Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation

We consider the inverse problem of identification of degenerate diffusion coefficient of the form $x^\alpha a(x)$ in a one dimensional parabolic equation by some extra data. We first prove by energy methods the uniqueness and Lipschitz stability results for the identification of a constant coefficient $a$ and the power $\alpha$ by knowing an interior data at some time. On the other hand, we obtain the uniqueness result for the identification of a general diffusion coefficients $a(x)$ and also the power $\alpha$ form a boundary data on one side of the space interval. The proof is based on global Carleman estimates for a hyperbolic problem and an inversion of the integral transform similar to the Laplace transform. Finally, the theoretical results are satisfactory verified by numerically experiments.