scholarly journals On the cauchy problem for dispersive equations with nonlinear terms involving high derivatives and with arbitrarily large initial data

1994 ◽  
Vol 22 (7) ◽  
pp. 835-845
Author(s):  
Thierry Colin
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Dispersive Equations ◽  
Large Initial Data ◽  
The Cauchy Problem

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.

scholarly journals The Cauchy Problem for a Fifth-Order Dispersive Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hongjun Wang ◽  
Yongqi Liu ◽  
Yongqiang Chen
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Initial Data ◽  
Order Equation ◽  
Dispersive Equations ◽  
Dispersive Equation ◽  
The Cauchy Problem ◽  
Fifth Order ◽  
Well Posed

This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev spaceHs(R)withs≥1/4. We also establish the ill-posedness for the initial data inHs(R)withs<1/4. Thus, the regularity requirement for the fifth-order dispersive equationss≥1/4is sharp.

Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem

1995 ◽  
Vol 125 (1) ◽  
pp. 115-131 ◽  
Author(s):  
Pierangelo Marcati ◽  
Roberto Natalini
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence Theorem ◽  
Weak Solutions ◽  
Hydrodynamic Model ◽  
Fractional Step ◽  
Large Initial Data ◽  
Global Weak Solutions ◽  
The Cauchy Problem

We investigate the Cauchy problem for a hydrodynamic model for semiconductors. An existence theorem of global weak solutions with large initial data is obtained by using the fractional step Lax—Friedrichs scheme and Godounov scheme.

scholarly journals Global well-posedness to the Cauchy problem of 2D inhomogeneous incompressible magnetic Bénard equations with large initial data and vacuum

2021 ◽  
Vol 6 (11) ◽  
pp. 12085-12103
Author(s):  
Zhongying Liu ◽  
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Initial Density ◽  
Well Posedness ◽  
Far Field ◽  
Large Initial Data ◽  
Global Strong Solution ◽  
Vacuum States ◽  
The Cauchy Problem ◽  
Field Decay

<abstract><p>In this paper, we are concerned with the Cauchy problem of inhomogeneous incompressible magnetic Bénard equations with vacuum as far-field density in $ \Bbb R^2 $. We prove that if the initial density and magnetic field decay not too slowly at infinity, the system admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even has compact support. Moreover, we extend the result of [16, 17] to the global one.</p></abstract>

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO DRIFT-DIFFUSION SYSTEM WITH GENERALIZED DISSIPATION

2009 ◽  
Vol 19 (06) ◽  
pp. 939-967 ◽  
Author(s):  
TAKAYOSHI OGAWA ◽  
MASAKAZU YAMAMOTO
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Initial Data ◽  
Global Existence ◽  
Asymptotic Behavior Of Solutions ◽  
Large Initial Data ◽  
Drift Diffusion ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

We show the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from semiconductor model. Our system has generalized dissipation given by a fractional order of the Laplacian. It is shown that the time global existence and decay of the solutions to the equation with large initial data. We also show the asymptotic expansion of the solution up to the second terms as t → ∞.

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