Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations
Francesca Bucci - Università degli Studi di Firenze, Dipartimento di Matematica Applicata, Via S. Marta 3, 50139 Firenze, Italy (email)
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.
Keywords: Coupled PDE system, global
attractor, finite fractal dimension,
nonlinear damping, critical exponent.
Received: January 2007; Revised: January 2008; Published: August 2008.
2012 Impact Factor1.005