Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 557 - 586, Volume 22, Issue 3, November 2008      doi:10.3934/dcds.2008.22.557

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Francesca Bucci - Università degli Studi di Firenze, Dipartimento di Matematica Applicata, Via S. Marta 3, 50139 Firenze, Italy (email)
Igor Chueshov - Kharkov National Universit, Department of Mathematics and Mechanics, 4 Svobody sq, 61077 Kharkov, Ukraine (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.
Mathematics Subject Classification:  37L30, 35M20, 74H40.

Received: January 2007;      Revised: January 2008;      Available Online: August 2008.