The Calderón problem for space-time fractional parabolic operators with variable coefficients

Abstract

We study an inverse problem for variable coefficient fractional parabolic operators of the form $({\partial }_t-\mathrm{div}(A(x){\nabla }_x{)}^s+q(x,t)$ for $s\in (0,1)$ and show the unique recovery of $q$ from exterior measured data. Similar to the fractional elliptic case, we use Runge type approximation argument which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift.

arXiv commentA further updated version with updated references and typographical corrections