Blow-up and global solutions to a new integrable model with two components

Abstract We will discuss a new integrable model which describes the motion of fluid. The present work is mainly concerned with global existence and blow-up phenomena which are largely due to the application of conservation laws for this integrable equations. Moreover, a new blow-up criterion for nonperiodic case is also established via the associated potential. Some interesting examples are also given to illustrate the application of our results. The precise blow-up rate is also investigated. Finally, we will emphasize the relations of classical Camassa–Holm equation and our model by analyzing the existence of global solutions.

[1]  H. McKean.  Breakdown of a shallow water equation , 1998 .

[2]  P. Olver,et al.  Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  Marcus Wunsch,et al.  The Generalized Hunter-Saxton System , 2010, SIAM J. Math. Anal..

[4]  G. Misiołek.  Classical solutions of the periodic Camassa—Holm equation , 2002 .

[5]  Giuseppe Maria Coclite,et al.  Global Weak Solutions for a Shallow Water Equation , 2008 .

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

[7]  Zhengguang Guo.  Some properties of solutions to the weakly dissipative Degasperis-Procesi equation , 2009 .

[8]  Rossen I. Ivanov,et al.  On an integrable two-component Camassa–Holm shallow water system , 2008, 0806.0868.

[9]  Zhengguang Guo.  Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa–Holm equation , 2008 .

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

[11]  Victor A. Galaktionov,et al.  The problem of blow-up in nonlinear parabolic equations , 2002 .

[12]  Zhengguang Guo,et al.  On a two-component Degasperis–Procesi shallow water system , 2010 .

[13]  The Cauchy problem for a two-component generalized θ-equations , 2010 .

[14]  Yong Zhou.  Local well‐posedness and blow‐up criteria of solutions for a rod equation , 2005 .

[15]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[16]  Yong Zhou,et al.  Blow-up of solutions to a nonlinear dispersive rod equation , 2006 .

[17]  J. Escher,et al.  Global existence and blow-up for a shallow water equation , 1998 .

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

[19]  Joachim Escher,et al.  Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation , 2007 .

[20]  Z. Popowicz.  A 2-component or N=2 supersymmetric Camassa–Holm equation , 2005, nlin/0509050.

[21]  A. Alexandrou Himonas,et al.  The Cauchy problem for an integrable shallow-water equation , 2001, Differential and Integral Equations.

[22]  L. Molinet.  On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey , 2004 .

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

[24]  Octavian G. Mustafa,et al.  On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system , 2009 .

[25]  Yong Zhou,et al.  On Solutions to a Two‐Component Generalized Camassa‐Holm Equation , 2010 .

[26]  Yong Zhou.  Wave breaking for a periodic shallow water equation , 2004 .

[27]  Ping Zhang,et al.  On the weak solutions to a shallow water equation , 2000 .

[28]  Yong Zhou,et al.  Blow-up of solutions to the DGH equation , 2007 .

[29]  Athanassios S. Fokas,et al.  Symplectic structures, their B?acklund transformation and hereditary symmetries , 1981 .

[30]  Joachim Escher,et al.  Global weak solutions and blow-up structure for the Degasperis–Procesi equation , 2006 .

[31]  R. Johnson,et al.  Camassa–Holm, Korteweg–de Vries and related models for water waves , 2002, Journal of Fluid Mechanics.

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

[33]  Youjin Zhang,et al.  A Two-component Generalization of the Camassa-Holm Equation and its Solutions , 2005, nlin/0501028.

[34]  G. Falqui,et al.  On a Camassa-Holm type equation with two dependent variables , 2005, nlin/0505059.

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

[36]  J. Escher,et al.  On the blow-up rate and the blow-up set of breaking waves for a shallow water equation , 2000 .

[37]  Yong Zhou,et al.  Wave breaking for a shallow water equation , 2004 .

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