Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations

TL;DR

This paper establishes the existence and properties of a finite-dimensional global attractor for a coupled nonlinear wave and thermoelastic plate system, analyzing stability, dimension bounds, and parameter dependence.

Contribution

It proves the existence of a compact global attractor for a coupled PDE system with nonlinear damping, including cases with and without rotational inertia, and studies its stability and parameter sensitivity.

Findings

Existence of a finite-dimensional global attractor.

Stability estimates show exponential convergence of trajectories.

Attractor's fractal dimension bounds are independent of key parameters.

Abstract

We prove the existence of a compact, finite dimensional, global attractor for a coupled PDE system comprising a nonlinearly damped semilinear wave equation and a nonlinear system of thermoelastic plate equations, without any mechanical (viscous or structural) dissipation in the plate component. The plate dynamics is modelled following Berger's approach; we investigate both cases when rotational inertia is included into the model and when it is not. A major part in the proof is played by an estimate--known as stabilizability estimate--which shows that the difference of any two trajectories can be exponentially stabilized to zero, modulo a compact perturbation. In particular, this inequality yields bounds for the attractor's fractal dimension which are independent of two key parameters, namely $\gamma$ and $\kappa$, the former related to the presence of rotational inertia in the plate model and the latter to the coupling terms. Finally, we show the upper semi-continuity of the attractor with respect to these parameters.