An Inverse Problem for the Moore Gibson Thompson Equation: Injectivity and Stability

Abstract: We study an inverse problem for the variable coefficient Moore Gibson Thompson (MGT) equation in a bounded domain, where the objective is to recover an unknown initial acceleration from boundary measurements of the acoustic field. We establish both injectivity and stability of the associated measurement operator. Injectivity is established through a unique continuation property by applying a theorem of Tataru for analytically principally normal operators. The argument relies on a factorization of the third order operator into wave type components, combined with Carleman estimates and propagation techniques in the spirit of Tataru. Stability is derived through a microlocal analysis of the forward operator, based on the construction of geometric optics approximate solutions and the identification of an elliptic pseudodifferential operator arising from microlocal back propagation of the data. These results connect the MGT inverse problem with the classical theory of hyperbolic inverse problems, such as photoacoustic tomography, while highlighting new features induced by the third order dynamics and damping.