Global uniqueness theorems for linear and nonlinear waves

Abstract

We prove a unique continuation from infinity theorem for regular waves of the form $[\square +\mathcal{V}(t,x)]\varphi =0$. 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.

31 pages; some minor corrections, additional clarifications in theorem statements and surrounding discussions

References in corpus (4)

• Uniqueness results for ill posed characteristic problems in curved space-times
• Unique continuation from infinity for linear waves
• On asymptotically flat solutions of Einstein's equations periodic in time II. Spacetimes with scalar-field sources
• On asymptotically flat solutions of Einstein's equations periodic in time I. Vacuum and electrovacuum solutions

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• The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimes
• Lorentzian Calderón problem near the Minkowski geometry
• A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimes
• Resolvent estimates for spacetimes bounded by Killing horizons
• Stationarity of asymptotically flat non-radiating electrovacuum spacetimes
• Control of waves on Lorentzian manifolds with curvature bounds
• Carleman Estimate for Ultrahyperbolic Operators and Improved Interior Control for Wave Equations
• On the rigidity from infinity for nonlinear Alfvén waves
• A uniqueness theorem for 3D semilinear wave equations satisfying the null condition