Electron. J. Diff. Equ., Vol. 2014 (2014), No. 127, pp. 1-12.

Energy decay for elastic wave equations with critical damping

Jaqueline Luiza Horbach, Naoki Nakabayashi

Abstract:
We show that the total energy decays at the rate $E_u(t) = O(t^{-2})$, as $t \to +\infty$, for solutions to the Cauchy problem of a linear system of elastic wave with a variable damping term. It should be mentioned that the the critical decay satisfies $V(x) \ge C_0(1+|x|)^{-1}$ for $C_0>2b$, where b represents the speed of propagation of the P-wave.

Submitted January 17, 2014. Published May 16, 2014.
Math Subject Classifications: 35L52, 35B45, 35A25, 35B33.
Key Words: Elastic wave equation; critical damping; multiplier method; total energy; compactly supported initial data; optimal decay.

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Jaqueline Luiza Horbach
Department of Mathematics
Federal University of Santa Catarina
88040-270 Florianópolis, Santa Catarina, Brazil
email: jaqueluizah@gmail.com
Naoki Nakabayashi
Department of Mathematics
Graduate School of Education, Hiroshima University
Higashi-Hiroshima 739-8524, Japan
email: n.naoki2655@gmail.com

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