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 ${\dot{H}}^{\frac{1}{2}}\times {\dot{H}}^{-\frac{1}{2}}$ 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 scatter to free waves as $t\to \pm \infty$.

Minor revisions. To appear in Analysis and PDE. 29 pages