Refined estimates for general initial data

Develop refined Carleman estimates adapted to the full variable-coefficient Moore–Gibson–Thompson operator in order to treat general initial data with nonzero initial displacement and initial velocity, rather than only initial conditions satisfying f_1=f_2=0.

Background

The unique-continuation argument factors the Moore–Gibson–Thompson operator into wave-type components and uses a weighted estimate controlling both P_0u and P_0u_t. The displayed estimate is derived under the restriction that the first two components of the initial data vanish, so that only the initial acceleration f contributes to the relevant boundary and initial terms.

The paper explicitly identifies extending this analysis to general initial data as unresolved. Such an extension would require estimates that directly accommodate nonzero initial displacement and initial velocity within the full third-order MGT operator, thereby broadening the unique-continuation framework beyond the inverse problem treated in the paper.

References

The above estimate requires the assumption $f_1 = f_2 = 0$ on the initial conditions. The treatment of general initial data will certainly necessitate refined estimates adapted to the full MGT operator and is therefore left for future investigation.

— An Inverse Problem for the Moore Gibson Thompson Equation: Injectivity and Stability  (2610.26760 - Rodriguez et al., 19 Oct 2026) in Remark following Lemma “bound for both norms,” Section 3, subsection “Preliminaries for the proof of Theorem [Boundary unique continuation for MGT]”