Scattering for the radial 3D cubic wave equation

Abstract

Consider the Cauchy problem for the radial cubic wave equation in $1+3$ dimensions with either the focusing or defocusing sign. This problem is critical in ${Ḣ}^{1∕2}\times {Ḣ}^{-1∕2}\left({ℝ}^3\right)$ and subcritical with respect to the conserved energy. Here we prove that if the critical norm of a solution remains bounded on the maximal time interval of existence, then the solution must in fact be global in time and must scatter to free waves as $t\to \pm \infty$.