Two parabolic inverse problems for an equation with unbounded zero-order coefficient

This work is composed of two parts. We prove in the first part the uniqueness of the determination of the unbounded zero-order coefficient in a parabolic equation from boundary measurements. The novelty of our result is that it covers the largest class of unbounded zero-order coefficients. We establish in the second part a multiple logarithm stability inequality for the problem of determining the initial condition from a single interior measurement. As by-product, we obtain an observability inequality for a parabolic equation with unbounded zero-order coefficient. The proof of this observability inequality is based on a new global quantitative unique continuation result for the Schrödinger equation with unbounded potential. For the sake of completeness, we provide in Appendix A a full proof of this result.