Inverse Problems for Subdiffusion from Observation at an Unknown Terminal Time

TL;DR

This paper investigates inverse subdiffusion problems where the terminal observation time is unknown, establishing uniqueness, stability, and demonstrating the simultaneous recovery of terminal time and spatial parameters through analytical and numerical methods.

Contribution

It provides the first analysis of inverse subdiffusion problems with unknown terminal time, including uniqueness, stability, and numerical validation.

Findings

Unique recovery of terminal time and parameters demonstrated.

Stability estimates established for inverse problems.

Numerical experiments confirm theoretical results.

Abstract

Inverse problems of recovering space-dependent parameters, e.g., initial condition, space-dependent source or potential coefficient, in a subdiffusion model from the terminal observation have been extensively studied in recent years. However, all existing studies have assumed that the terminal time at which one takes the observation is exactly known. In this work, we present uniqueness and stability results for three canonical inverse problems, e.g., backward problem, inverse source and inverse potential problems, from the terminal observation at an unknown time. The subdiffusive nature of the problem indicates that one can simultaneously determine the terminal time and space-dependent parameter. The analysis is based on explicit solution representations, asymptotic behavior of the Mittag-Leffler function, and mild regularity conditions on the problem data. Further, we present several one- and two-dimensional numerical experiments to illustrate the feasibility of the approach.