Singular Limit and Long-Time Dynamics of Bresse Systems

Abstract

The Bresse system is a valid model for arched beams which reduces to the classical Timoshenko system when the arch curvature $\ell$ is zero. 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 the upper-semicontinuity of their attractors as $\ell \to 0$.