The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds

Abstract

We consider the fractional anisotropic Calderón problem for the nonlocal parabolic equation $({\partial }_t-{\Delta }_g{)}^{s}u=f$ ($0<s<1$) on closed Riemannian manifolds. More concretely, we can determine the Riemannian manifold $(M,g)$ up to isometry by using the local source-to-solution map in an arbitrarily small open cylinder in the spacetime domain. This can be regarded as a nonlocal analog of the anisotropic Calderón problem in the parabolic setting. We also study several useful properties for nonlocal parabolic operators by using comprehensive spectrum analysis with semigroup theory.

15 pages. All comments are welcome!