Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds

TL;DR

This paper establishes Hölder stability estimates for recovering a time-dependent potential in a semi-linear wave equation on Lorentzian manifolds using boundary measurements, employing higher order linearization and Gaussian beams.

Contribution

It introduces a novel stability analysis for inverse problems on Lorentzian manifolds without boundary convexity assumptions, utilizing advanced geometric constructions.

Findings

Potential q can be stably recovered from boundary data.

The method does not require convex boundary or unique lightlike geodesic intersections.

The approach extends inverse problem techniques to more general Lorentzian geometries.

Abstract

This paper concerns an inverse boundary value problem of recovering a zeroth order time-dependent term of a semi-linear wave equation on a globally hyperbolic Lorentzian manifold. We show that an unknown potential $q$ in the non-linear wave equation $\square_g u +q u^m=0$, $m\geq 4$, can be recovered in a H\"older stable way from the Dirichlet-to-Neumann map. Our proof is based on the higher order linearization method and the use of Gaussian beams. Unlike some related works, we do not assume that the boundary is convex or that pairs of lightlike geodesics can intersect only once. For this, we introduce some general constructions in Lorentzian geometry. We expect these constructions to be applicable to studies of related problems as well.