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Acta Mathematica Sinica, English Series

, Volume 30, Issue 5, pp 861–871 | Cite as

Instability of standing wave for the Klein-Gordon-Hartree equation

  • Xiao Guang Li
  • Jian Zhang
  • Yong Hong Wu
Article

Abstract

The instability property of the standing wave u ω (t, x) = eiωt φ(x) for the Klein-Gordon-Hartree equation
$$\frac{{\partial ^2 u}} {{\partial t^2 }} - \Delta u + u - \left( {\left| x \right|^{ - \gamma } *\left| u \right|^2 } \right)u = 0, x \in \mathbb{R}^N , 0 < \gamma < \min \left\{ {N,4} \right\}$$
is investigated. For the case N ≥ 3 and \(\omega ^2 < \tfrac{2} {{N + 4 - \gamma }}\), it is shown that the standing wave eiωt φ(X) is strongly unstable by blow-up in finite time.

Keywords

Klein-Gordon-Hartree equation standing waves strong instability 

MR(2010) Subject Classification

35L70 35B35 35A15 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Visual Computing and Virtual Reality Key Laboratory of Sichuan ProvinceSichuan Normal UniversityChengduP. R. China
  2. 2.College of Mathematics and Software ScienceSichuan Normal UniversityChengduP. R. China
  3. 3.Department of Mathematics and StatisticsCurtin University of TechnologyPerthAustralia

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