The Defocusing Energy-Critical Wave Equation with a Cubic Convolution

TL;DR

This paper establishes the global well-posedness and scattering for an energy-critical wave equation with a cubic convolution nonlinearity in dimensions five and higher, overcoming the lack of finite speed of propagation using advanced analytical techniques.

Contribution

It introduces a novel approach combining extended causality and concentration compactness to handle the nonlocal cubic convolution nonlinearity in energy-critical wave equations.

Findings

Proves global well-posedness and scattering in dimensions d ≥ 5.

Prevents finite time blowup, soliton solutions, and energy cascades.

Utilizes Morawetz estimates and potential energy concentration methods.

Abstract

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.