Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

We consider inverse problems in space–time ( M , g ), a 4-dimensional Lorentzian manifold. For semilinear wave equations □ g u + H ( x , u ) = f , where □ g denotes the usual Laplace–Beltrami operator, we prove that the source-to-solution map L : f → u | V , where V is a neighborhood of a time-like geodesic μ , determines the topological, differentiable structure and the conformal class of the metric of the space–time in the maximal set, where waves can propagate from μ and return back. Moreover, on a given space–time ( M , g ), the source-to-solution map determines some coefficients of the Taylor expansion of H in u .