Extension to space–time-dependent coefficients

Determine whether Lipschitz stability can be established for recovering space–time-dependent friction and relaxation coefficients in the Kelvin–Voigt and Moore–Gibson–Thompson equations, including the prescribed-profile cases b(x,t)=b_*(x,t)+ρ(t)β(x) and α(x,t)=α_*(x,t)+μ(t)κ(x), and identify the additional observations or Carleman strategy required for general space–time coefficients.

Background

The paper proves Lipschitz stability only for a spatial friction coefficient b(x) in the Kelvin–Voigt equation and a spatial relaxation coefficient α(x) in the Moore–Gibson–Thompson equation. Extending these results to coefficients depending on both space and time is unresolved because time differentiation of the coefficient-comparison equation introduces the additional unknowns ∂t(b¹−b²) and ∂t(α¹−α²), whereas the fixed-time identities recover only a single temporal slice of each coefficient.

The authors propose a more structured intermediate problem in which the temporal dependence is known and only spatial amplitudes are unknown. They also indicate that recovering a general space–time coefficient may require additional measurements or a substantially different Carleman argument.

References

The analysis also raises several questions that remain open. We briefly discuss three of them. Space--time-dependent coefficients. It is natural to ask whether the stability results remain valid when $b=b(x,t)$ or $\alpha=\alpha(x,t)$. This is not a direct extension of the arguments developed here. Indeed, differentiating the difference equation in time introduces the additional unknowns $\partial_t(b1-b2)$ and $\partial_t(\alpha1-\alpha2)$, while the fixed-time identities recover only one temporal slice of the coefficient. A plausible intermediate problem is to consider a prescribed temporal profile, for example $b(x,t)=b_\ast(x,t)+\rho(t)\beta(x)$, $\alpha(x,t)=\alpha_\ast(x,t)+\mu(t)\kappa(x)$, where $\rho$ and $\mu$ are known and the spatial amplitudes $\beta$ and $\kappa$ are unknown. Recovering a general space--time coefficient is expected to require additional observations or a different Carleman strategy.

— A unified Carleman approach to inverse problems for coupled parabolic and hyperbolic-ODE equations  (2610.26157 - Apraiz et al., 19 Oct 2026) in Section 5, “Concluding remarks and open problems”

The analysis also raises several questions that remain open. We briefly discuss three of them. Space--time-dependent coefficients. It is natural to ask whether the stability results remain valid when $b=b(x,t)$ or $\alpha=\alpha(x,t)$. This is not a direct extension of the arguments developed here. Indeed, differentiating the difference equation in time introduces the additional unknowns $\partial_t(b1-b2)$ and $\partial_t(\alpha1-\alpha2)$, while the fixed-time identities recover only one temporal slice of the coefficient. A plausible intermediate problem is to consider a prescribed temporal profile, for example $b(x,t)=b_\ast(x,t)+\rho(t)\beta(x)$, $\alpha(x,t)=\alpha_\ast(x,t)+\mu(t)\kappa(x)$, where $\rho$ and $\mu$ are known and the spatial amplitudes $\beta$ and $\kappa$ are unknown. Recovering a general space--time coefficient is expected to require additional observations or a different Carleman strategy. Variable principal coefficients in the MGT equation. Another question is whether the diffusivity $b$ or the sound speed $c$ can depend on space or time and, possibly, be recovered from measurements. Such dependence changes the principal part of the equation and produces commutator terms in the wave--ODE decomposition, since $c2/b$ is no longer constant. It would be interesting to determine which factorizations and geometric conditions survive in this setting, as well as the minimal number of experiments needed to identify more than one coefficient. Numerical reconstruction. The Lipschitz estimates proved in this paper provide a natural starting point for reconstructing $b(x)$ and $\alpha(x)$. A further problem is to design and analyze a stable numerical procedure, for instance through a Carleman-weighted functional or an iterative regularization method, and to study its behavior under discretization and noisy observations.

— A unified Carleman approach to inverse problems for coupled parabolic and hyperbolic-ODE equations  (2610.26157 - Apraiz et al., 19 Oct 2026) in Section 5, “Concluding remarks and open problems”