Given $\rho\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{ tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{ 0}(t)+\varepsilon ^{ -\rho }g_{ 1}(t/\varepsilon ) $$ together with the { \it averaged} equation $$ \partial_{ tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s +f(u)=g_{ 0}(t). $$ Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\varepsilon(t,\tau)$ acting on the natural weak energy space ${ \mathcal H}$ are shown to possess uniform attractors ${ \mathcal A}^\varepsilon$. Within the further assumption $\rho<1$, 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.
Categories: math.AP math.DS