The defocusing energy-critical klein–Gordon–Hartree equation

We study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution u_tt — ∆u + u + (|x|^-4 * |u|²)u = 0 in spatial dimension d ≥ 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.