Global uniqueness in an inverse problem for time fractional diffusion equations

Given (M,g), a compact connected Riemannian manifold of dimension d⩾2, with boundary ∂M, we consider an initial boundary value problem for a fractional diffusion equation on (0,T)×M, T>0, with time-fractional Caputo derivative of order α∈(0,1)∪(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 solutions on a subset of ∂M at fixed time. In the “flat” case where M is a compact subset of Rd, two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation ρ∂tαu−div(a∇u)+qu=0 on (0,T)×M are recovered simultaneously.