Global attractors and synchronization of coupled critical Lamé systems with nonlinear damping

We investigate the global attractors and synchronization phenomenon of a coupled critical Lamé system defined on a smooth bounded domain Ω⊂R3 with nonlinear damping and nonlinear forces of critical growth. The existence of a unique and finitely dimensional global attractor Aϰ is proved in the natural energy space H=[D((−Δe)12)]2×[(L2(Ω))3]2, where ϰ is the coupling parameter and Δe is the Lamé operator. This attractor is further proved to be smooth in the regular space [(H2(Ω))3∩(H01(Ω))3]2×[(H01(Ω))3]2. We also show that the coupled Lamé system can be reduced to a single one when ϰ tends to infinity. Then we can compare dynamics of the coupled and single systems by proving the upper-semicontinuity of their attractors in Hδ=[(H2−δ(Ω))3∩(H01−δ(Ω))3]2×[(H01−δ(Ω))3]2 for any δ∈(0,1) as ϰ→∞. These results are finally used to study the asymptotic and exponential synchronization phenomena for the coupled Lamé system.