Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation with cubic nonlinearity

2017 ◽  
Vol 151 ◽  
pp. 208-226 ◽  
Author(s):  
Min Li ◽  
Zhaoyang Yin
Keyword(s):  
Blow Up ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Holm Equation

scholarly journals On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

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.

Well-posedness and blow-up solution for the stochastic Dullin-Gottwald-Holm equation

2019 ◽  
Vol 60 (8) ◽  
pp. 083513 ◽  
Author(s):  
Wujun Lv ◽  
Ping He ◽  
Qinghua Wang
Keyword(s):  
Blow Up ◽  
Well Posedness ◽  
Holm Equation

scholarly journals Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Profile ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.

scholarly journals On the blow-up of solutions to the integrable modified Camassa–Holm equation

2014 ◽  
Vol 12 (04) ◽  
pp. 355-368 ◽  
Author(s):  
Yue Liu ◽  
Peter J. Olver ◽  
Changzheng Qu ◽  
Shuanghu Zhang
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Momentum Density ◽  
Initial Energy ◽  
Cubic Nonlinearity ◽  
Initial Momentum ◽  
Holm Equation

We derive conditions on the initial data, including cases where the initial momentum density is not of one sign, that produce blow-up of the induced solution to the modified integrable Camassa–Holm equation with cubic nonlinearity. The blow-up conditions are formulated in terms of the initial momentum density and the average initial energy.

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