Best rates of decay for coupled waves with different propagation speeds

Best rates of decay for coupled waves with different propagation speeds We consider an abstract system of two coupled evolution equations. One of these equations has an internal damping, and the other is simply elastic. When both equations have the same propagation speed, Alabau et al. (J Evol Equ 2:127–150, 2002) showed that the semigroup of this system decays polynomially in time with the rate $$t^{-1/2}$$ t - 1 / 2 . In this work, we consider this coupled system when the propagation speeds of the equations are different, and we study the asymptotic behavior of the semigroup. For this case, we show that the semigroup still decays polynomially with a slower rate as $$t^{-1/4}$$ t - 1 / 4 . Moreover, we prove that this rate of decay is the best. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Zeitschrift für angewandte Mathematik und Physik Springer Journals

Best rates of decay for coupled waves with different propagation speeds

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Engineering; Theoretical and Applied Mechanics; Mathematical Methods in Physics
ISSN
0044-2275
eISSN
1420-9039
D.O.I.
10.1007/s00033-017-0821-9
Publisher site
See Article on Publisher Site

Abstract

We consider an abstract system of two coupled evolution equations. One of these equations has an internal damping, and the other is simply elastic. When both equations have the same propagation speed, Alabau et al. (J Evol Equ 2:127–150, 2002) showed that the semigroup of this system decays polynomially in time with the rate $$t^{-1/2}$$ t - 1 / 2 . In this work, we consider this coupled system when the propagation speeds of the equations are different, and we study the asymptotic behavior of the semigroup. For this case, we show that the semigroup still decays polynomially with a slower rate as $$t^{-1/4}$$ t - 1 / 4 . Moreover, we prove that this rate of decay is the best.

Journal

Zeitschrift für angewandte Mathematik und PhysikSpringer Journals

Published: Jun 13, 2017

References

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