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 the sense that at least one of the following is true: (1) $\int_{|x|> |t|} \left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x \ge \eta_1(u) > 0, \ \mathrm{for} \ \mathrm{all} \ t \ge 0 \ \mathrm{or} \ \mathrm{all} \ t \le 0,$ (2) $\int_{|x|> -\varepsilon +|t|} \left( |\partial_t u|^2 + |\nabla u|^2 \right) \mathrm{d}x \ge \eta_2(\varepsilon, u) > 0, \ \mathrm{for} \ \mathrm{all} \ t \in \mathbb{R}, \varepsilon > 0.$ 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.