Smooth Dynamics of Singularly Perturbed Lamé Systems: Quasi-Stability and Continuity of Global Attractors

We investigate the long-term smooth evolution of a semilinear Lamé system within bounded regions of \(\mathbb {R}^3\) ${\mathbb{R}}^3$ , subject to Dirichlet boundary conditions and nonlinear forces of critical nature. The challenge lies in addressing the fact that the semigroups exhibit singular behavior as \(\varepsilon \) $\epsilon$ approaches 0. This parameter \(\varepsilon \) $\epsilon$ appears in the second-order time derivative of the displacement vector, introducing complexities into the model. Our focus is on understanding the dynamics of solutions as \(\varepsilon \rightarrow 0^+\) $\epsilon \to 0^{+}$ . Results regarding the well-posedness of the systems, quasi-stability, existence, finite dimensionality, regularity, and the robustness of attractors are proven.