A unified Carleman approach to inverse problems for coupled parabolic and hyperbolic-ODE equations
Abstract
We study inverse coefficient problems for two evolution equations with relaxation: a Kelvin--Voigt viscoelastic equation and the Moore--Gibson--Thompson (MGT) equation arising in nonlinear acoustics. In both cases, we establish the Lipschitz stability for recovering a space-dependent coefficient from a single measurement. Our approach exploits an exact factorization of each operator into a classical evolution operator (the heat operator for the viscoelastic system, or the wave operator for the MGT equation) coupled with a first-order differential operator in time. This makes it possible to combine global Carleman estimates with weighted ODE inequalities. For the Kelvin--Voigt model, the data consist of an interior dynamic observation and a Cauchy snapshot at a prescribed positive time. For the MGT equation, a single boundary trace is sufficient under a standard geometric condition. The resulting argument provides a common framework for coefficient inverse problems in evolution equations with relaxation effects.
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BibTeX
@misc{apraiz2026unified,
title = {{A unified Carleman approach to inverse problems for coupled parabolic and hyperbolic-ODE equations}},
author = {Jone Apraiz and Rodrigo Lecaros and Sebastián Zamorano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26157},
eprint = {2610.26157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26157},
}
RIS
TY - UNPB
TI - A unified Carleman approach to inverse problems for coupled parabolic and hyperbolic-ODE equations
AU - Jone Apraiz
AU - Rodrigo Lecaros
AU - Sebastián Zamorano
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26157
UR - https://arxiv.org/abs/2610.26157
AB - We study inverse coefficient problems for two evolution equations with relaxation: a Kelvin--Voigt viscoelastic equation and the Moore--Gibson--Thompson (MGT) equation arising in nonlinear acoustics. In both cases, we establish the Lipschitz stability for recovering a space-dependent coefficient from a single measurement. Our approach exploits an exact factorization of each operator into a classical evolution operator (the heat operator for the viscoelastic system, or the wave operator for the MGT equation) coupled with a first-order differential operator in time. This makes it possible to combine global Carleman estimates with weighted ODE inequalities. For the Kelvin--Voigt model, the data consist of an interior dynamic observation and a Cauchy snapshot at a prescribed positive time. For the MGT equation, a single boundary trace is sufficient under a standard geometric condition. The resulting argument provides a common framework for coefficient inverse problems in evolution equations with relaxation effects.
ER -