Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa–Holm equation

2008 ◽  
Vol 49 (3) ◽  
pp. 033516 ◽  
Author(s):  
Zhengguang Guo
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Propagation Speed ◽  
Holm Equation ◽  
Infinite Propagation Speed

scholarly journals On the Cauchy Problem for theb-Family Equations with a Strong Dispersive Term

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Xuefei Liu ◽  
Mingxuan Zhu ◽  
Zaihong Jiang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Blow Up ◽  
Propagation Speed ◽  
Persistence Property ◽  
The Cauchy Problem ◽  
Infinite Propagation Speed ◽  
Dispersive Term

In this paper, we considerb-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.

scholarly journals On the Cauchy Problem for a Class of Weakly Dissipative One-Dimensional Shallow Water Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jingjing Xu ◽  
Zaihong Jiang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Blow Up ◽  
Propagation Speed ◽  
One Dimensional ◽  
Dissipative Term ◽  
The Cauchy Problem ◽  
Infinite Propagation Speed

We investigate a more general family of one-dimensional shallow water equations with a weakly dissipative term. First, we establish blow-up criteria for this family of equations. Then, global existence of the solution is also proved. Finally, we discuss the infinite propagation speed of this family of equations.

scholarly journals Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation

2014 ◽  
Vol 12 ◽  
pp. 10-20
Author(s):  
Jiang Bo Zhou ◽  
Jun De Chen ◽  
Wen Bing Zhang
Keyword(s):  
Global Existence ◽  
Shallow Water ◽  
Blow Up ◽  
Shallow Water Equation ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Holm Equation ◽  
Special Cases ◽  
Long Time

We first establish the local well-posedness for a weakly dissipative shallow water equation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. Then two blow-up results are derived for certain initial profiles. Finally, We study the long time behavior of the solutions.

scholarly journals The dynamic properties of solutions for a nonlinear shallow water equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yeqin Su ◽  
Shaoyong Lai ◽  
Sen Ming
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Shallow Water ◽  
Wave Breaking ◽  
Dynamic Properties ◽  
Shallow Water Equation ◽  
Propagation Speed ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Infinite Propagation Speed

Abstract The local well-posedness for the Cauchy problem of a nonlinear shallow water equation is established. The wave-breaking mechanisms, global existence, and infinite propagation speed of solutions to the equation are derived under certain assumptions. In addition, the effects of coefficients λ, β, a, b, and index k in the equation are illustrated.

scholarly journals Wave Breaking and Propagation Speed for a Class of One-Dimensional Shallow Water Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Zaihong Jiang ◽  
Sevdzhan Hakkaev
Keyword(s):  
Initial Data ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Time ◽  
Wave Breaking ◽  
Blow Up ◽  
Propagation Speed ◽  
One Dimensional ◽  
General Family ◽  
Infinite Propagation Speed

We investigate a more general family of one-dimensional shallow water equations. Analogous to the Camassa-Holm equation, these new equations admit blow-up phenomenon and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data. It turns out that it is the shape instead of the size and smoothness of the initial data which influences breakdown in finite time. Then, infinite propagation speed for the shallow water equations is proved in the following sense: the corresponding solutionu(t,x)with compactly supported initial datumu0(x)does not have compactx-support any longer in its lifespan.

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