Time-Dependent Potential Recovery from Local Data on Conformally Transversally Anisotropic Manifolds

Abstract

We study the recovery of a time-dependent potential in a parabolic equation on a conformally transversally anisotropic manifold. The initial state and a lateral Dirichlet datum supported near the front face are used as inputs, while the terminal state and the Neumann trace near the back face are observed. Assuming injectivity of the attenuated geodesic ray transform on the transversal manifold for all sufficiently small constant attenuations, we prove that these partial input-output data uniquely determine the potential. No simplicity assumption is imposed on the transversal manifold. The proof combines a conformal reduction, transversal Gaussian beam quasimodes, boundary Carleman estimates, and geometric optics solutions with prescribed lateral supports.