  The paper is concerned with the exponential attractors for the viscoelastic
wave model in $\Omega\subset \mathbb R^3$:
  $$u_{tt}-h_t(0)\Delta u-\int_0^\infty\partial_sh_t(s)\Delta u(t-s)\mathrm
ds+f(u)=h,$$ with time-dependent memory kernel $h_t(\cdot)$ which is used to
model aging phenomena of the material. Conti et al [Amer. J. Math., 2018]
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, Pata and Temam [J.
Differential Equations, 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. Dynam.
Diff.Eqs.,2004], and on the basis of the theory and technique developed in
[Amer. J. Math., 2018] 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.
