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The defocusing energy-critical Klein–Gordon–Hartree equation

Volume 140 / 2015

Qianyun Miao, Jiqiang Zheng Colloquium Mathematicum 140 (2015), 31-58 MSC: Primary 35P25; Secondary 35B40, 35Q40, 81U99. DOI: 10.4064/cm140-1-4


We study the scattering theory for the defocusing energy-critical Klein–Gordon equation with a cubic convolution $u_{tt}-\varDelta u+u+(|x|^{-4}\ast |u|^2)u=0$ in spatial dimension $d \geq 5$. We utilize the strategy of Ibrahim et al. (2011) derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering can be reduced to disproving the existence of a soliton-like solution. Employing the technique of Pausader (2010), we consider a virial-type identity in the direction orthogonal to the momentum vector to exclude such a solution.


  • Qianyun MiaoSchool of Mathematics
    and System Sciences
    Beihang University
    Beijing 100191, China
  • Jiqiang ZhengLaboratoire de Mathématiques J. A. Dieudonné
    Université Nice Sophia-Antipolis
    06108 Nice Cedex 02, France

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