Radiation of the energy-critical wave equation with compact support

TL;DR

This paper establishes lower bounds on the exterior energy for solutions to the energy-critical nonlinear wave equation with compact support, showing such solutions must radiate energy unless trivial, and explores implications for the rigidity conjecture.

Contribution

It proves exterior energy lower bounds for nonradial solutions with compact support and addresses the rigidity conjecture, including a new proof for global existence of solutions with the compactness property.

Findings

Nontrivial solutions with compact support must radiate energy.

Established energy lower bounds in space dimensions 3 to 5.

Provided partial results and new proofs related to the rigidity conjecture.

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.