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June 1991 Asymptotic behavior of solutions to the Klein-Gordon equation with a nonlinear dissipative term
Takahiro Motai
Tsukuba J. Math. 15(1): 151-160 (June 1991). DOI: 10.21099/tkbjm/1496161576

Abstract

We study the asymptotic behavior of the Klein-Gordon equation with a nonlinear dissipative term $|\partial_{t}w(t)|^{p- 1}\partial_{t}w(t) (p\gt1)$ in $x\in \textbf{R}^{n} (n\geqq 1)$ and $t\geqq 0$. We prove that the energy of solutions does not converge to $0$ as $ t \rightarrow \infty$ for $p\gt1+2/n$ if Cauchy data are suffciently small. We also prove that solutions of the above equation converge to suitable solutions of the linear Klein-Gordon equation in the energy space as $t \rightarrow \infty$ for $p\gt1+4/n$ if $1\leqq n\leqq 6$ and $1+4/n \lt p \lt n/(n-6)$ if $n\geqq 7$.

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Takahiro Motai. "Asymptotic behavior of solutions to the Klein-Gordon equation with a nonlinear dissipative term." Tsukuba J. Math. 15 (1) 151 - 160, June 1991. https://doi.org/10.21099/tkbjm/1496161576

Information

Published: June 1991
First available in Project Euclid: 30 May 2017

zbMATH: 0776.35058
MathSciNet: MR1118591
Digital Object Identifier: 10.21099/tkbjm/1496161576

Rights: Copyright © 1991 University of Tsukuba, Institute of Mathematics

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Vol.15 • No. 1 • June 1991
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