Analysis and approximation for inverse potential problem of space-dependent variable-exponent sub-diffusion in heterogeneous medium

Abstract

Anomalous transport in complex heterogeneous medium is often characterized by non-Markovian memory effects and spatially varying trapping properties. From the point of view of continuous-time random walks, such heterogeneity can be described by a position-dependent heavy-tailed waiting-time density $q(t \mid x)\sim C(x)t^{-1-α(x)}$, where $0<α(x)<1$ is the local anomalous exponent. This microscopic interpretation naturally leads to the sub-diffusion with a space-dependent variable exponent $α(x)$. Compared with constant-exponent models, the variable-exponent fractional operators generally do not commute with the spatial Laplacian, which makes both the forward and inverse analysis substantially more delicate. This paper investigates the inverse problem of recovering a spatially dependent radiative or potential coefficient in variable-exponent sub-diffusion from terminal observation data. We formulate the inverse problem as a nonlinear fixed-point equation and prove that the associated fixed-point operator is monotone such that the fixed-point iteration converges decreasingly to the unique solution. We further establish a Lipschitz-type conditional stability estimate for the terminal observation map, which also implies uniqueness of the inverse problem for sufficiently large terminal time. For numerical reconstruction, we develop a fully discrete finite element scheme and derive an error estimate for the inverse problem. Numerical simulations are conducted to validate the effectiveness of the method, even in the presence of noisy data.