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

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