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

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 Mathematical Modeling of Linear Fractional Oscillators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1879 ◽  
Author(s):  
Roman Parovik
Keyword(s):  
Cauchy Problem ◽  
Fractional Derivatives ◽  
Model Equation ◽  
Initial Conditions ◽  
Memory Function ◽  
Analytical Formula ◽  
Forced Oscillations ◽  
Caputo Fractional Derivatives ◽  
Specific Test ◽  
The Cauchy Problem

In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out.

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.

The Cauchy problem for an equation of Sobolev type with power non-linearity

2005 ◽  
Vol 69 (1) ◽  
pp. 59-111 ◽  
Author(s):  
E I Kaikina ◽  
P I Naumkin ◽  
I A Shishmarev
Keyword(s):  
Cauchy Problem ◽  
Sobolev Type ◽  
The Cauchy Problem
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