Plate models with state-dependent damping coefficient and their quasi-static limits

We study long-time dynamics of a class of plate models with a state-dependent damping coefficient and their quasi-static limits. We first present the problem in abstract form and then prove the existence of finite-dimensional global attractors and their upper semicontinuity in the quasi-static limit, i.e., in the case when the mass density of plate tends to zero. Our proofs involve a recently developed method based on “compensated” compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, von Karman and Berger plate models with different types of boundary conditions and damping coefficients. Our results can be also applied to the nonlinear wave equations in an arbitrary dimension.