The Calderón problem for a nonlocal diffusion equation with time-dependent coefficients

We investigate the Calderón problem for a nonlocal diffusion equation depending on a globally unknown isotropic coefficient \gamma(x,t) . The forward problem is posed on \Omega\times (0,T) for a domain \Omega that is bounded in one direction. We first show that the Dirichlet-to-Neumann map \Lambda_{\gamma} determines \gamma in the measurement set. By studying various properties of the related nonlocal Neumann derivatives \mathcal{N}_{\gamma} , we prove that both quantities \langle\Lambda_{\gamma} f,g\rangle and \langle \mathcal{N}_{\gamma} f,g\rangle carry the same information as long as f,g\colon \mathbb{R}^{n}\setminus\overline{\Omega}\to\mathbb{R} have disjoint supports and \gamma is known in \operatorname{supp}(f)\cup\operatorname{supp}(g) . We obtain the desired global uniqueness theorem using a suitable integral identity for \mathcal{N}_{\gamma} and the Runge approximation property. The results hold for any spatial dimension n\geq 1 . In conclusion, the main observations of this article are twofold: (1) the information of \Lambda_{\gamma} is needed for exterior determination for \gamma , (2) the knowledge of \mathcal{N}_{\gamma} and \gamma in the measurement set is enough to recover \gamma in the interior.