Correct problem for the wave equation without initial data

1969 ◽  
Vol 20 (6) ◽  
pp. 692-702
Author(s):  
L. P. Nizhnik
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Correct Problem

scholarly journals Decay of solutions of the wave equation in expanding cosmological spacetimes

2019 ◽  
Vol 16 (01) ◽  
pp. 35-58
Author(s):  
João L. Costa ◽  
José Natário ◽  
Pedro F. C. Oliveira
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Decay Rate ◽  
Time Derivative ◽  
Iteration Scheme ◽  
Higher Order ◽  
Accelerated Expansion ◽  
De Sitter ◽  
Decay Of Solutions ◽  
Partial Energy

We study the decay of solutions of the wave equation in some expanding cosmological spacetimes, namely flat Friedmann–Lemaître–Robertson–Walker (FLRW) models and the cosmological region of the Reissner–Nordström–de Sitter (RNdS) solution. By introducing a partial energy and using an iteration scheme, we find that, for initial data with finite higher order energies, the decay rate of the time derivative is faster than previously existing estimates. For models undergoing accelerated expansion, our decay rate appears to be (almost) sharp.

Global well-posedness and self-similarity for semilinear wave equations in a time-weighted framework of Besov type

2020 ◽  
Vol 17 (01) ◽  
pp. 123-139
Author(s):  
Lucas C. F. Ferreira ◽  
Jhean E. Pérez-López
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Besov Spaces ◽  
Wave Equations ◽  
Well Posedness ◽  
Self Similarity ◽  
Semilinear Wave Equations ◽  
Semilinear Wave Equation

We show global-in-time well-posedness and self-similarity for the semilinear wave equation with nonlinearity [Formula: see text] in a time-weighted framework based on the larger family of homogeneous Besov spaces [Formula: see text] for [Formula: see text]. As a consequence, in some cases of the power [Formula: see text], we cover a initial-data class larger than in some previous results. Our approach relies on dispersive-type estimates and a suitable [Formula: see text]-product estimate in Besov spaces.

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.

HIGH FREQUENCY ANALYSIS OF FAMILIES OF SOLUTIONS TO THE EQUATION OF VISCOELASTICITY OF KELVIN–VOIGT

2004 ◽  
Vol 01 (04) ◽  
pp. 789-812 ◽  
Author(s):  
AMEL ATALLAH-BARAKET ◽  
CLOTILDE FERMANIAN KAMMERER
Keyword(s):  
Wave Equation ◽  
Energy Density ◽  
Initial Data ◽  
Lower Bounds ◽  
Frequency Analysis ◽  
High Frequency ◽  
Weak Limit ◽  
Negative Viscosity

In this paper, we study the evolution of the energy density of a sequence of solutions to the Kelvin–Voigt viscoelasticity equation. We do not suppose lower bounds on the non-negative viscosity matrix. We prove that, in the zone where the viscosity matrix is invertible, this term prevents propagation of concentation and oscillation effects contrary to what happens in the wave equation. We calculate precisely the weak limit of the energy density in terms of microlocal defect measures associated with the initial data under the assumption that the oscillations of the data are not microlocally localized on directions which are in the kernel of the viscosity matrix.

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