Uniqueness and stability for inverse source problem for fractional diffusion-wave equations

Abstract

This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at subboundary. Uniqueness result is obtained by using the analyticity and the new established unique continuation principle provided that the coefficients are all temporally independent. We also derive a Lipschitz stability of our inverse source problem under a suitable topology whose norm is given via the adjoint system of the fractional diffusion-wave equation.

Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations

“…While several authors have already addressed the inverse problem of retrieving the space-varying part of the source term in a fractional diffusion equation, see e.g., [12,34], only uniqueness results are available in the mathematical literature, see e.g., [16,19,22,25,33] (see also [20] for some related inverse problem) with the exception of the recent stability result of [9] stated with specific norms. This is not surprising since the classical methods used to build a stability estimate for the source of parabolic (α = 1) or hyperbolic (α = 2) systems, see e.g., [10,14,24,3], do not apply in a straightforward way to time-fractional diffusion equations.…”

“…As far as we know, the only mathematical work dealing with the stability issue of the inverse source problem under consideration in the present article, can be found in [9,18,26]. While the authors of [18,26] studied this problem in the peculiar framework of a cylindrical domain Ω ′ × (−ℓ, ℓ), where Ω ′ is an open subset of R d−1 and ℓ ∈ R + , the Lipschitz stability estimate presented in [9] is derived for α ∈ (1, 2) under a specifically designed topology induced by the adjoint system of the fractional wave equation.…”