# Talk by Enzo Vitillaro

On February 10th, 2021, Prof. Dr. Enzo Vitillaro (University of Perugia) gave a talk about "**The wave equation with acoustic boundary conditions**" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.

## Abstract

The aim of the talk is to give some recent results, obtained in a collaboration (still in progress) with Delio Mugnolo, on the wave equation with acoustic boundary conditions, which has been subject of a large literature starting from the original analysis of Beale and Rosencrans in the 70’s and continuing up today, where the effect of nonlinear perturbations of any type is studied.

In particular, after summarizing well-posedness and regularity results in the associated energy space, we shall show that for a bounded domain some physically inexplicable stationary solutions make the problem not asymptotically stable, even with an effective damping, while it is asymptotically stable provided that the initial data are restricted to a 1-codimensional subspace, which is invariant under the flow.

This mathematical result leads to a physical re-thinking of the derivation of the model itself in Theoretical Acoustic. In particular, starting from Newton’s Second Law, we shall show that the PDEs appearing in it need to be integrated with an integral condition which is exactly the one found in the stability analysis of it, a fact never observed in the existing literature.