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

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.