On the Cauchy Problem for the b-Family Equations with a Strong Dispersive Term

In this paper, we consider b-family equations with a strong dispersive term. First, we present a criterion on blow-up. Then global existence and persistence property of the solution are also established. Finally, we discuss infinite propagation speed of this equation.

[1]  Yong Zhou,et al.  Wave breaking and propagation speed for a class of nonlocal dispersive θ-equations , 2011 .

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

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

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

[5]  Xiuming Li.  The Infinite Propagation Speed and the Limit Behavior for the B-family Equation with Dispersive Term , 2009 .

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

[7]  Yin Jiu-li.  Stability of solitary waves for a rod equation , 2006 .

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

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

[10]  M. Birkner,et al.  Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach , 2002 .

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

[12]  Yong Zhou,et al.  A new asymptotic behavior of solutions to the Camassa-Holm equation , 2012 .

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

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

[15]  Darryl D. Holm,et al.  Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary pde , 2002, nlin/0203007.

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

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