The Defocusing Energy-critical Klein-Gordon-Hartree Equation

TL;DR

This paper establishes scattering for the defocusing energy-critical Klein-Gordon equation with a cubic convolution in dimensions five and higher, using concentration compactness and virial identities to rule out soliton solutions.

Contribution

It extends scattering theory to a new class of energy-critical Klein-Gordon equations with convolution nonlinearities in higher dimensions.

Findings

Proves global well-posedness and scattering for the equation.

Develops a method to exclude soliton-like solutions.

Adapts concentration compactness and virial techniques for this setting.

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 $d \geq 5$. 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.