Instantaneous blow‐up versus local solvability of solutions to the Cauchy problem for the equation of a semiconductor in a magnetic field

2018 ◽  
Vol 41 (17) ◽  
pp. 8070-8099 ◽  
Author(s):  
Maxim O. Korpusov ◽  
Alexey V. Ovchinnikov ◽  
Alexander A. Panin
Keyword(s):  
Magnetic Field ◽  
Cauchy Problem ◽  
Blow Up ◽  
Local Solvability ◽  
The Cauchy Problem

scholarly journals Landau-Lifshitz equation of ferromagnetism with external magnetic field

2002 ◽  
Vol 72 (3) ◽  
pp. 299-316 ◽  
Author(s):  
P. Y. H. Pang ◽  
J. Xiao ◽  
F. Zhou
Keyword(s):  
Magnetic Field ◽  
Cauchy Problem ◽  
External Magnetic Field ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Uniqueness Of Solution ◽  
Energy Identity ◽  
Lifshitz Equation ◽  
The Cauchy Problem

AbstractIn this article, we prove the existence and uniqueness of solution for the Cauchy problem of the Landau-Lifshitz equation of ferromagnetism with external magnetic field. We also show that the solution is globally regular with the exception of at most finitely many blow-up points. An energy identity at blow-up points is presented.

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.

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.

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 blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohammad Kafini
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Blow Up ◽  
Initial Energy ◽  
Higher Order ◽  
Viscoelastic Wave Equation ◽  
Nonlinear Viscoelastic ◽  
Viscoelastic Wave ◽  
The Cauchy Problem ◽  
Whole Space

<p style='text-indent:20px;'>In this paper we consider the Cauchy problem for a higher-order viscoelastic wave equation with finite memory and nonlinear logarithmic source term. Under certain conditions on the initial data with negative initial energy and with certain class of relaxation functions, we prove a finite-time blow-up result in the whole space. Moreover, the blow-up time is estimated explicitly. The upper bound and the lower bound for the blow up time are estimated.</p>

Existence of strong solutions to the rotating shallow water equations with degenerate viscosities

2019 ◽  
Vol 18 (02) ◽  
pp. 333-358
Author(s):  
Ben Duan ◽  
Zhen Luo ◽  
Yan Zhou
Keyword(s):  
Cauchy Problem ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Blow Up ◽  
Local Existence ◽  
Strong Solutions ◽  
Force Term ◽  
The Cauchy Problem ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established.

On the critical exponent “instantaneous blow-up” versus “local solubility” in the Cauchy problem for a model equation of Sobolev type

2021 ◽  
Vol 85 (1) ◽  
Author(s):  
Maxim Olegovich Korpusov ◽  
Alexander Anatolyevich Panin ◽  
Andrey Evgenievich Shishkov
Keyword(s):  
Cauchy Problem ◽  
Critical Exponent ◽  
Model Equation ◽  
Blow Up ◽  
Sobolev Type ◽  
The Cauchy Problem
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