Determining evolutionary equations by a single passive boundary observation

This work presents a comprehensive study of inverse boundary problems for evolutionary equations with a single passive boundary observation, focusing on hyperbolic and parabolic equations. We establish unique identifiability results for simultaneously determining several key parameters, including the wave/diffusion speed $c$, causal sources $f$ and $h$, and conductivity tensor $\sigma$, under generic conditions. As a central application, we provide a complete resolution to the joint reconstruction problem in thermoacoustic and photoacoustic tomography (TAT/PAT), proving for the first time that both the initial pressure distribution $f$ and the heterogeneous sound speed $c$ can be uniquely and simultaneously determined from just a single, passive, partial boundary observation. We develop a novel, state-of-the-art framework that exploits the full spectral content of the wave field, combining both low-frequency and high-frequency asymptotics. Our approach avoids artificial decoupling assumptions and extends to general domain geometries and general evolutionary equations. This result resolves a long-standing open problem in coupled-physics imaging and provides a rigorous mathematical framework for addressing similar inverse problems in more sophisticated evolutionary setups.