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 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 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.

The Cauchy problem of a weakly dissipative shallow water equation

2018 ◽  
Vol 98 (8) ◽  
pp. 1387-1402 ◽  
Author(s):  
Sen Ming ◽  
Shaoyong Lai ◽  
Yeqin Su
Keyword(s):  
Cauchy Problem ◽  
Shallow Water ◽  
Shallow Water Equation ◽  
The Cauchy Problem

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 Cauchy Problem for a Dissipative Periodic 2-Component Degasperis-Procesi System

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Ls Yong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Transport Equation ◽  
Wave Breaking ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Well Posedness ◽  
Priori Estimates ◽  
The Cauchy Problem

The dissipative periodic 2-component Degasperis-Procesi system is investigated. A local well-posedness for the system in Besov space is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking criterions for strong solutions to the system with certain initial data are derived.

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