Uniform observability under vanishing auxiliary damping

Determine whether the observability constant for the viscoelastic wave equation with auxiliary damping term $j^{-1}\beta(x)\partial_tu_j$ remains bounded as the auxiliary damping vanishes, in order to justify passage to the unperturbed problem with no frictional damping.

Background

The paper reviews an approach in which an auxiliary frictional damping term $j^{-1}\beta(x)\partial_tu_j$ is added to the viscoelastic wave equation and then removed by taking the limit $j\to\infty$. Passing to this limit requires an observability estimate whose constant is uniform with respect to $j$.

The authors state that the cited proofs do not establish whether this constant remains bounded as the auxiliary damping disappears. Consequently, obtaining the conclusion for the original problem without frictional damping appears to require an additional argument. This is distinct from the main result of the paper, which proves exponential stability directly under the geometric control condition.