Z. Yin, Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation, J. Evol. Equ., 4 (2004), 391-419.  doi: 10.1007/s00028-004-0166-7.

[24]

J. YinL. Tian and X. Fan, Orbital stability of floating periodic peakons for the Camassa-Holm equation, Nonlinear Anal. Real World Appl., 11 (2010), 4021-4026.  doi: 10.1016/j.nonrwa.2010.03.008.

show all references

References:
[1]

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.

[2]

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.

[3]

M. ChenS. Liu and Y. Zhang, A 2-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15.  doi: 10.1007/s11005-005-0041-7.

[4]

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

[5]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation, Ann. Inst. Fourier, 50 (2000), 321-362. 

[6]

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

[7]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Commun. Pure Appl. Math., 51 (1998), 475-504.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[8]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91.  doi: 10.1007/PL00004793.

[9]

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

[10]

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.

[11]

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

[12]

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

[13]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278.  doi: 10.1016/j.jfa.2010.02.008.

[14]

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.

[15]

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

[16]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Vol. 448, (1975), 25–70.

[17]

T. Kato, On the Korteweg-de Vries equation, Manuscr. Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.

[18]

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

[19]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer Science Business Media, 2012. doi: 10.1007/978-1-4612-5561-1.

[20]

G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1980.

[21]

S. Wu and Z. Yin, Blow-up, blow-up rate and decay of the weakly dissipative Camassa-Holm equation, J. Math. Phys., 47 (2006), Art. 013504. doi: 10.1063/1.2158437.

[22]

Z. Yin, Well-posedness, blowup and global existence for an integrable shallow water equation, Discrete Contin. Dyn. Syst., 11 (2004), 393-411.  doi: 10.3934/dcds.2004.11.393.

[23]

Z. Yin, Well-posedness, global solutions and blowup phenomena for a nonlinearly dispersive wave equation, J. Evol. Equ., 4 (2004), 391-419.  doi: 10.1007/s00028-004-0166-7.

[24]

J. YinL. Tian and X. Fan, Orbital stability of floating periodic peakons for the Camassa-Holm equation, Nonlinear Anal. Real World Appl., 11 (2010), 4021-4026.  doi: 10.1016/j.nonrwa.2010.03.008.

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