Logarithmic stability for determining the damping coefficient for the time-fractional damped wave equation

Abstract

This paper investigates the inverse problem of determining the spatially dependent damping coefficient in a time-fractional damped wave equation, where the damping term is given by a Caputo derivative of order \(α\in(0,1)\). We first prove the well-posedness of the direct problem in exponentially weighted Sobolev spaces. Then, by means of the Fourier--Laplace transform in time, the nonstationary problem is reduced to a family of stationary elliptic equations with complex frequencies. Based on complex geometrical optics solutions and a suitable integral identity, we estimate the Fourier transform of the difference of two damping coefficients in terms of the difference of their Dirichlet-to-Neumann maps. Combining low-frequency estimates with a high-frequency decay argument, we obtain a conditional logarithmic stability result.