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

TL;DR

This paper proves the existence of a finite-dimensional global attractor for a coupled wave and plate system with localized damping, extending previous results that required full damping, thus advancing understanding of long-term dynamics in acoustic-structure models.

Contribution

It demonstrates that a finite-dimensional global attractor exists for a coupled wave-plate system with localized damping, generalizing prior results that assumed full damping.

Findings

Existence of a finite-dimensional global attractor for the system.

Attractor regularity and dimensionality are established.

Results hold even with localized damping, not just full damping.

Abstract

The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are: 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, Chueshov and Lasiecka (Comm. Pure Appl. Anal., 2007) only in the presence of full interior acoustic damping -- holds even in the case of localised dissipation. This nontrivial generalization is inspired by and consistent with the recent advances in the study of wave equations with nonlinear localised damping.