Inverse problems for heat equation and space–time fractional diffusion equation with one measurement

Abstract Given a connected compact Riemannian manifold ( M , g ) without boundary, dim ⁡ M ≥ 2 , we consider a space–time fractional diffusion equation with an interior source that is supported on an open subset V of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order α ∈ ( 0 , 1 ] , and the space fractional part by ( − Δ g ) β , where β ∈ ( 0 , 1 ] and Δ g is the Laplace–Beltrami operator on the manifold. The case α = β = 1 , which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on V determines the manifold up to a Riemannian isometry.