Exponential Attractor for the Viscoelastic Wave Model with Time-Dependent Memory Kernels

Abstract

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

$$\begin{aligned} u_{tt}-h_t(0)\varDelta u-\int _0^\infty \partial _sh_t(s)\varDelta u(t-s)\mathrm ds+f(u)=g, \end{aligned}$$

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.