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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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