Longtime dynamics of a semilinear Lam\'e system

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