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.