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

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  • 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.

    Mathematics Subject Classification: 35B44, 35G25, 35Q35.


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  • [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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