Variable principal coefficients in the MGT equation

Determine which factorizations and geometric conditions remain valid for the Moore–Gibson–Thompson equation when the diffusivity b or sound speed c depends on space or time, and establish the minimal number of experiments required to recover one or more such variable principal coefficients together with the relaxation coefficient.

Background

The MGT analysis relies on the factorization of the operator into a constant-speed wave operator and a first-order operator in time, with the constant ratio c²/b appearing in the latter. If b or c varies in space or time, the principal part changes and the ratio c²/b is no longer constant, generating commutator terms in the wave–ODE decomposition.

The unresolved issue is therefore both structural and inverse: it concerns the survival of suitable factorizations and geometric conditions, as well as the number of measurements needed when more than one coefficient must be identified.

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. 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.

— 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”