Singular limit and dynamics of the Timoshenko system with second sound and past history

This paper is concerned with the Timoshenko system with second sound and past history. We first prove that the Timoshenko system is the singular limit of the system we considered as δ → 0. Then by showing that the system is dissipative, asymptotically compact, and quasi stable, we establish the existence of the global attractor Aδ with finite fractal dimension. Through deep analysis, we also get the further regularity of the attractors, the existence of the generalized exponential attractor. Finally, we prove that the global attractor Aδ is supper-semicontinuous.