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July  2020, 40(7): 4565-4576. doi: 10.3934/dcds.2020191

On the blow up solutions to a two-component cubic Camassa-Holm system with peakons

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

2. 

Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

Received  December 2019 Revised  January 2020 Published  April 2020

This paper is concerned with the Cauchy problem for a two- component cubic Camassa-Holm system with peakons, which is a natural multi-component extension of the single Fokas-Olver-Rosenau-Qiao equation. By sufficiently exploiting the fine structure of the system, we derive two useful conservation laws which turns out an exponential increase estimate for the $ L^\infty $-norm of the strong solution within its lifespan. As a result, two new blow-up solutions with certain initial profiles are established.

Citation: Kai Yan. On the blow up solutions to a two-component cubic Camassa-Holm system with peakons. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4565-4576. doi: 10.3934/dcds.2020191
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. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[3]

R. M. ChenF. GaoY. Liu and C. Qu, Analysis on the blow-up of solutions to a class of integrable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374.  doi: 10.1016/j.jfa.2016.01.017.

[4]

A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.

[5]

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

[6]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328. 

[7]

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. 

[8]

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

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.

[10]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. 

[11]

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.

[12]

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

[13]

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), 194501, 4501–4504. doi: 10.1103/PhysRevLett.87.194501.

[14]

A. Fokas, On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150.  doi: 10.1016/0167-2789(95)00133-O.

[15]

G. GuiY. LiuP. 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.

[16]

A. A. Himonas and D. Mantzavinos, Hölder continuity for the Fokas-Olver-Rosenau-Qiao equation, J. Nonlinear Sci., 24 (2014), 1105-1124.  doi: 10.1007/s00332-014-9212-y.

[17]

X. LiuY. Liu and C. Qu, Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation, Adv. Math., 255 (2014), 1-37.  doi: 10.1016/j.aim.2013.12.032.

[18]

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.

[19]

P. J. 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.

[20]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9 pp. doi: 10.1063/1.2365758.

[21]

C. QuX. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity, Comm. Math. Phys., 322 (2013), 967-997.  doi: 10.1007/s00220-013-1749-3.

[22]

J. Song, C. Qu and Z. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 013503, 9 pp. doi: 10.1063/1.3530865.

[23]

G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[24]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. 

[25]

K. YanZ. Qiao and Y. Zhang, Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions, J. Differential Equations, 259 (2015), 6644-6671.  doi: 10.1016/j.jde.2015.08.004.

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. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[3]

R. M. ChenF. GaoY. Liu and C. Qu, Analysis on the blow-up of solutions to a class of integrable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374.  doi: 10.1016/j.jfa.2016.01.017.

[4]

A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  doi: 10.5802/aif.1757.

[5]

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

[6]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328. 

[7]

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. 

[8]

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

[9]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568.  doi: 10.4007/annals.2011.173.1.12.

[10]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. 

[11]

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.

[12]

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

[13]

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), 194501, 4501–4504. doi: 10.1103/PhysRevLett.87.194501.

[14]

A. Fokas, On a class of physically important integrable equations, Phys. D, 87 (1995), 145-150.  doi: 10.1016/0167-2789(95)00133-O.

[15]

G. GuiY. LiuP. 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.

[16]

A. A. Himonas and D. Mantzavinos, Hölder continuity for the Fokas-Olver-Rosenau-Qiao equation, J. Nonlinear Sci., 24 (2014), 1105-1124.  doi: 10.1007/s00332-014-9212-y.

[17]

X. LiuY. Liu and C. Qu, Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation, Adv. Math., 255 (2014), 1-37.  doi: 10.1016/j.aim.2013.12.032.

[18]

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.

[19]

P. J. 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.

[20]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701, 9 pp. doi: 10.1063/1.2365758.

[21]

C. QuX. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity, Comm. Math. Phys., 322 (2013), 967-997.  doi: 10.1007/s00220-013-1749-3.

[22]

J. Song, C. Qu and Z. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 013503, 9 pp. doi: 10.1063/1.3530865.

[23]

G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.

[24]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. 

[25]

K. YanZ. Qiao and Y. Zhang, Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions, J. Differential Equations, 259 (2015), 6644-6671.  doi: 10.1016/j.jde.2015.08.004.

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