Title: The defocusing energy-critical wave equation with a cubic convolution

Authors: Changxing Miao, Junyong Zhang and Jiqiang Zheng

Issue: Volume 63 (2014), Issue 4, 993-1015


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.