Inverse heat source problem from terminal data on a set of low Hausdorff dimension

We study the recovery of a spatial source in the heat equation from terminal observations on a compact interior set E with positive $β$ dimensional Hausdorff content for some $β>d-1$. The prescribed temporal forcing vanishes on a nontrivial interval immediately preceding the observation time. Under a spectral Sobolev prior and a quantitative noncancellation condition on the terminal spectral multipliers, we establish conditional logarithmic stability. For scattered observations on such sets, we derive a sampling inequality under a quasi uniformity assumption. Under explicit sampling and independent noise assumptions, we prove a logarithmic convergence rate in probability for the corresponding Tikhonov reconstructions with a data driven parameter choice. Two numerical experiments using a finite candidate approximation of a fixed Hausdorff set assess discrete spectral sampling stability and illustrate the effects of the waiting interval and measurement noise on reconstruction.