Longtime Dynamics of a Semilinear Lamé System

This paper is concerned with longtime dynamics of semilinear Lamé systems ∂ t 2 u - μ Δ u - ( λ + μ ) ∇ div u + α ∂ t u + f ( u ) = b , defined in bounded domains of R 3 with Dirichlet boundary condition. Firstly, we establish the existence of finite dimensional global attractors subjected to a critical forcing f ( u ). Writing λ + μ as a positive parameter ε , we discuss some physical aspects of the limit case ε → 0 . Then, we show the upper-semicontinuity of attractors with respect to the parameter when ε → 0 . To our best knowledge, the analysis of attractors for dynamics of Lamé systems has not been studied before.