Unique and Stable Recovery of Space-Variable Order in Multidimensional Subdiffusion

TL;DR

This paper addresses the problem of uniquely identifying and stably recovering a spatially variable order in a multidimensional subdiffusion model using boundary flux measurements, with new theoretical results on identifiability and stability.

Contribution

It provides new unique identifiability results and a conditional Lipschitz stability estimate for recovering the variable order from boundary data, employing resolvent estimates and Laplace transform techniques.

Findings

Unique identifiability from boundary observations.

Conditional Lipschitz stability estimate.

Novel asymptotic expansions of Laplace transform of boundary flux.

Abstract

In this work we investigate the unique identifiability and stable recovery of a spatially dependent variable-order in the subdiffusion model from the boundary flux measurement. We establish several new unique identifiability results from the observation at one point on the boundary without / with the knowledge of medium properties, and a conditional Lipschitz stability estimate when the observation is available on the whole boundary. The analysis crucially employs resolvent estimates in the $L^r(\Omega)$ ($r>d$) spaces, solution representation in the Laplace domain and novel asymptotic expansions of the Laplace transform of the boundary flux at $p= 0$ and $p=1$.