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**The local and global existence of solutions for a generalized Camassa-Holm equation.**
*(English)*
Zbl 1237.35139

Summary: A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of the pseudoparabolic regularization technique, its local well posedness in Sobolev space \(H^s(\mathbb{R})\) with \(s > 3/2\) is established via a limiting procedure. Provided that the initial value \(u_0\) satisfies the sign condition and \(u_0 \in H^s (\mathbb{R})\) (\(s > 3/2\)), it is shown that there exists a unique global solution for the equation in space \(C([0, \infty); H^s(\mathbb{R})) \cap C^1([0, \infty); H^{s-1}(\mathbb{R}))\).

### MSC:

35Q53 | KdV equations (Korteweg-de Vries equations) |

35Q51 | Soliton equations |

35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |

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\textit{N. Li} et al., Abstr. Appl. Anal. 2012, Article ID 532369, 26 p. (2012; Zbl 1237.35139)

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### References:

[1] | R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661-1664, 1993. · Zbl 0972.35521 |

[2] | A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 165-186, 2009. · Zbl 1169.76010 |

[3] | R. S. Johnson, “Camassa-Holm, Korteweg-de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, no. 1, pp. 63-82, 2002. · Zbl 1037.76006 |

[4] | D. Ionescu-Kruse, “Variational derivation of the Camassa-Holm shallow water equation,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 3, pp. 303-312, 2007. · Zbl 1157.76005 |

[5] | R. S. Johnson, “The Camassa-Holm equation for water waves moving over a shear flow,” Japan Society of Fluid Mechanics. Fluid Dynamics Research. An International Journal, vol. 33, no. 1-2, pp. 97-111, 2003. · Zbl 1032.76519 |

[6] | H.-H. Dai, “Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,” Acta Mechanica, vol. 127, no. 1-4, pp. 193-207, 1998. · Zbl 0910.73036 |

[7] | A. Constantin and W. A. Strauss, “Stability of a class of solitary waves in compressible elastic rods,” Physics Letters. A, vol. 270, no. 3-4, pp. 140-148, 2000. · Zbl 1115.74339 |

[8] | M. Lakshmanan, “Integrable nonlinear wave equations and possible connections to tsunami dynamics,” in Tsunami and Nonlinear Waves, A. Kundu, Ed., pp. 31-49, Springer, Berlin, Germany, 2007. · Zbl 1310.76044 |

[9] | A. Constantin and R. S. Johnson, “Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,” Fluid Dynamics Research, vol. 40, no. 3, pp. 175-211, 2008. · Zbl 1135.76007 |

[10] | A. Constantin, “The trajectories of particles in Stokes waves,” Inventiones Mathematicae, vol. 166, no. 3, pp. 523-535, 2006. · Zbl 1108.76013 |

[11] | A. Constantin and J. Escher, “Particle trajectories in solitary water waves,” Bulletin of American Mathematical Society, vol. 44, no. 3, pp. 423-431, 2007. · Zbl 1126.76012 |

[12] | A. Constantin and W. A. Strauss, “Stability of peakons,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp. 603-610, 2000. · Zbl 1049.35149 |

[13] | A. Constantin and J. Escher, “Analyticity of periodic traveling free surface water waves with vorticity,” Annals of Mathematics, vol. 173, no. 1, pp. 559-568, 2011. · Zbl 1228.35076 |

[14] | K. El Dika and L. Molinet, “Stability of multipeakons,” Annales de l’Institut Henri Poincare. Analyse Non Lineaire, vol. 26, no. 4, pp. 1517-1532, 2009. · Zbl 1171.35459 |

[15] | A. Constantin and W. A. Strauss, “Stability of the Camassa-Holm solitons,” Journal of Nonlinear Science, vol. 12, no. 4, pp. 415-422, 2002. · Zbl 1022.35053 |

[16] | A. Constantin and H. P. McKean, “A shallow water equation on the circle,” Communications on Pure and Applied Mathematics, vol. 52, no. 8, pp. 949-982, 1999. · Zbl 0940.35177 |

[17] | A. Constantin, “On the inverse spectral problem for the Camassa-Holm equation,” Journal of Functional Analysis, vol. 155, no. 2, pp. 352-363, 1998. · Zbl 0907.35009 |

[18] | A. Constantin, V. S. Gerdjikov, and R. I. Ivanov, “Inverse scattering transform for the Camassa-Holm equation,” Inverse Problems, vol. 22, no. 6, pp. 2197-2207, 2006. · Zbl 1105.37044 |

[19] | H. P. McKean, “Integrable systems and algebraic curves,” in Global Analysis, vol. 755 of Lecture Notes in Mathematics, pp. 83-200, Springer, Berlin, Germany, 1979. · Zbl 0449.35080 |

[20] | A. Constantin, T. Kappeler, B. Kolev, and P. Topalov, “On geodesic exponential maps of the Virasoro group,” Annals of Global Analysis and Geometry, vol. 31, no. 2, pp. 155-180, 2007. · Zbl 1121.35111 |

[21] | G. Misiolek, “A shallow water equation as a geodesic flow on the Bott-Virasoro group,” Journal of Geometry and Physics, vol. 24, no. 3, pp. 203-208, 1998. · Zbl 0901.58022 |

[22] | Z. Xin and P. Zhang, “On the weak solutions to a shallow water equation,” Communications on Pure and Applied Mathematics, vol. 53, no. 11, pp. 1411-1433, 2000. · Zbl 1048.35092 |

[23] | Z. Xin and P. Zhang, “On the uniqueness and large time behavior of the weak solutions to a shallow water equation,” Communications in Partial Differential Equations, vol. 27, no. 9-10, pp. 1815-1844, 2002. · Zbl 1034.35115 |

[24] | G. M. Coclite, H. Holden, and K. H. Karlsen, “Global weak solutions to a generalized hyperelastic-rod wave equation,” SIAM Journal on Mathematical Analysis, vol. 37, no. 4, pp. 1044-1069, 2005. · Zbl 1100.35106 |

[25] | A. Bressan and A. Constantin, “Global conservative solutions of the Camassa-Holm equation,” Archive for Rational Mechanics and Analysis, vol. 183, no. 2, pp. 215-239, 2007. · Zbl 1105.76013 |

[26] | A. Bressan and A. Constantin, “Global dissipative solutions of the Camassa-Holm equation,” Analysis and Applications, vol. 5, no. 1, pp. 1-27, 2007. · Zbl 1139.35378 |

[27] | H. Holden and X. Raynaud, “Dissipative solutions for the Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 24, no. 4, pp. 1047-1112, 2009. · Zbl 1178.65099 |

[28] | H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation-a Lagrangian point of view,” Communications in Partial Differential Equations, vol. 32, no. 10-12, pp. 1511-1549, 2007. · Zbl 1136.35080 |

[29] | Y. A. Li and P. J. Olver, “Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,” Journal of Differential Equations, vol. 162, no. 1, pp. 27-63, 2000. · Zbl 0958.35119 |

[30] | A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229-243, 1998. · Zbl 0923.76025 |

[31] | R. Beals, D. H. Sattinger, and J. Szmigielski, “Multi-peakons and a theorem of Stieltjes,” Inverse Problems, vol. 15, no. 1, pp. L1-L4, 1999. · Zbl 0923.35154 |

[32] | D. Henry, “Persistence properties for a family of nonlinear partial differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 4, pp. 1565-1573, 2009. · Zbl 1170.35509 |

[33] | S. Hakkaev and K. Kirchev, “Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 30, no. 4-6, pp. 761-781, 2005. · Zbl 1076.35098 |

[34] | G. Rodríguez-Blanco, “On the Cauchy problem for the Camassa-Holm equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 46, no. 3, pp. 309-327, 2001. · Zbl 0980.35150 |

[35] | Z. Yin, “On the blow-up scenario for the generalized Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 867-877, 2004. · Zbl 1068.35030 |

[36] | Z. Yin, “On the Cauchy problem for an integrable equation with peakon solutions,” Illinois Journal of Mathematics, vol. 47, no. 3, pp. 649-666, 2003. · Zbl 1061.35142 |

[37] | Y. Zhou, “Wave breaking for a periodic shallow water equation,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 591-604, 2004. · Zbl 1042.35060 |

[38] | S. Lai and Y. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations, vol. 248, no. 8, pp. 2038-2063, 2010. · Zbl 1187.35179 |

[39] | S. Y. Wu and Z. Yin, “Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4309-4321, 2009. · Zbl 1195.35072 |

[40] | T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, vol. 448 of Lecture notes in Mathematics, pp. 25-70, Springer, Berlin, 1975. · Zbl 0315.35077 |

[41] | T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891-907, 1988. · Zbl 0671.35066 |

[42] | W. Walter, Differential and Integral Inequalities, Springer-Verlag, New York, NY, USA, 1970. · Zbl 0252.35005 |

[43] | B. Kolev, “Lie groups and mechanics: an introduction,” Journal of Nonlinear Mathematical Physics, vol. 11, no. 4, pp. 480-498, 2004. · Zbl 1069.35070 |

[44] | A. Constantin, “Existence of permanent and breaking waves for a shallow water equation: a geometric approach,” Annales de l’Institut Fourier, vol. 50, no. 2, pp. 321-362, 2000. · Zbl 0944.35062 |

[45] | B. Kolev, “Poisson brackets in hydrodynamics,” Discrete and Continuous Dynamical Systems, vol. 19, no. 3, pp. 555-574, 2007. · Zbl 1139.53040 |

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