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

We investigate the global attractors and synchronization phenomenon of a coupled critical Lame system defined on a smooth bounded domain Omega subset of R-3 with nonlinear damping and nonlinear forces of critical growth. The existence of a unique and finitely dimensional global attractor A(x) is proved in the natural energy space H = [D((-Delta(e))(1/2) )](2) x [(L-2(Omega))(3)](2), where x is the coupling parameter and Delta(e) is the Lame operator. This attractor is further proved to be smooth in the regular space [(H-2(Omega))(3) boolean AND (H-0(1) (Omega))(3)](2) x [(H-0(1) (Omega))(3)](2). We also show that the coupled Lame system can be reduced to a single one when x tends to infinity. Then we can compare dynamics of the coupled and single systems by proving the upper-semicontinuity of their attractors in H-delta = [(H2-delta(Omega))3 boolean AND (H-0(1-delta) (Omega))(3)](2) x [(H-0(1-delta) (Omega))(3)](2) for any delta is an element of (0, 1) as x -> infinity. These results are finally used to study the asymptotic and exponential synchronization phenomena for the coupled Lame system. (c) 2023 Elsevier Inc. All rights reserved.