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Blow-up, global existence and persistence properties for the coupled Camassa-Holm equations. (English) Zbl 1245.35115
Summary: In this paper, we consider the coupled Camassa-Holm equations. First, we present some new criteria on blow-up. Then global existence and blow-up rate of the solution are also established. Finally, we discuss persistence properties of this system.
35Q53KdV-like (Korteweg-de Vries) equations
35B44Blow-up (PDE)
35A01Existence problems for PDE: global existence, local existence, non-existence
[1]Constantin, A.: On the Cauchy problem for the periodic Camassa–Holm equation. J. Differ. Equ. 141, 218–235 (1997) · Zbl 0889.35022 · doi:10.1006/jdeq.1997.3333
[2]Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier 50, 321–362 (2000) · Zbl 0944.00010 · doi:10.5802/aif.1757
[3]Constantin, A.: Finite propagation speed for the CamassaHolm equation. J. Math. Phys. 46, 023506 (2005) · Zbl 1076.35109 · doi:10.1063/1.1845603
[4]Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math 166, 523–535 (2006) · Zbl 1108.76013 · doi:10.1007/s00222-006-0002-5
[5]Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equation. Acta Math. 181, 229–243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[6]Constantin, A., Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233, 75–91 (2000) · Zbl 0954.35136 · doi:10.1007/PL00004793
[7]Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007) · Zbl 1126.76012 · doi:10.1090/S0273-0979-07-01159-7
[8]Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Math. Ann. 173, 559–568 (2011) · Zbl 1228.35076 · doi:10.4007/annals.2011.173.1.12
[9]Fu, Y., Liu, Y., Qu, C.: Well-possdness and blow-up solution for a modified two-component Camassa–Holm system with peakons. Math. Ann. 348, 415–448 (2010) · Zbl 1207.35074 · doi:10.1007/s00208-010-0483-9
[10]Fu, Y., Qu, C.: Well-possdness and blow-up solution for a new coupled Camassa–Holm equations with peakons. J. Math. Phys. 50, 012906 (2009) · Zbl 1189.35273 · doi:10.1063/1.3064810
[11]Guo, Z.: Blow up and global solutions to a new integrable model with two components. J. Math. Anal. Appl. 372, 316–327 (2010) · Zbl 1205.35045 · doi:10.1016/j.jmaa.2010.06.046
[12]Guo, Z., Zhou, Y.: On solutions to a two-component generalized Camassa–Holm equation. Stud. Appl. Math. 124, 307–322 (2009) · Zbl 1189.35255 · doi:10.1111/j.1467-9590.2009.00472.x
[13]Henry, D.: Compactly supported solutions of the Camassa–Holm equation. J. Nonlin. Math. Phys. 12, 342–347 (2005) · Zbl 1086.35079 · doi:10.2991/jnmp.2005.12.3.3
[14]Henry, D.: Persistence properties for a family of nonlinear partial differential equations. Nonlinear Anal. 70, 1565–1573 (2009) · Zbl 1170.35509 · doi:10.1016/j.na.2008.02.104
[15]Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007) · Zbl 1142.35078 · doi:10.1007/s00220-006-0172-4
[16]Jin, L., Guo, Z.: On a two-component Degasperis-Procesi shallow water system. Nonlinear Anal. 11, 4164–4173 (2010) · Zbl 1203.35063 · doi:10.1016/j.nonrwa.2010.05.003
[17]Jin, L., Liu, Y., Zhou, Y.: Blow-up of solutions to a periodic nonlinear dispersive rod equation. Doc. Math. 15, 267–283 (2010)
[18]Mckean, H.P.: Breakdown of a shallow water equation. Asian J. Math. 2, 767–774 (1998)
[19]Ni, L.: The Cauchy problem for a two-component generalized θ-equations. Nonlinear Anal. 73, 1338–1349 (2010) · Zbl 1195.37052 · doi:10.1016/j.na.2010.04.064
[20]Ni, L., Zhou, Y.: A new asymptotic behavior of solutions to the Camassa–Holm equation. Proc. Am. Math. Soc. (2011). doi: 10.1007/s11040-011-9094-2
[21]Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)
[22]Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57, 137–152 (2004) · Zbl 1106.35070 · doi:10.1016/j.na.2004.02.004
[23]Zhou, Y.: Blow-up phenomenon for a periodic rod equation. Phys. Lett. A 353, 479–486 (2006) · Zbl 1181.35287 · doi:10.1016/j.physleta.2006.01.042
[24]Zhou, Y.: Blow-up of solutions to the DGH equation. J. Funct. Anal. 250(1), 227–248 (2007) · Zbl 1124.35079 · doi:10.1016/j.jfa.2007.04.019