The Defocusing Energy-Critical Wave Equation with a Cubic Convolution
  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 $u_{tt}-\Delta u+(|x|^{-4}\ast|u|^2)u=0$ in spatial dimension $d \geq 5$. 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 the global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity $u_{tt}-\Delta u+|u|^\frac4{d-2}u=0$. To compensate it, we resort to the extended causality and utilize the strategy derived from concentration compactness ideas. Then, the proof of the global well-posedness and scattering is reduced to show the nonexistence of the three enemies: finite time blowup; soliton-like solutions and low-to-high cascade. We will utilize the Morawetz estimate, the extended causality and the potential energy concentration to preclude the above three enemies. 
Changxing Miao, Junyong Zhang and Jiqiang Zheng
math.AP math-ph math.MP