The paper is concerned with the exponential attractors for the viscoelastic wave model in \(\varOmega \subset \mathbb R^3\):

with time-dependent memory kernel \(h_t(\cdot )\) which is used to model aging phenomena of the material. Conti et al. (Am J Math 140(2):349–389, 2018a; Am J Math 140(6):1687–1729, 2018b) recently provided the correct mathematical setting for the model and a well-posedness result within the novel theory of dynamical systems acting on time-dependent spaces, recently established by Conti et al. (J Differ Equ 255:1254–1277, 2013), and proved the existence and the regularity of the time-dependent global attractor. In this work, we further study the existence of the time-dependent exponential attractors as well as their regularity. We establish an abstract existence criterion via quasi-stability method introduced originally by Chueshov and Lasiecka (J Dyn Differ Equ 16:469–512, 2004), and on the basis of the theory and technique developed in Conti et al. (2018a, b) we further provide a new method to overcome the difficulty of the lack of further regularity to show the existence of the time-dependent exponential attractor. And these techniques can be used to tackle other hyperbolic models.

该论文涉及\(\varOmega \subset \mathbb R^3\) 中粘弹性波模型的指数吸引子：

与时间相关的内存内核\(h_t(\cdot )\)用于模拟材料的老化现象。康蒂等人。（Am J Math 140(2):349–389, 2018a; Am J Math 140(6):1687–1729, 2018b）最近为模型提供了正确的数学设置，并在新的动力学理论中提供了适定性结果最近由 Conti 等人建立的作用于时间相关空间的系统。(J Differ Equ 255:1254–1277, 2013)，并证明了瞬态全局吸引子的存在性和规律性。在这项工作中，我们进一步研究了瞬态指数吸引子的存在及其规律性。我们通过最初由 Chueshov 和 Lasiecka (J Dyn Differ Equ 16:469–512, 2004) 引入的准稳定性方法建立了一个抽象的存在标准，并基于 Conti 等人开发的理论和技术。(2018a, b) 我们进一步提供了一种新的方法来克服缺乏进一步规律性的困难来证明时间相关指数吸引子的存在。这些技术可用于处理其他双曲线模型。