Analytical Properties for the Fifth Order Camassa-Holm (FOCH) Model

This paper devotes to present analysis work on the fifth order Camassa-Holm (FOCH) model which recently proposed by Liu and Qiao. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, we study the property of the infinite propagation speed. Finally, we discuss the long time behavior of the support of momentum density with a compactly supported initial data.

[1]  Y.-S. Kwon,et al.  Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation , 2008, 0811.0549.

[2]  J. Escher,et al.  Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation , 1998 .

[3]  J. Escher,et al.  Well-posedness, blow-up phenomena, and global solutions for the b-equation , 2008 .

[4]  J. Lenells.  Conservation laws of the Camassa–Holm equation , 2005 .

[5]  Zheng-rong Liu,et al.  Well-posedness of the modified Camassa–Holm equation in Besov spaces , 2015 .

[6]  Peng Zhao,et al.  Algebro-geometric Solutions for the Degasperis-Procesi Hierarchy , 2012, SIAM J. Math. Anal..

[7]  R. McLachlan,et al.  Well-posedness of modified Camassa-Holm equations , 2009 .

[8]  H. P. MCKEANt.  BREAKDOWN OF A SHALLOW WATER EQUATION* , 2016 .

[9]  Yong Zhou,et al.  Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation , 2006 .

[10]  Wenjun Cui,et al.  Infinite propagation speed and asymptotic behavior for a generalized fifth-order Camassa–Holm equation , 2019 .

[11]  Z. Qiao,et al.  Fifth order Camassa–Holm model with pseudo-peakons and multi-peakons , 2018, International Journal of Non-Linear Mechanics.

[12]  Zhaoyang Yin,et al.  Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation , 2006 .

[13]  A. Bressan,et al.  GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION , 2007 .

[14]  J. Escher,et al.  Wave breaking for nonlinear nonlocal shallow water equations , 1998 .

[15]  I. Sibgatullin.  well-posedness , 2020 .

[16]  Shengtai Li,et al.  A New Integrable Hierarchy, Parametric Solutions and Traveling Wave Solutions , 2002 .

[17]  L. Tian,et al.  Global existence for the higher-order CamassaHolm shallow water equation , 2011 .

[18]  W. Strauss,et al.  Stability of peakons , 2000 .

[19]  Z. Qiao,et al.  On the Cauchy problem for a higher-order $\mu$-Camassa-Holm equation , 2017, 1712.09583.

[20]  Darryl D. Holm,et al.  A New Integrable Equation with Peakon Solutions , 2002, nlin/0205023.

[21]  Darryl D. Holm,et al.  Nonintegrability of a Fifth-Order Equation with Integrable Two-Body Dynamics , 2003 .

[22]  Enrique G. Reyes,et al.  Geometric Integrability of the Camassa–Holm Equation , 2002 .

[23]  A. Constantin.  Existence of permanent and breaking waves for a shallow water equation: a geometric approach , 2000 .

[24]  Danping Ding.  Traveling solutions and evolution properties of the higher order Camassa–Holm equation , 2017 .

[25]  Yong Zhou,et al.  Wave Breaking of the Camassa–Holm Equation , 2012, J. Nonlinear Sci..

[26]  L. Tian,et al.  Global existence and blow-up phenomena for the peakon b-family of equations , 2008 .

[27]  Yong Zhou,et al.  Blow-up phenomenon for the integrable Degasperis¿Procesi equation , 2004 .

[28]  H. Holden,et al.  WELL-POSEDNESS OF HIGHER-ORDER CAMASSA-HOLM EQUATIONS , 2009 .

[29]  Z. Qiao,et al.  Well-posedness and peakons for a higher-orderμ-Camassa–Holm equation , 2017, Nonlinear Analysis.

[30]  Yong Zhou.  On solutions to the Holm–Staley b-family of equations , 2010 .

[31]  S. Hakkaev,et al.  On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis-Procesi equation , 2009 .

[32]  K. Karlsen,et al.  Periodic solutions of the Degasperis–Procesi equation: Well-posedness and asymptotics☆ , 2015 .

[33]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

[34]  Penghui Lv,et al.  Conservative solutions for higher-order Camassa–Holm equations , 2010 .

[35]  Z. Qiao.  Communications in Mathematical Physics The Camassa-Holm Hierarchy , N-Dimensional Integrable Systems , and Algebro-Geometric Solution on a Symplectic Submanifold , 2003 .

[36]  Jonatan Lenells,et al.  Traveling wave solutions of the Degasperis-Procesi equation , 2005 .

[37]  P. Olver,et al.  Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation , 2000 .

[38]  V. Gerdjikov,et al.  Inverse scattering transform for the Camassa–Holm equation , 2006, Inverse Problems.

[39]  Adrian Constantin,et al.  A shallow water equation on the circle , 1999 .

[40]  O. Mustafa.  A Note on the Degasperis-Procesi Equation , 2005 .

[41]  Z. Qiao.  Integrable Hierarchy, 3×3 Constrained Systems, and Parametric Solutions , 2004 .

[42]  Yong Zhou,et al.  Large time behavior for the support of momentum density of the Camassa-Holm equation , 2013 .

[43]  A. Bressan,et al.  Global Conservative Solutions of the Camassa–Holm Equation , 2007 .