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We continue the analysis on the model equation arising in the theory ofviscoelasticity $$ \partial_{tt} u(t)-\big[1+k_t(0)\big]\Delta u(t)-\int_0^\infty k'_t(s)\Delta u(t-s) d s + f(u(t)) = g $$ in the presence of a(convex, nonnegative and summable) memory kernel $k_t(\cdot)$ explicitlydepending on time. Such a model is apt to describe, for instance, the dynamicsof aging viscoelastic materials. The earlier paper [4] was concerned with thecorrect mathematical setting of the problem, and provided a well-posednessresult within the novel theory of dynamical systems acting on time-dependentspaces, recently established by Di Plinio {\it et al.}\ [14] In this secondwork, we focus on the asymptotic properties of the solutions, proving theexistence and the regularity of the time-dependent global attractor for thedynamical process generated by the equation. In addition, when $k_t$ approachesa multiple $m\delta_0$ of the Dirac mass at zero as $t\to\infty$, we show thatthe asymptotic dynamics of our problem is close to the one of its formal limit$$\partial_{tt} u(t)-\Delta u(t) -m\Delta\partial_t u(t)+ f(u(t)) = g$$describing viscoelastic solids of Kelvin-Voigt type.

Viscoelasticity with time-dependent memory kernels. Part II: asymptotic behavior of solutions