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Averaging of equations of viscoelasticity with singularly oscillating external forces

Given $ρ\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t)+\varepsilon ^{-ρ}g_{1}(t/\varepsilon ) $$ together with the {\it averaged} equation $$ \partial_{tt} u-κ(0)Δu - \int_0^\infty κ'(s)Δu(t-s) d s +f(u)=g_{0}(t). $$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,τ)$ acting on the natural weak energy space ${\mathcal H}$ are shown to possess uniform attractors ${\mathcal A}^\varepsilon$. Within the further assumption $ρ<1$, the family ${\mathcal A}^\varepsilon$ turns out to be bounded in ${\mathcal H}$, uniformly with respect to $\varepsilon\in[0,1]$. The convergence of the attractors ${\mathcal A}^\varepsilon$ to the attractor ${\mathcal A}^0$ of the averaged equation as $\varepsilon\to 0$ is also established.