Longtime Dynamics of a Semilinear Lamé System

Abstract

This paper is concerned with longtime dynamics of semilinear Lamé systems ∂2 t u − μ u − (λ + μ)∇divu + α∂tu + f (u) = b, defined in bounded domains ofR3 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.