Abstract
This paper presents a short survey of recent results pertaining to stability and long time behavior of von Karman thermoelastic plates. Questions such as uniform stability - and associated exponential decay rates for the energy function, existence of attractors in the case of internally/externally forced plates along with properties of attractors such as smoothness and dimensionality will be presented. The model considered consists of undamped oscillatory plate equation strongly coupled with heat equation. There are no other sources of dissipation. Nevertheless it will be shown that that the long-time behavior of the nonlinear evolution is ultimately finite dimensional and "smooth". In addition, the obtained estimate for the dimension and the size of the attractor are independent of the rotational inertia parameter \gamma, which is known to change the character of dynamics from hyperbolic (\gamma > 0) to parabolic like (\gamma= 0). Other properties such as additional smoothness of attractors, upper-semicontinuity with respect to parameter \gamma and existence of inertial manifolds are also presented.
Highlights
In what follows below we shall describe model under consideration which is thermoelastic von Karman plate subjected to an external and internal forcing
- for the sake of concretness we limit ourselves to von Karman nolinearities which are representative of major mathematical difficulties encountered
The main aim in this paper is to provide a survey of results pertinent to wellposedness and long time behavior of the thermal von Karman evolutions described by (1), (2). , Particular emphasis will be placed on dependence of regularity and von Karman evolutions with thermal effects long time characteristics with respect to varying parameters 0 ≤ γ ≤ Mγ for some positive constants Mγ
Summary
In what follows below we shall describe model under consideration which is thermoelastic von Karman plate subjected to an external and internal forcing. Recent progress in the area of control theory and inverse problems, [1,3,8,9,36,41,43,51,54,57] has provided a stimulus to the field and produced an array of results on controllability, analyticity (when γ = 0) and uniform stability without any mechanical dissipation It was shown in [2] that linear thermoelastic plates with either hinged or clamped boundary conditions are exponentially stable without any mechanical dissipation, and that the decay rates are independent on the values of rotational parameter γ ≥ 0. R∗ depends on F0 W 2,∞(Ω) and p L2(Ω) only
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call