Singular limit and long-time dynamics of Bresse systems

The Bresse system is a valid model for arched beams which reduces to the classical Timoshenko system when the arch curvature $\mathrm{\ell }=0$. Our first result shows the Timoshenko system as a singular limit of the Bresse system as $\ell \to 0$. The remaining results are concerned with the long-time dynamics of Bresse systems. In a general framework, allowing nonlinear damping and forcing terms, we prove the existence of a smooth global attractor with finite fractal dimension and exponential attractors as well. We also compare the Bresse system with the Timoshenko system, in the sense of upper semicontinuity of their attractors as $\mathrm{\ell }\to 0$.