Averaging of equations of viscoelasticity with singularly oscillating external forces

Abstract

Given ρ∈[0,1], we consider for ε∈(0,1] the nonautonomous viscoelastic equation with a singularly oscillating external force∂ttu−κ(0)Δu−∫0∞κ′(s)Δu(t−s)ds+f(u)=g0(t)+ε−ρg1(t/ε) together with the averaged equation∂ttu−κ(0)Δu−∫0∞κ′(s)Δu(t−s)ds+f(u)=g0(t). Under suitable assumptions on the nonlinearity and on the external force, the related solution processes Sε(t,τ) acting on the natural weak energy space H are shown to possess uniform attractors Aε. Within the further assumption ρ<1, the family Aε turns out to be bounded in H, uniformly with respect to ε∈[0,1]. The convergence of the attractors Aε to the attractor A0 of the averaged equation as ε→0 is also established.