ON A DECAY RATE OF SOLUTIONS TO ONE DIMENSIONAL THERMOELASTIC EQUATIONS ON A HALF LINE; LINEAR PART

1998 ◽  
pp. 198-291
Author(s):  
YOSHIHIRO SHIBATA
Keyword(s):  
Decay Rate ◽  
Linear Part ◽  
Half Line ◽  
One Dimensional

Landau–Ginzburg-type equations on the half-line in the critical case

2005 ◽  
Vol 135 (6) ◽  
pp. 1241-1262 ◽  
Author(s):  
Elena I. Kaikina ◽  
Hector F. Ruiz-Paredes
Keyword(s):  
Critical Case ◽  
Linear Part ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Image Position ◽  
Initial Boundary Value

We study nonlinear Landau–Ginzburg-type equations on the half-line in the critical case where β ∈ C, ρ > 2. The linear operator K is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol K(p) = αpρ, M = [1/2ρ]. The aim of this paper is to prove the global existence of solutions to the initial–boundary-value problem and to find the main term of the asymptotic representation of solutions in the critical case, when the time decay of the nonlinearity has the same rate as that of the linear part of the equation.

scholarly journals Optimal Polynomial Decay to Coupled Wave Equations and Its Numerical Properties

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. F. C. Lobato ◽  
S. M. S. Cordeiro ◽  
M. L. Santos ◽  
D. S. Almeida Júnior
Keyword(s):  
Decay Rate ◽  
Spectral Methods ◽  
Finite Differences ◽  
Wave Equations ◽  
Coupled System ◽  
Polynomial Decay ◽  
One Dimensional ◽  
Coupled Wave Equations ◽  
Optimal Polynomial ◽  
The One

In this work we consider a coupled system of two weakly dissipative wave equations. We show that the solution of this system decays polynomially and the decay rate is optimal. Computational experiments are conducted in the one-dimensional case in order to show that the energies properties are preserved. In particular, we use finite differences and also spectral methods.

The analytic structure of the reflection coefficient, a sum rule and a complete description of the Weyl m-function of half-line Schrödinger operators with L2-type potentials

2006 ◽  
Vol 136 (3) ◽  
pp. 615-632 ◽  
Author(s):  
Alexei Rybkin
Keyword(s):  
Reflection Coefficient ◽  
Schrödinger Operators ◽  
Half Line ◽  
Analytic Structure ◽  
Schrodinger Operators ◽  
Trace Inequality ◽  
Sum Rule ◽  
One Dimensional ◽  
Upper Half Plane ◽  
Image Position

We prove that the reflection coefficient of one-dimensional Schrödinger operators with potentials supported on a half-line can be represented in the upper half-plane as the quotient of a contractive analytic function and a properly regularized Blaschke product. We apply this fact to obtain a new trace formula and trace inequality for the reflection coefficient that yields a description of the Weyl m-function of Dirichlet half-line Schrödinger operators with slowly decaying potentials q subject to Among others, we also refine the 3/2-Lieb-Thirring inequality.

The statistics of waves propagating in a one-dimensional random medium

1985 ◽  
Vol 398 (1815) ◽  
pp. 341-363 ◽  
Keyword(s):  
Brownian Motion ◽  
Decay Rate ◽  
Normal Distribution ◽  
Random Medium ◽  
Transmitted Intensity ◽  
One Dimensional ◽  
Time Harmonic ◽  
Log Normal ◽  
The Waves ◽  
Log Normal Distribution

We consider the one-dimensional scattering of waves in a time-independent random medium. The waves considered are time-harmonic. It is assumed that the wavelength of the waves and the correlation length of the scatterers are small compared with the distance required for significant scattering. Stochastic process theory is used to investigate the statistics of the wavefield. The problem of a wave incident on a length of random medium is investigated in two cases. The first is where the medium is backed by a perfectly reflecting boundary. Here the intensity is shown to be a product of two factors; an exponential term that decays into the medium and the exponential of a ‘Brownian motion’ that describes the fluctuations of intensity with different realizations of the random medium. Because a Brownian motion has a normal distribution, the intensity has a log–normal distribution at any fixed point in the interior of the medium. For a typical realization of the random medium the exponential decay leads to most of the wave energy being near the front of a long medium. However, it is shown that the average intensity is independent of position in the medium. This is because of the long tail of the log–normal distribution and comes about because the average is heavily weighted by exceptional realizations of the medium. Thus the average value of the intensity, unlike the average of the logarithm of the intensity, is not representative of the intensity in a typical realization. The exponential decay of intensity is a result of the phenomenon of Anderson localization, which has received much attention in solid-state physics. The second case considered is where there is no barrier at the back of the medium. For large lengths of the random medium, it is shown that the transmitted intensity has an approximately log–normal distribution. The typical transmitted intensity is exponentially small as a result of localization, the decay rate with length being the same as the decay rate for the previous case. The average transmitted intensity is also exponentially small, but with a different decay rate because of weighting by exceptional realizations. The third problem discussed is that of the response of a random medium to time-harmonic forcing in the interior. The boundaries are taken to be perfectly reflecting and the response is found to be localized near the source for a typical realization. This result is related to the existence of localized normal modes in a long medium.

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