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Asymptotics for the nonlinear dissipative wave equation

Author: Tokio Matsuyama
Journal: Trans. Amer. Math. Soc. 355 (2003), 865-899
MSC (2000): Primary 35L05; Secondary 35L10
Published electronically: November 1, 2002
MathSciNet review: 1938737
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Abstract: We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like $a_1 (1+\vert x \vert)^{-\delta} \vert u_t \vert^{\beta} u_t$ $(a_1$, $\beta$, $\delta>0)$ and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.

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Additional Information

Tokio Matsuyama
Affiliation: Department of Mathematics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan

Received by editor(s): November 6, 2001
Received by editor(s) in revised form: July 4, 2002
Published electronically: November 1, 2002
Additional Notes: Supported in part by a Grant-in-Aid for Scientific Research (C)(2)(No.11640213), Japan Society for the Promotion of Science.
The author would like to express his sincere gratitude to Professors K. Mochizuki, M. Nakao and M. Yamaguchi for several useful comments. He is also indebted to Professors Y. Shibata, N. Hayashi and T. Kobayashi, who pointed out the uniform decay estimate to him. The author thanks Doctor H. Nakazawa for advising him of the existence of scattering states. The author also thanks the referee for a careful reading of the manuscript.
Dedicated: Dedicated to Professor Kunihiko Kajitani on the occasion of his sixtieth birthday
Article copyright: © Copyright 2002 American Mathematical Society