Longtime dynamics of a semilinear Lamé system
This paper is concerned with longtime dynamics of semilinear Lamé systems $$ \partial^2_t u - μΔu - (λ+ μ) \nabla {\rm div} u + α\partial_t u + f(u) = 0, $$ 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 $λ+ μ$ 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 Lamé systems has not been studied before.
math.AP↗