Longtime Dynamics of a Semilinear Lame System

This paper is concerned with longtime dynamics of semilinear Lame systems partial derivative(2)(t)u - mu Delta u-(lambda+mu) del div u + alpha partial derivative(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 lambda+mu as a positive parameter epsilon, we discuss some physical aspects of the limit case epsilon -> 0. Then, we show the upper-semicontinuity of attractors with respect to the parameter when epsilon -> 0. To our best knowledge, the analysis of attractors for dynamics of Lame systems has not been studied before.