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

Abstract

We consider the inverse problem of identification of degenerate diffusion coefficient of the form x α 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 α by knowing 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 α form 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.