Finite dimensional attractor for a composite system of wave/plate equations with localised damping

Abstract: The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system consists of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are the existence of a global attractor for the dynamics generated by this composite system as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension -- established in a previous work by Bucci et al. (2007 Commun. Pure Appl. Anal. 6 113-140) only in the presence of full-interior acoustic damping -- holds even in the case of localized dissipation. This nontrivial generalization is inspired by, and consistent with, the recent advances in the study of wave equations with nonlinear localized damping.