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 R d , two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation ρ ∂ t α u − div ( a ∇ u ) + q u = 0 on ( 0 , T ) × M are recovered simultaneously.