scholarly journals Wave-breaking phenomena and global existence for the weakly dissipative generalized Novikov equation

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Shuguan Ji ◽  
Yonghui Zhou
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Global Solution ◽  
Wave Breaking ◽  
Blow Up ◽  
Momentum Density ◽  
Well Posedness ◽  
Novikov Equation ◽  
Blow Up Rate

In this paper, we mainly study several problems on the weakly dissipative generalized Novikov equation. We first establish the local well-posedness of solutions. We then give the precise blow-up scenarios for the generalized Novikov equation provided the momentum density associated with their initial data changes sign, and obtain the blow-up rate of blow-up solutions. Finally, we prove that the equation has a global solution provided the momentum density associated with their initial data do not change sign.

scholarly journals The Cauchy Problem for a Weakly Dissipative 2-Component Camassa-Holm System

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Yonghong Wu
Keyword(s):  
Wave Breaking ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Well Posedness ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Blow Up Rate ◽  
Global Existence Result

The weakly dissipative 2-component Camassa-Holm system is considered. A local well-posedness for the system in Besov spaces is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking mechanisms and the exact blow-up rate of strong solutions to the system are presented. Moreover, a global existence result for strong solutions is derived.

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.

Sharp condition for blow-up and global existence in a two species chemotactic Keller–Segel system in 2

2012 ◽  
Vol 24 (2) ◽  
pp. 297-313 ◽  
Author(s):  
ELIO ESPEJO ◽  
KARINA VILCHES ◽  
CARLOS CONCA
Keyword(s):  
Global Existence ◽  
Global Solution ◽  
Blow Up ◽  
Critical Mass ◽  
Single Species ◽  
Radial Symmetry ◽  
Initial Mass ◽  
Threshold Curve ◽  
Sharp Condition ◽  
Additional Question

For the parabolic–elliptic Keller–Segel system in 2 it has been proved that if the initial mass is less than 8π/χ, a global solution exists, and in case the initial mass is larger than 8π/χ, blow-up happens. The case of several chemotactic species introduces an additional question: What is the analog for the critical mass obtained for the single species system? We find a threshold curve in the two species case that allows us to determine if the system is a blow-up or a global in time solution. No radial symmetry is assumed.

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 Systems with Local and Nonlocal Diffusions, Mixed Boundary Conditions, and Reaction Terms

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Mauricio Bogoya ◽  
Julio D. Rossi
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Asymptotic Bounds ◽  
Blowing Up ◽  
Blow Up Rate ◽  
Blowing Up Solutions

We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.

Global existence and blow-up phenomena for the weakly dissipative Novikov equation

2012 ◽  
Vol 75 (4) ◽  
pp. 2464-2473 ◽  
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Yimin Zhang
Keyword(s):  
Global Existence ◽  
Blow Up ◽  
Novikov Equation

scholarly journals Global Analysis for Rough Solutions to the Davey-Stewartson System

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Han Yang ◽  
Xiaoming Fan ◽  
Shihui Zhu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Equation ◽  
Global Solution ◽  
Global Analysis ◽  
Space Time ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
New Space ◽  
Time Bounds

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is inHswiths> 2/5, then there exists a global solution in time, and theHsnorm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-timeLt,x4estimate for nonlinear equation with the relatively general defocusing power nonlinearity.

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.

scholarly journals Blow-Up Criteria for the Modified Novikov Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Caochuan Ma ◽  
Wujun Lv
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Finite Time ◽  
Blow Up ◽  
Solution Blowing ◽  
Blowing Up ◽  
The Cauchy Problem ◽  
Novikov Equation

We investigate the Cauchy problem for the modified Novikov equation. We establish blow-up criteria on the initial data to guarantee the corresponding solution blowing up in finite time.

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