Longtime Dynamics of a Semilinear Lamé System

This paper is concerned with longtime dynamics of semilinear Lamé systems $$\begin{aligned} \partial ^2_t u - \mu \Delta (\lambda + ) \nabla \mathrm{div} \alpha _t f(u) = b, \end{aligned}$$ defined in bounded domains $${\mathbb {R}}^3$$ Dirichlet boundary condition. Firstly, we establish the existence finite dimensional global attractors subjected to a critical forcing f(u). Writing $$\lambda $$ as positive parameter $$\varepsilon , discuss some physical aspects limit case \rightarrow 0$$ . Then, show upper-semicontinuity respect when To our best knowledge, analysis for has not been studied before.