The Defocusing Energy-critical Klein-Gordon-Hartree Equation

Abstract

In this paper, we study the scattering theory for the defocusing energy-critical Klein-Gordon equation with a cubic convolution $u_{tt}-\Delta u+u+(|x{|}^{-4}\ast |u{|}^2)u=0$ in the spatial dimension $d\ge 5$. We utilize the strategy in [S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation. Analysis and PDE., 4 (2011), 405-460.] derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the soliton-like solution. Employing technique from [B. Pausader, Scattering for the Beam Equation in Low Dimensions. Indiana Univ. Math. J., 59 (2010), 791-822.], we consider a virial-type identity in the direction orthogonal to the momentum vector so as to exclude such solution.

23 pages. arXiv admin note: substantial text overlap with arXiv:math/0612028

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