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A note on the lifespan of solutions to the semilinear damped wave equation


Authors: Masahiro Ikeda and Yuta Wakasugi
Journal: Proc. Amer. Math. Soc. 143 (2015), 163-171
MSC (2010): Primary 35L71
DOI: https://doi.org/10.1090/S0002-9939-2014-12201-5
Published electronically: August 19, 2014
MathSciNet review: 3272741
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Abstract: This paper concerns estimates of the lifespan of solutions to the semilinear damped wave equation $ \square u+\Phi (t,x)u_t=\vert u\vert^p$ in $ (t,x)\in [0,\infty )\times \mathbf {R}^n$, where the coefficient of the damping term is $ \Phi (t,x)=\langle x\rangle ^{-\alpha }(1+t)^{-\beta }$ with $ \alpha \in [0,1),\ \beta \in (-1,1)$ and $ \alpha \beta =0$. Our novelty is to prove an upper bound of the lifespan of solutions in subcritical cases $ 1<p<2/(n-\alpha )$.


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Masahiro Ikeda
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Email: mikeda@math.kyoto-u.ac.jp

Yuta Wakasugi
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
Email: y-wakasugi@cr.math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2014-12201-5
Keywords: Semilinear damped wave equation, lifespan, upper bound
Received by editor(s): February 2, 2013
Received by editor(s) in revised form: February 28, 2013
Published electronically: August 19, 2014
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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