Uniqueness in determining multidimensional domains with unknown initial data

This paper addresses several geometric inverse problems for some linear parabolic systems where the initial data (and sometimes also the coefficients of the equations) are unknown. The goal is to identify a subdomain within a multidimensional set. The non-homogeneous part of the equation is expressed as a function satisfying some specific assumptions near a positive time. We establish uniqueness results by incorporating observations that can be on a part of the boundary or in an interior (small) domain. Through this process, we also derive information about the initial data. The main tools required for the proofs include semigroup theory, unique continuation and time analyticity results.