| Abstract: |
| We deal with the wave equation in a suitably regular open domain of the Euclidean space, supplied with an acoustic boundary condition on a part of the boundary and a homogeneous Neumann boundary condition on the (possibly empty) remaining part of it. This problem has been proposed a long time ago by Beale and Rosencrans, to model acoustic wave propagation with locally reacting boundary, and it has been the object of a wide literature. The case of non--locally reacting boundaries, when the homogeneous Neumann boundary condition is replaced by the mathematically more attracting homogeneous Dirichlet boundary condition, has been studied as well. The physical derivation of the problem is treated in the talk by the author in SS96. In this talk, we focus on non-locally reacting boundaries without any Dirichlet boundary condition. We first give well-posedness results in the natural energy space and regularity results. Hence, we shall give precise qualitative results for solutions when the domain is bounded and sufficiently regular. The results presented will appear in the Memoirs of the American Mathematical Society and are available at the address https://doi.org/10.48550/arXiv.2105.09219. |
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