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Article

Averaging of equations of viscoelasticity with singularly oscillating external forces

Journal de Mathématiques Pures et Appliquées. 2017. Vol. 108. P. 841-868.
Chepyzhov V. V., Pata V., Conti M.

We consider for $\eps\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force together with the averaged equation. Under suitable assumptions on the nonlinearity and on the external force, the related solution processes $S_\eps(t,\tau)$ acting on the natural weak energy space $\H$  are shown to possess uniform attractors $\A^\eps$. Within the further assumption, the family $\A^\eps$ turns out to be bounded in $\H$, uniformly with respect to $\eps\in[0,1]$. The convergence of the attractors $\A^\eps$ to the attractor $\A^0$ of the averaged equation as $\eps\to 0$ is also established.