Global uniqueness in an inverse problem for time fractional diffusion equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) \times M$, $T>0$, with time-fractional Caputo derivative of order $α\in (0,1) \cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $\partial M$ at fixed time. In the "flat" case where $M$ is a compact subset of $\mathbb R^d$, two out the three coefficients $ρ$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $ρ\partial_t^αu-\textrm{div}(a \nabla u)+ q u=0$ on $(0,T)\times Ω$ are recovered simultaneously.

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