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.

Reconstruction of degenerate conductivity region for parabolic equations

“…In this section we will perform the explicit computation of the normal derivative ∂ x u a (1, t), where u a (x, t) is the solution to (3). In order to do this we will need an explicit expression of solutions to the associated eigenfunctions and eigenvalues in terms of the Bessel functions of the first kind.…”

“…In the sequel, we will concentrate on the analysis on the lateral problem (3). For each a, let us call u a = u a (x, t) the corresponding solution to (3).…”

“…Given u 0 ∈ C 1 ([0, 1]) let us introduce the map µ u0 : [0, 1) × [0, +∞) → R given by µ u0 (a, t) := ∂ x u a (1, t) , (4) where u a satisfies (3).…”

“…The inverse problem of recovering the first-order coefficient of degenerate parabolic equations is analyzed in Deng and Yang [8] and Kamynin [17]. In a recent paper [3] Cannarsa, Doubova, Yamamoto have analyzed several inverse problems of reconstruction of degenerate diffusion coefficient in a parabolic equation.…”

“…An effective approach to accomplish this, as illustrated here below, is to reformulate the search for the degeneracy as an extremal problem. This is nowadays classical and has been widely implemented in many situations; see for instance Lavrentiev et al [18], Samarskii and Vabishchevich [26], Vogel [29] and Cannarsa et al [3].…”