scholarly journals Logarithmic Decay of Wave Equation with Kelvin-Voigt Damping

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 715 ◽  
Author(s):  
Luc Robbiano ◽  
Qiong Zhang
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Resolvent Operator ◽  
Carleman Estimate ◽  
Proper Class ◽  
Geometric Assumption ◽  
Logarithmic Decay ◽  
Longtime Behavior

In this paper, we analyze the longtime behavior of the wave equation with local Kelvin-Voigt Damping. Through introducing proper class symbol and pseudo-diff-calculus, we obtain a Carleman estimate, and then establish an estimate on the corresponding resolvent operator. As a result, we show the logarithmic decay rate for energy of the system without any geometric assumption on the subdomain on which the damping is effective.

scholarly journals Energy decay rate for a quasi-linear wave equation with localized strong dissipation

2011 ◽  
Vol 62 (1) ◽  
pp. 164-172 ◽  
Author(s):  
Daewook Kim ◽  
Yong Han Kang ◽  
Mi Jin Lee ◽  
Il Hyo Jung
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Energy Decay ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Energy Decay Rate

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.

DECAY RATES FOR THE SHIFTED WAVE EQUATION ON A SYMMETRIC SPACE OF NONCOMPACT TYPE

2013 ◽  
Vol 10 (04) ◽  
pp. 677-701
Author(s):  
CARLOS ALMADA
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Symmetric Space ◽  
Decay Rate ◽  
Wave Operator ◽  
Symmetric Spaces ◽  
Decay Rates ◽  
Noncompact Type ◽  
Killing Fields ◽  
The Cauchy Problem

We derive L∞–L1 decay rate estimates for solutions of the shifted wave equation on certain symmetric spaces (M, g). The Cauchy problem for the shifted wave operator on these spaces was studied by Helgason, who obtained a closed form for its solution. Our results extend to this new context the classical estimates for the wave equation in ℝn. Then, following an idea from Klainerman, we introduce a new norm based on Lie derivatives with respect to Killing fields on M and we derive an estimate for the case that n = dim M is odd.

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