The radiation theory of radial solutions to 3D energy critical wave equations

TL;DR

This paper develops a radiation theory for radial solutions to 3D energy-critical wave equations, classifies solutions with exterior scattering, and proves scattering for all finite-energy radial solutions in the defocusing case.

Contribution

It introduces a comprehensive classification of radial solutions exhibiting exterior scattering and applies this to establish global scattering results for defocusing energy-critical wave equations.

Findings

Classified all solutions with exterior scattering for radial data.

Proved scattering for all finite-energy radial solutions in the defocusing case.

Provided applications to the global behavior of radial solutions.

Abstract

In this work we consider a wide range of energy critical wave equation in 3-dimensional space with radial data. We are interested in exterior scattering phenomenon, in which the asymptotic behaviour of a solutions $u$ to the non-linear wave equation is similar to that of a linear free wave $v_L$ in an exterior region $\{x: |x|>R+|t|\}$, i.e. \[ \lim_{t\rightarrow \pm \infty} \int_{|x|>R+|t|} (|\nabla(u-v_L)|^2 + |u_t-\partial_t v_L|^2) dx = 0. \] We classify all such solutions for a given linear free wave $v_L$ in this work. We also give some applications of our theory on the global behaviours of radial solutions to this kind of equations. In particular we show the scattering of all finite-energy radial solutions to the defocusing energy critical wave equations.