Long time dynamics of von Karman evolutions with thermal effects

Abstract

This paper presents a short survey of recent results pertaining tostability 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 equationstrongly 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 °, 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.

Keywords

• Von Karman evolutions
• thermoelastic plates
• attractors
• dimension
• rate of attraction.