  We consider an inverse problem of reconstructing a degeneracy point in the
diffusion coefficient in a one-dimensional parabolic equation by measuring the
normal derivative on one side of the domain boundary. We analyze the
sensitivity of the inverse problem to the initial data. We give sufficient
conditions on the initial data for uniqueness and stability for the one-point
measurement and show some examples of positive and negative results. On the
other hand, we present more general uniqueness results, also for the
identification of an initial data by measurements distributed over time. The
proofs are based on an explicit form of the solution by means of Bessel
functions of the first type. Finally, the theoretical results are supported by
numerical experiments.
