Lipschitz stability in the simultaneous recovery of wave speed and initial source
Abstract
We consider a inverse problem for a hyperbolic equation. The objective is to simultaneous recover both the spatially varying coefficient, whose square root represents the wave speed, and the initial displacement from a single pair of Dirichlet-Neumann Cauchy data. By introducing a suitable transformation and differentiating the resulting equation in time, we derive a coupled system for the differences of the unknown coefficient and initial displacement. Combining weighted energy estimates with Carleman estimates, we establish a Lipschitz stability estimate for both unknown quantities in appropriate Sobolev norms. The proof relies on a priori regularity assumptions, non-degeneracy conditions on the initial data, and structural inequalities controlling the interaction between the coefficient and initial-displacement differences. Our result extends coefficient-only stability results to a simultaneous coefficient--initial-displacement identification problem.
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BibTeX
@misc{ma2026lipschitz,
title = {{Lipschitz stability in the simultaneous recovery of wave speed and initial source}},
author = {Shiqi Ma and Qi Zhai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30767},
eprint = {2609.30767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30767},
}
RIS
TY - UNPB
TI - Lipschitz stability in the simultaneous recovery of wave speed and initial source
AU - Shiqi Ma
AU - Qi Zhai
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30767
UR - https://arxiv.org/abs/2609.30767
AB - We consider a inverse problem for a hyperbolic equation. The objective is to simultaneous recover both the spatially varying coefficient, whose square root represents the wave speed, and the initial displacement from a single pair of Dirichlet-Neumann Cauchy data. By introducing a suitable transformation and differentiating the resulting equation in time, we derive a coupled system for the differences of the unknown coefficient and initial displacement. Combining weighted energy estimates with Carleman estimates, we establish a Lipschitz stability estimate for both unknown quantities in appropriate Sobolev norms. The proof relies on a priori regularity assumptions, non-degeneracy conditions on the initial data, and structural inequalities controlling the interaction between the coefficient and initial-displacement differences. Our result extends coefficient-only stability results to a simultaneous coefficient--initial-displacement identification problem.
ER -