Reconstruction of degeneracy region and power for parabolic equations and systems

Abstract

We address the inverse problem of recovering a degeneracy point within the diffusion coefficient of a one-dimensional complex parabolic equation by observing the normal derivative at one point of the boundary. The strongly degenerate case is analyzed. In particular, we derive sufficient conditions on the initial data that guarantee the stability and uniqueness of the solution obtained from a one-point measurement. Moreover, we present more general uniqueness theorems, which also cover the identification of the initial data, the coefficient of the zero order term and the degeneracy power, using measurements taken over time. Our method is based on a careful analysis of the spectral problem and relies on an explicit form of the solution in terms of Bessel functions. Our investigation also covers the case of real 1-D degenerate parabolic systems of equations coupled with a specific structure. Theoretical results are also supported by numerical simulations.