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

The paper is concerned with the exponential attractors for viscoelastic wave model in $$\varOmega \subset \mathbb R^3$$ : $$\begin{aligned} u_{tt}-h_t(0)\varDelta u-\int _0^\infty \partial _sh_t(s)\varDelta u(t-s)\mathrm ds+f(u)=g, \end{aligned}$$ time-dependent memory kernel $$h_t(\cdot )$$ which used to aging phenomena of material. Conti et al. (Am J Math 140(2):349–389, 2018a; Am 140(6):1687–1729, 2018b) recently provided correct mathematical setting and a well-posedness result within novel theory dynamical systems acting on spaces, established by (J Differ Equ 255:1254–1277, 2013), proved existence regularity global attractor. In this work, we further study as well their regularity. We establish an abstract criterion via quasi-stability method introduced originally Chueshov Lasiecka Dyn 16:469–512, 2004), basis technique developed (2018a, b) provide new overcome difficulty lack show And these techniques can be tackle other hyperbolic models.