Determination of source terms and coefficients dependent on $n-1$ spatial variable of parabolic and Schr\"odinger equations

Abstract

We establish the uniqueness in the determination of a source term or a coefficient of the zeroth order term of a second-order parabolic equation. Moreover we consider the determination of a potential of the Schr\"odinger equation. For a parabolic equation, an unknown source term and coefficient depend on the time and $n-1$ spatial variables. For the Schr\"odinger equation, we assume that a potential depends on $n-1$ spatial variables. The data are taken on a part of the boundary satisfying some geometric condition.