scholarly journals Space-Time Estimates on Damped Fractional Wave Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Jiecheng Chen ◽  
Dashan Fan ◽  
Chunjie Zhang
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Linear Equations ◽  
Space Time ◽  
Regularity Conditions ◽  
Fractional Wave Equation ◽  
Time Estimates ◽  
Almost Everywhere ◽  
The Cauchy Problem

We obtain space-time estimates on the solutionu(t,x)to the Cauchy problem of damped fractional wave equation. We mainly focus on the linear equation. The almost everywhere convergence of the solution to linear equations ast→0+is also studied, with the initial data satisfying certain regularity conditions.

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 Global Analysis for Rough Solutions to the Davey-Stewartson System

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Han Yang ◽  
Xiaoming Fan ◽  
Shihui Zhu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Nonlinear Equation ◽  
Global Solution ◽  
Global Analysis ◽  
Space Time ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
New Space ◽  
Time Bounds

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is inHswiths> 2/5, then there exists a global solution in time, and theHsnorm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-timeLt,x4estimate for nonlinear equation with the relatively general defocusing power nonlinearity.

scholarly journals THE CAUCHY PROBLEM FOR THE WAVE EQUATION WITH LÉVY NOISE INITIAL DATA

2006 ◽  
Vol 09 (02) ◽  
pp. 249-270
Author(s):  
BERNT ØKSENDAL ◽  
FRANK PROSKE ◽  
MIKAEL SIGNAHL
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Explicit Form ◽  
Lévy Noise ◽  
Hermite Transform ◽  
The Cauchy Problem ◽  
Levy Noise

In this paper we study the Cauchy problem for the wave equation with spacetime Lévy noise initial data in the Kondratiev space of stochastic distributions. We prove that this problem has a strong and unique C2-solution, which takes an explicit form. Our approach is based on the use of the Hermite transform.

The Analytic Continuation of the Riemannliouville Integral in the Hyperbolic Case

1961 ◽  
Vol 13 ◽  
pp. 37-47 ◽  
Author(s):  
Marcel Riesz
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Analytic Continuation ◽  
Present Article ◽  
Space Time ◽  
Liouville Type ◽  
Hyperbolic Case ◽  
The Cauchy Problem ◽  
Multiple Integrals

In 1949 I published in the Acta Mathematica (vol. 81) a rather long paper: “L'intégrale de Riemann-Liouville et le problème de Cauchy.” This work will be quoted in the sequel as Acta paper. Only minor local references to this paper will be made here, and knowledge of it is not required for the reading of the present article. The notations used here are slightly different from those used in my former paper.In the Acta paper I introduce multiple integrals and of the Riemann- Liouville type depending on a parameter α and converging for sufficiently large values of α. I give the solution of the Cauchy problem for the wave equation in a unique formula, the same for space-time of odd or even dimensions, implying an analytic continuation with respect to the parameter α.

scholarly journals A TIME-INDEPENDENT ENERGY ESTIMATE FOR OUTGOING SCALAR WAVES IN THE KERR GEOMETRY

2008 ◽  
Vol 05 (01) ◽  
pp. 221-255 ◽  
Author(s):  
FELIX FINSTER ◽  
JOEL SMOLLER
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Wave Packet ◽  
Energy Estimate ◽  
Outgoing Wave ◽  
Mathematical Tool ◽  
Scalar Wave ◽  
Kerr Geometry ◽  
The Cauchy Problem

The Cauchy problem for the scalar wave equation in the Kerr geometry is considered, with initial data which is smooth and compactly supported outside the event horizon. A time-independent energy estimate for the outgoing wave is obtained. As an application, the outgoing energy for wave-packet initial data is estimated, uniformly as the support of the initial data is shifted to infinity. The main mathematical tool is the previously derived integral representation of the wave propagator.

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