Averaging of equations of viscoelasticity with singularly oscillating external forces

Abstract

Given $\rho \in [0,1]$, we consider for $\epsilon \in (0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $${\partial }_{tt}u-\kappa (0)\Delta u-{\int }_0^{\infty }{\kappa }^{\prime }(s)\Delta u(t-s)ds+f(u)=g_0(t)+{\epsilon }^{-\rho }{g}_1(t/\epsilon )$$ together with the {\it averaged} equation $${\partial }_{tt}u-\kappa (0)\Delta u-{\int }_0^{\infty }{\kappa }^{\prime }(s)\Delta u(t-s)ds+f(u)=g_0(t).$$ 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.