The present talk is concerned with long-time dynamics of wave equations, defined on compact Riemannian manifolds, with boundary, and featuring localized damping and nonlinear forcing terms with supercritical Sobolev growth. The main objective is to construct optimal damping regions with arbitrarily small summed interior/boundary measure that imply the existence of a regular finite-dimensional global attractor. To this end, among other results, we prove a supercritical extension of a unique continuation theorem of Triggiani and Yao.
