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

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.