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A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy

Author: Yanjin Wang
Journal: Proc. Amer. Math. Soc. 136 (2008), 3477-3482
MSC (2000): Primary 35L05, 35L15
Published electronically: May 23, 2008
MathSciNet review: 2415031
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Abstract: In this paper we consider the nonexistence of global solutions of a Klein-Gordon equation of the form

$\displaystyle u_{tt}-\Delta u+m^2u=f(u),$   $\displaystyle (t,x)\in [0,T)\times\mathbb{R}^n.$  

Here $ m\neq 0$ and the nonlinear power $ f(u)$ satisfies some assumptions which will be stated later. We give a sufficient condition on the initial datum with arbitrarily high initial energy such that the solution of the above Klein-Gordon equation blows up in finite time.

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Additional Information

Yanjin Wang
Affiliation: Graduate School of Mathematics, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan
Address at time of publication: Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-15, Beijing, 100088, People’s Republic of China

Keywords: Klein-Gordon equation, blow up, positive initial energy
Received by editor(s): November 10, 2006
Received by editor(s) in revised form: November 23, 2006
Published electronically: May 23, 2008
Additional Notes: This work was supported by a Japanese government scholarship. The author wishes to express his deep gratitude to Professor Hitoshi Kitada for his constant encouragement and kind guidance. Thanks also go to the referees for their comments and careful reading of the manuscript
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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