Speaker
Description
Degenerate evolution equations arise when the diffusion or propagation mechanism vanishes on part of the spatial domain. A natural inverse problem is then to determine the location and the strength of the degeneracy from suitable observations of the solution. In this talk I will discuss a class of inverse problems for strongly degenerate evolution equations, starting from one-dimensional parabolic models in which the diffusion coefficient vanishes at an unknown interior point. Boundary measurements can be used to recover the location of the degeneracy and, under suitable assumptions, to obtain uniqueness and stability results. The analysis also leads to the simultaneous identification of other parameters, including the degeneracy exponent and, in some cases, the initial datum. Spectral methods play a central role and also provide a basis for effective numerical reconstruction. I will then describe some recent developments concerning multidimensional parabolic equations and degenerate wave equations. These extensions illustrate how the reconstruction problem changes when explicit spectral representations are no longer available or when finite propagation introduces a critical observation time.