The defocusing energy-critical Klein–Gordon–Hartree equation
Volume 140 / 2015
Colloquium Mathematicum 140 (2015), 31-58
MSC: Primary 35P25; Secondary 35B40, 35Q40, 81U99.
DOI: 10.4064/cm140-1-4
Abstract
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.
