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.

EXPLOSIVE INSTABILITY DUE TO 3-WAVE OR 4-WAVE MIXING, WITH OR WITHOUT DISSIPATION

2008 ◽  
Vol 06 (04) ◽  
pp. 413-428 ◽  
Author(s):  
HARVEY SEGUR
Keyword(s):  
Initial Data ◽  
Nonlinear Waves ◽  
Finite Time ◽  
Blow Up ◽  
Wave Mixing ◽  
Explosive Instability ◽  
Effective Threshold ◽  
Blowing Up ◽  
Gain Energy

It is known that an "explosive instability" can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. In that context, three resonantly interacting wavetrains all gain energy from a background source, and all blow up together, in finite time. A recent paper [17] showed that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing, and results in four resonantly interacting wavetrains all blowing up in finite time. In both cases, the instability occurs in systems with no dissipation. This paper reviews the earlier work, and shows that adding a common form of dissipation to the system, with either 3-wave or 4-wave mixing, provides an effective threshold for blow-up. Only initial data that exceed the respective thresholds blow up in finite time.

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.

The critical exponents of parabolic equations and blow-up in Rn

1998 ◽  
Vol 128 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Yuan-wei Qi
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Initial Value ◽  
The Cauchy Problem

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

scholarly journals On the Cauchy Problem for the Two-Component Novikov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Transport Equation ◽  
Besov Spaces ◽  
Well Posedness ◽  
Two Component ◽  
The Cauchy Problem ◽  
Novikov Equation

We are concerned with the Cauchy problem of two-component Novikov equation, which was proposed by Geng and Xue (2009). We establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory which is motivated by that in Danchin's cerebrated paper (2001). Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend some results of Himonas (2003) to more general equations.

scholarly journals Blow-up for the sixth-order multidimensional generalized Boussinesq equation with arbitrarily high initial energy

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jianghao Hao ◽  
Aiyuan Gao
Keyword(s):  
Cauchy Problem ◽  
Fourier Transform ◽  
Finite Time ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Initial Energy ◽  
Sixth Order ◽  
The Cauchy Problem ◽  
Generalized Boussinesq Equation

AbstractIn this paper, we consider the Cauchy problem for the sixth-order multidimensional generalized Boussinesq equation with double damping terms. By using the improved convexity method combined with Fourier transform, we show the finite time blow-up of solution with arbitrarily high initial energy.

scholarly journals Properties at potential blow-up times for Navier-Stokes

2017 ◽  
Vol 35 (2) ◽  
pp. 127 ◽  
Author(s):  
Paulo R. Zingano ◽  
Jens Lorenz
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Incompressible Flows ◽  
Blow Up ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Solution Properties ◽  
Maximal Interval ◽  
Navier Stokes Equations ◽  
The Cauchy Problem

In this paper we consider the Cauchy problem for the 3D navier-Stokes equations for incompressible flows. The initial data are assume d to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solution can develop singularities in finite time. Assuming the maximal interval of existence to be finite, we give a unified discussion of various known solution properties as time approaches the blow-up time.

scholarly journals A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion

2020 ◽  
Vol 18 (1) ◽  
pp. 194-203 ◽  
Author(s):  
Zaiyun Zhang ◽  
Limei Li ◽  
Chunhua Fang ◽  
Fan He ◽  
Chuangxia Huang ◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Type Equation ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Math Phys ◽  
Blow Up Criterion

Abstract In this paper, we investigate the Cauchy problem for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion. Furthermore, we establish the blow-up result of the positive solutions in finite time under certain conditions on the initial datum. This result complements the early one in the literature, such as [E. Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation, J. Math. Phys. 54 (2013), no. 9, 092703, DOI 10.1063/1.4820786] and [Z.Y. Zhang, J.H. Huang, and M.B. Sun, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited, J. Math. Phys. 56 (2015), no. 9, 092701, DOI 10.1063/1.4930198].

scholarly journals On the Cauchy Problem of a Quasilinear Degenerate Parabolic Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-12
Author(s):  
Zongqi Liang ◽  
Huashui Zhan
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Classical Solution ◽  
Finite Time ◽  
Numerical Experiments ◽  
Degenerate Parabolic Equation ◽  
Blow Up ◽  
Strong Solutions ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

By Oleinik's line method, we study the existence and the uniqueness of the classical solution of the Cauchy problem for the following equation in[0,T]×R2:∂xxu+u∂yu−∂tu=f(⋅,u), provided thatTis suitable small. Results of numerical experiments are reported to demonstrate that the strong solutions of the above equation may blow up in finite time.

scholarly journals Qualitative analysis for a new generalized 2-component Camassa-Holm system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Shouming Zhou ◽  
Shanshan Zheng
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Blow Up ◽  
Well Posedness ◽  
Energy Methods ◽  
Bernoulli Beam ◽  
String Problem ◽  
The Cauchy Problem ◽  
Novel Method ◽  
Euler Bernoulli Beam

<p style='text-indent:20px;'>This paper considers the Cauchy problem for a 2-component Camassa-Holm system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} m_t = ( u m)_x+ u _xm- v m, \ \ n_t = ( u n)_x+ u _xn+ v n, \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n+m = \frac{1}{2}( u _{xx}-4 u ) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n-m = v _x $\end{document}</tex-math></inline-formula>, this model was proposed in [<xref ref-type="bibr" rid="b2">2</xref>] from a novel method to the Euler-Bernoulli Beam on the basis of an inhomogeneous matrix string problem. The local well-posedness in Sobolev spaces <inline-formula><tex-math id="M3">\begin{document}$ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ s&gt;\frac{5}{2} $\end{document}</tex-math></inline-formula> of this system was investigated through the Kato's theory, then the blow-up criterion for this system was described by the technique on energy methods. Finally, we established the analyticity in both time and space variables of the solutions for this system with a given analytic initial data.</p>

LOCAL WELL POSEDNESS OF CAUCHY PROBLEM FOR VISCOUS DIFFUSION EQUATIONS

2009 ◽  
Vol 20 (04) ◽  
pp. 509-519
Author(s):  
YACHENG LIU ◽  
RUNZHANG XU
Keyword(s):  
Cauchy Problem ◽  
Integral Equations ◽  
Finite Time ◽  
Blow Up ◽  
Diffusion Equations ◽  
Well Posedness ◽  
The Cauchy Problem

In this paper, we study the Cauchy problem of multi-dimensional viscous diffusion equations. By using an equivalent integral equations, we get the existence of local Wk,p solutions. And we prove the finite time blow up of solutions under appropriate conditions.

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