The Cauchy problem and blow-up phenomena of a new integrable two-component Camassa–Holm system

2016 ◽  
Vol 132 ◽  
pp. 25-46 ◽  
Author(s):  
Xiuting Li
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Two Component ◽  
The Cauchy Problem

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 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.

Well-posedness and blow-up criterion for the Chaplygin gas equations in ℝN

2019 ◽  
Vol 16 (04) ◽  
pp. 639-661 ◽  
Author(s):  
Zhen Wang ◽  
Xinglong Wu
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Initial Density ◽  
Chaplygin Gas ◽  
Bmo Space ◽  
Well Posedness ◽  
Dimension Formula ◽  
The Cauchy Problem ◽  
Bounded Below ◽  
Blow Up Criterion

We establish a well-posedness theory and a blow-up criterion for the Chaplygin gas equations in [Formula: see text] for any dimension [Formula: see text]. First, given [Formula: see text], [Formula: see text], we prove the well-posedness property for solutions [Formula: see text] in the space [Formula: see text] for the Cauchy problem associated with the Chaplygin gas equations, provided the initial density [Formula: see text] is bounded below. We also prove that the solution of the Chaplygin gas equations depends continuously upon its initial data [Formula: see text] in [Formula: see text] if [Formula: see text], and we state a blow-up criterion for the solutions in the classical BMO space. Finally, using Osgood’s modulus of continuity, we establish a refined blow-up criterion of the solutions.

BLOW-UP OF THE SOLUTION FOR SEMILINEAR DAMPED WAVE EQUATION WITH TIME-DEPENDENT DAMPING

2012 ◽  
Vol 14 (05) ◽  
pp. 1250034
Author(s):  
JIAYUN LIN ◽  
JIAN ZHAI
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Upper Bound ◽  
Blow Up ◽  
Time Dependent ◽  
Damped Wave Equation ◽  
Large Initial Data ◽  
Power Type ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped wave equation with time-dependent damping and a power-type nonlinearity |u|ρ. For some large initial data, we will show that the solution to the damped wave equation will blow up within a finite time. Moreover, we can show the upper bound of the life-span of the solution.

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