Longtime dynamics of a semilinear Lam\'e system

This paper is concerned with longtime dynamics of semilinear Lam\'e systems $$\partial^2_t u - \mu \Delta u - (\lambda + \mu) \nabla {\rm div} u + \alpha \partial_t u + f(u) = 0,$$ 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 $\varepsilon$, we discuss some physical aspects of the limit case $\varepsilon \to 0$. Then, we show the upper-semicontinuity of attractors with respect to the parameter when $\varepsilon \to 0$. To our best knowledge, the analysis of attractors for dynamics of Lam\'e systems has not been studied before.