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Non decay of the total energy for the wave equation with the dissipative term of spatial anisotropy

Published online by Cambridge University Press:  22 January 2016

Mishio Kawashita ,
Hideo Nakazawa  and
Hideo Soga
Mishio Kawashita
Affiliation:
Department of Mathematics, Hiroshima University, 1-3-1, Kagamiyama, Higashi-Hiroshima, Hiroshima, 739-8526, Japan, [email protected]
Hideo Nakazawa*
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji Tokyo, 192-0397, Japan, [email protected]
Hideo Soga
Affiliation:
Ibaraki University, 2-1-1, Bunkyo, Mito Ibaraki, 310-8521, Japan, [email protected]
*
Department of Mathematics, Chiba Institute of Technology, 2-1-1, Shibazono, Narashino, Chiba, 275-0023, Japan, [email protected]
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Abstract

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We consider the behavior of the total energy for the wave equation with the dissipative term. When the dissipative term works well uniformly in every direction, several authors obtain uniform decay estimates of the total energy. On the other hand, if the dissipative term is small enough uniformly in every direction, it is known that there exists a solution whose total energy does not decay. We examine the case that the dissipative term vanishes only in a neighborhood of a half-line. We introduce a uniform decay property, which is a natural generalization of the uniform decay estimates, and show that this property does not hold in our case. We prove this by constructing asymptotic solutions supported in the place where the dissipative term vanishes.

Keywords

35L0535L15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 174 , 2004 , pp. 115 - 126
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

Footnotes

*

Partly supported by Grant-in-Aid for Sci. Research (C) 14540176 from JSPS.

**

Partly supported by Grant-in-Aid for Sci. Research (C) 15540152 from JSPS.

References

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