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September  2008, 22(3): 557-586. doi: 10.3934/dcds.2008.22.557

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

Francesca Bucci 1, and  Igor Chueshov 2,

1. 

Università degli Studi di Firenze, Dipartimento di Matematica Applicata, Via S. Marta 3, 50139 Firenze
2. 

Kharkov National Universit, Department of Mathematics and Mechanics, 4 Svobody sq, 61077 Kharkov, Ukraine

Received  January 2007 Revised  January 2008 Published  August 2008

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: nonlinear damping, Coupled PDE system, global attractor, finite fractal dimension, critical exponent..
Mathematics Subject Classification: 37L30, 35M20, 74H4.
Citation: Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557
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