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arXiv · 2610.35014

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

Kai Yu·Zhiyuan Li·Zhidong Zhang
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Abstract

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.

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Kai Yu, Zhiyuan Li, Zhidong Zhang. 2026-10-26. Inverse heat source problem from terminal data on a set of low Hausdorff dimension. https://arxiv.org/abs/2610.35014

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