The Wave Equation with Computable Initial Data Whose Unique Solution Is Nowhere Computable

1997 ◽  
Vol 43 (4) ◽  
pp. 499-509 ◽  
Author(s):  
Marian B. Pour-El ◽  
Ning Zhong
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Unique Solution

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.

LANDAU–GINZBURG TYPE EQUATIONS IN THE SUBCRITICAL CASE

2003 ◽  
Vol 05 (01) ◽  
pp. 127-145 ◽  
Author(s):  
NAKAO HAYASHI ◽  
ELENA I. KAIKINA ◽  
PAVEL I. NAUMKIN
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Decay Rate ◽  
Unique Solution ◽  
Linear Case ◽  
Decay Estimates ◽  
Time Decay ◽  
Subcritical Case ◽  
Small Norm ◽  
The Cauchy Problem

We study the Cauchy problem for the nonlinear Landau–Ginzburg equation [Formula: see text] where α, β ∈ C with dissipation condition ℜα > 0. We are interested in the subcritical case [Formula: see text]. We assume that θ = | ∫ u0(x) dx| ≠ 0 and ℜδ (α, β) > 0, where [Formula: see text] Furthermore we suppose that the initial data u0 ∈ L1 are such that (1+|x|)au0 ∈ L1, with sufficiently small norm ε = ‖(1 + |x|)a u0 ‖1, where a ∈ (0,1). Also we assume that σ is sufficiently close to [Formula: see text]. Then there exists a unique solution of the Cauchy problem (*) such that [Formula: see text] satisfying the following time decay estimates for large t > 0[Formula: see text] Note that in comparison with the corresponding linear case the decay rate of the solutions of (*) is more rapid.

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.

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