Reconstruction of strong degeneracy region for parabolic equations and systems
Piermarco Cannarsa, Veronica Danesi, Anna Doubova
Published: 2025/9/17
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 and the coefficient of the zero order term, 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.