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Averaging of equations of viscoelasticity with singularly oscillating external forces

Abstract

Given ρ[0,1]\rho\in[0,1], we consider for ε(0,1]\varepsilon\in(0,1] the nonautonomous viscoelastic equation with a singularly oscillating external force ttuκ(0)Δu0κ(s)Δu(ts)ds+f(u)=g0(t)+ερg1(t/ε) \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 ttuκ(0)Δu0κ(s)Δu(ts)ds+f(u)=g0(t). \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 Sε(t,τ)S_\varepsilon(t,\tau) acting on the natural weak energy space H{\mathcal H} are shown to possess uniform attractors Aε{\mathcal A}^\varepsilon. Within the further assumption ρ<1\rho<1, the family Aε{\mathcal A}^\varepsilon turns out to be bounded in H{\mathcal H}, uniformly with respect to ε[0,1]\varepsilon\in[0,1]. The convergence of the attractors Aε{\mathcal A}^\varepsilon to the attractor A0{\mathcal A}^0 of the averaged equation as ε0\varepsilon\to 0 is also established

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