scholarly journals Wave Breaking Phenomenon for DGH Equation with Strong Dissipation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Zhengguang Guo ◽  
Min Zhao
Keyword(s):  
Finite Time ◽  
Wave Breaking ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Dissipative Term

The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipative term. We establish some sufficient conditions to guarantee finite time blow-up of strong solutions.

scholarly journals Blow-Up Analysis for the Periodic Two-Component μ-Hunter-Saxton System

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yunxi Guo ◽  
Tingjian Xiong
Keyword(s):  
Transport Equation ◽  
Wave Breaking ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Blow Up Analysis ◽  
Classic Method ◽  
Two Component ◽  
Localization Analysis ◽  
Spatially Periodic ◽  
Periodic Setting

The two-component μ-Hunter-Saxton system is considered in the spatially periodic setting. Firstly, a wave-breaking criterion is derived by employing the localization analysis of the transport equation theory. Secondly, several sufficient conditions of the blow-up solutions are established by using the classic method. The results obtained in this paper are new and different from those in previous works.

scholarly journals Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2075
Author(s):  
Ratinan Boonklurb ◽  
Tawikan Treeyaprasert ◽  
Aong-art Wanna
Keyword(s):  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Initial Conditions ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Heat Equations ◽  
Quenching Rate ◽  
Linear Heat ◽  
Quenching Set

This article deals with finite-time quenching for the system of coupled semi-linear heat equations ut=uxx+f(v) and vt=vxx+g(u), for (x,t)∈(0,1)×(0,T), where f and g are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given.

Wave breaking and shock waves for a periodic shallow water equation

2007 ◽  
Vol 365 (1858) ◽  
pp. 2281-2289 ◽  
Author(s):  
Joachim Escher
Keyword(s):  
Shock Waves ◽  
Shallow Water ◽  
Wave Breaking ◽  
Blow Up ◽  
Shallow Water Equation ◽  
Strong Solutions ◽  
Global Weak Solutions ◽  
Initial Profile ◽  
And Shock Waves ◽  
Periodic Shock

This paper is devoted to the study of a recently derived periodic shallow water equation. We discuss in detail the blow-up scenario of strong solutions and present several conditions on the initial profile, which ensure the occurrence of wave breaking. We also present a family of global weak solutions, which may be viewed as global periodic shock waves to the equation under discussion.

scholarly journals Blow-Up of Solutions to a Novel Two-Component Rod System

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Shengrong Liu ◽  
Zhengguang Guo ◽  
Weiming Wang
Keyword(s):  
Shallow Water ◽  
Finite Time ◽  
Blow Up ◽  
Strong Solutions ◽  
Shallow Water Theory ◽  
Initial Value ◽  
Rod System ◽  
Two Component ◽  
Special Classes

We consider a novel two-component rod system which is closely connected to the shallow water theory. The present work is mainly concerned with the blow-up mechanism of strong solutions; we establish new conditions in view of some special classes of initial value to guarantee finite time blow-up of solutions.

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 Solution concepts, well-posedness, and wave breaking for the Fornberg–Whitham equation

2020 ◽  
Author(s):  
Günther Hörmann
Keyword(s):  
Cauchy Problem ◽  
Traveling Waves ◽  
Wave Breaking ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Solution Concepts ◽  
Nonlinear Operators ◽  
The Cauchy Problem ◽  
Whitham Equation

AbstractWe discuss concepts and review results about the Cauchy problem for the Fornberg–Whitham equation, which has also been called Burgers–Poisson equation in the literature. Our focus is on a comparison of various strong and weak solution concepts as well as on blow-up of strong solutions in the form of wave breaking. Along the way we add aspects regarding semiboundedness at blow-up, from semigroups of nonlinear operators to the Cauchy problem, and about continuous traveling waves as weak solutions.

scholarly journals Arrest of blow up for the 3D semi-relativistic Schrödinger–Poisson system with pseudo-relativistic diffusion

2015 ◽  
Vol 27 (10) ◽  
pp. 1550023
Author(s):  
W. Abou Salem ◽  
T. Chen ◽  
V. Vougalter
Keyword(s):  
Coulomb Interaction ◽  
Finite Time ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Negative Energy ◽  
Energy Norm ◽  
Well Posedness ◽  
System Of Equations ◽  
Initial Condition ◽  
Poisson System

We show global well-posedness in energy norm of the semi-relativistic Schrödinger–Poisson system of equations with attractive Coulomb interaction in [Formula: see text] in the presence of pseudo-relativistic diffusion. We also discuss sufficient conditions to have well-posedness in [Formula: see text]. In the absence of dissipation, we show that the solution corresponding to an initial condition with negative energy blows up in finite time, which is as expected, since the nonlinearity is critical.

BLOW-UP IN TWO-COMPONENT NONLINEAR SCHRÖDINGER SYSTEMS WITH AN EXTERNAL DRIVEN FIELD

2013 ◽  
Vol 23 (09) ◽  
pp. 1699-1727 ◽  
Author(s):  
ANSGAR JÜNGEL ◽  
RADA-MARIA WEISHÄUPL
Keyword(s):  
Rabi Frequency ◽  
Blow Up ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Einstein Condensate ◽  
Model Parameters ◽  
Nonlinear Schrödinger ◽  
Bose Einstein Condensate ◽  
Nonlinear Schrödinger Systems

A system of two nonlinear Schrödinger equations in up to three space dimensions is analyzed. The equations are coupled through cubic mean-field terms and a linear term which models an external driven field described by the Rabi frequency. The intraspecific mean-field expressions may be non-cubic. The system models, for instance, two components of a Bose–Einstein condensate in a harmonic trap. Sufficient conditions on the various model parameters for global-in-time existence of strong solutions are given. Furthermore, the finite-time blow-up of solutions is proved under suitable conditions on the parameters and in the presence of at least one focusing nonlinearity. Numerical simulations in one and two space dimensions verify and complement the theoretical results. It turns out that the Rabi frequency of the driven field may be used to control the mass transport and hence to influence the blow-up behavior of the system.

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.

scholarly journals Blow-Up of Solutions to Fractional-in-Space Burgers-Type Equations

2021 ◽  
Vol 5 (4) ◽  
pp. 249
Author(s):  
Munirah Alotaibi ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Finite Time ◽  
Function Method ◽  
Burgers Equation ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Bounded Domains ◽  
Initial Value ◽  
Test Function ◽  
Burgers Type Equations ◽  
Test Function Method

We consider fractional-in-space analogues of Burgers equation and Korteweg-de Vries-Burgers equation on bounded domains. Namely, we establish sufficient conditions for finite-time blow-up of solutions to the mentioned equations. The obtained conditions depend on the initial value and the boundary conditions. Some examples are provided to illustrate our obtained results. In the proofs of our main results, we make use of the test function method and some integral inequalities.

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