Radiation of the energy-critical wave equation with compact support

Abstract

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions $3\le d\le 5$, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in a sharp sense. In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.