Cooper, S; Savostianov, A; (2019) Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations. Advances in Nonlinear Analysis , 9 (1) pp. 745-787. 10.1515/anona-2020-0024.

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Abstract

Homogenisation of global A ε and exponential Mε attractors for the damped semi-linear anisotropic wave equation ∂ 2 t u ε + γ∂tu ε − div a x ε � ∇u ε � + f(u ε ) = g, on a bounded domain Ω ⊂ R 3 , is performed. Order-sharp estimates between trajectories u ε (t) and their homogenised trajectories u 0 (t) are established. These estimates are given in terms of the operator-norm di�erence between resolvents of the elliptic operator div a x ε � ∇ � and its homogenised limit div � a h∇ � . Consequently, norm-resolvent estimates on the Hausdor� distance between the anisotropic attractors and their homogenised counter-parts A 0 and M0 are established. These results imply error estimates of the form distX(A ε , A 0 ) ≤ Cεκ and dists X(Mε , M0 ) ≤ Cεκ in the spaces X = L 2 (Ω) × H −1 (Ω) and X = (C β (Ω))2 . In the natural energy space E := H 1 0 (Ω) × L 2 (Ω), error estimates distE(A ε , TεA 0 ) ≤ C √ ε κ and dists E(Mε , TεM0 ) ≤ C √ ε κ are established where Tε is �rst-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditio

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