Asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity

In this study we consider the von Karman evolution equations accounting for rotational inertial forces along with a nonlinear localized damping. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact attractor Aα,R. The main result of the paper is upper semicontinuity of attractors with respect to rotational inertia terms. The main obstacles are: criticality of the nonlinear term of the equation combined with geometrically constrained dissipation. The flux multiplier used for propagation of the damping does not cooperate with the nonlinear term which is of critical exponent. A new method based on uniform compactness is developed in order to handle such cases. The method allows to establish uniform   smoothness of the elements from the attractor which is the key ingredient in proving that the attractor is upper semicontinuous with respect to the rotational inertia α>0α>0 and conclude that it “converges” to the attractor AR obtained (in the case α=0α=0) in Geredeli et al. (2013)  [23] when α→0α→0.