Global uniqueness theorems for linear and nonlinear waves

TL;DR

This paper establishes unique continuation theorems for linear and nonlinear waves, demonstrating that under certain decay conditions and no radiation assumptions, solutions must vanish, using new Carleman estimates.

Contribution

Introduces a novel family of global Carleman estimates and proves optimal unique continuation results for linear and nonlinear waves.

Findings

Solutions vanish under no radiation and decay conditions.

Results are optimal and extend to certain nonlinear wave equations.

New Carleman estimates are developed for the exterior of a null cone.

Abstract

We prove a unique continuation from infinity theorem for regular waves of the form $[ \Box + \mathcal{V} (t, x) ]\phi=0$. Under the assumption of no incoming and no outgoing radiation on specific halves of past and future null infinities, we show that the solution must vanish everywhere. The "no radiation" assumption is captured in a specific, finite rate of decay which in general depends on the $L^\infty$-profile of the potential $\mathcal{V}$. We show that the result is optimal in many regards. These results are then extended to certain power-law type nonlinear wave equations, where the order of decay one must assume is independent of the size of the nonlinear term. These results are obtained using a new family of global Carleman-type estimates on the exterior of a null cone. A companion paper to this one explores further applications of these new estimates to such nonlinear waves.