The defocusing energy-critical wave equation with a cubic convolution Changxing MiaoJian ZhangJIqiang Zheng 35P2535B4035Q4081U99.wave-Hartree equationConcentration compactnessMorawetz estimateExtended causalityScattering In this paper, we study the theory of the global well-posedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity $u_{tt}-Delta u+(|x|^{-4}*|u|^2)u=0$ in spatial dimension $d\geq5$. The main difficulties are the absence of the classical finite speed of propagation (i.e., the monotonic local energy estimate on the light cone), which is a fundamental property to show global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity $u_{tt}-\Delta u+|u|^{4/(d-2)}u=0$. To compensate for this, we resort to the extended causality and use the strategy derived from concentration compactness ideas. Then, the proof of global well-posedness and scattering is reduced to show the nonexistence of three enemies: finite-time blowup, soliton-like solutions, and low-to-high cascade. We use the Morawetz estimate, the extended causality, and the potential energy concentration to preclude the above three enemies. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5271 10.1512/iumj.2014.63.5271 en Indiana Univ. Math. J. 63 (2014) 993 - 1015 state-of-the-art mathematics http://iumj.org/access/