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

We continue the analysis on the model equation arising in the theory of viscoelasticity $$ \partial_{tt} u(t)-\big[1+k_t(0)\big]\Delta u(t) - \int_0^\infty k'_t(s)\Delta u(t-s)\d s + f(u(t)) = g $$ in the presence of a (convex, nonnegative and summable) memory kernel $k_t(\cdot)$ explicitly depending on time. Such a model is apt to describe, for instance, the dynamics of aging viscoelastic materials. Our earlier paper (with C. Giorgi) was concerned with the correct mathematical setting of the problem, and provided a well-posedness result within the novel theory of dynamical systems acting on time-dependent spaces, recently established by Di Plinio et al. In this second work, we focus on the asymptotic properties of the solutions, proving the existence and the regularity of the time-dependent global attractor for the dynamical process generated by the equation. In addition, when $k_t$ approaches a multiple $m\delta_0$ of the Dirac mass at zero as $t\to\infty$, we show that the 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.