Attractors and long time behavior of von Karman thermoelastic plates
This paper undertakes a study of asymptotic behavior of solutions corresponding to von Karman thermoelastic plates. A distinct feature of the work is that the model considered has no added dissipation-particularly mechanical dissipation typically added to plate equation when long time-behavior is considered. Thus, the model consists of undamped oscillatory plate equation strongly coupled with heat equation. Nevertheless we are able to show that the ultimate (asymptotic) behavior of the von Karman evolution is described by finite dimensional global attractor. In addition, the obtained estimate for the dimension and the size of the attractor are independent of the rotational inertia parameter γ and heat/thermal capacity κ, where the former is known to change the character of dynamics from hyperbolic (γ>0) to parabolic like (γ=0). Other properties of attractors such as additional smoothness and upper-semicontinuity with respect to parameters γ and κ are also established. The main ingredients of the proofs are (i) sharp regularity of Airy's stress function, and (ii) newly developed (Chueshov and Lasiecka in Memoirs of AMS, in press) "compensated" compactness methods applicable to non-compact dynamics. © 2007 Springer Science+Business Media, LLC.
Applied Mathematics and Optimization
Chueshov, I., & Lasiecka, I. (2008). Attractors and long time behavior of von Karman thermoelastic plates. Applied Mathematics and Optimization, 58 (2), 195-241. https://doi.org/10.1007/s00245-007-9031-8