The Lamé system with fully coupled vector nonlinearity in the critical regime

We investigate the long-time dynamics of a damped semilinear Lamé system in a bounded three-dimensional domain under the action of a fully coupled vector-valued nonlinearity with Sobolev-critical growth. In contrast with the existing literature, where either decomposed or fully uncoupled structures are assumed to the nonlinear term, we address the critical regime in its general coupled vectorial form. By developing new symmetry properties involving the first and second Fréchet derivatives of the nonlinear vector field, we prove that the corresponding dynamical system possesses a regular compact global attractor with finite fractal dimension and additional qualitative properties. The core of this work relies in the treatment of critical fully coupled vector nonlinearities in the Lamé problem.