Averaging of equations of viscoelasticity with singularly oscillating external forces

Given $\rho \in [0,1]$, we consider for $\epsilon \in (0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $${\mathrm{\partial }}_{tt}u-\kappa (0)\mathrm{\Delta }u-{\int }_0^{\mathrm{\infty }}{\kappa }^{\mathrm{\prime }}(s)\mathrm{\Delta }u(t-s)ds+f(u)=g_0(t)+{\epsilon }^{-\rho }{g}_1(t\mathrm{/}\epsilon )$$ together with the {\it averaged} equation $${\mathrm{\partial }}_{tt}u-\kappa (0)\mathrm{\Delta }u-{\int }_0^{\mathrm{\infty }}{\kappa }^{\mathrm{\prime }}(s)\mathrm{\Delta }u(t-s)ds+f(u)=g_0(t)\mathrm{.}$$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_{\epsilon }(t,\tau )$ acting on the natural weak energy space $\mathcal{H}$ are shown to possess uniform attractors ${\mathcal{A}}^{\epsilon }$. Within the further assumption $\rho <1$, the family ${\mathcal{A}}^{\epsilon }$ turns out to be bounded in $\mathcal{H}$, uniformly with respect to $\epsilon \in [0,1]$. The convergence of the attractors ${\mathcal{A}}^{\epsilon }$ to the attractor ${\mathcal{A}}^0$ of the averaged equation as $\epsilon \to 0$ is also established.