Boundary Damping of Wave Equations for Dummies

A research presentation with title "Stability of Waves with Boundary Damping: Sharpening the Observability Knife" which given in late January 2025 in an international online seminar.
Presented by Professor Lassi Paunonen (paunonenmath.com/ Tampere University, Finland)
Abstract of the presentation:
Stability properties of multi-dimensional wave equations with viscous damping have been studied actively in the literature. There are also important results for the case where the damping of the equation acts on the boundary of the spatial domain, but this case has been studied in fewer references. In particular, there is only little information on stability properties of equations whose boundary damping region fails to satisfy the Geometric Control Condition. In this case the wave equation is not exponentially stable. The observability type criteria in the reference [CPSST, 2023] can be used as a tool to derive polynomial or other sub-exponential decay rates for the classical solutions of the equation. However, these observability-type results need to take into account the possible overdamping and this typically leads to sub-optimality in the decay rates. This situation becomes pronounced in the case of boundary dampings. In this talk I will discuss selected features of this sub-optimality and talk about situations where the decay rates can be made sharper using the improvements introduced recently in [PSV, arXiv 2024, Section 7].
[CPSST, 2023]
Ralph Chill, Lassi Paunonen, David Seifert, Reinhard Stahn, and Yuri Tomilov. Non-uniform stability of damped contraction semigroups. Analysis & PDE, 16(5): 1089-1132, 2023. arxiv.org/abs/...
[PSV, arXiv 2024]
Lassi Paunonen, David Seifert and Nicolas Vanspranghe. Admissibility theory in abstract Sobolev scales and transfer function growth at high frequencies. arXiv:2407.2412.14786, 2024. arxiv.org/abs/...