The defocusing energy-critical wave equation with a cubic convolution

In this paper, we study the theory of the global wellposedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity u_tt - Δu + (|x|^-4∗|u|²)u = 0 in spatial dimension d ≥ 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 global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity u_tt - Δu +|u|^4/(d2)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.