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Abstract

We investigate the inverse problem of recovering an initial source for the wave equation with fractional attenuation, motivated by photoacoustic tomography (PAT). The attenuation is modeled by a Caputo fractional derivative of order $\alpha\in(0,1)$. We establish uniqueness under a geometric foliation condition via an adaptation of two types of Carleman estimates to the fractional setting, prove stability through continuity inequalities for fractional time-derivatives of wave solutions, and derive a reconstruction scheme based on a Neumann series. While our results apply directly to PAT, we expect that the analytic approach and tools employed might be of broader relevance to the analysis of PDEs with memory effects and to inverse problems for attenuated wave models.