Homogenisation of global ?ε and exponential ?ε attractors for the damped semi-linear anisotropic wave equation ∂t2uε+y∂tuε−divaxε∇uε+f(uε)=g,$egin{array}{}displaystylepartial_t ^2u^arepsilon + y partial_t u^arepsilon-operatorname{div} left(aleft( frac{x}{arepsilon} ight)abla u^arepsilon ight)+f(u^arepsilon)=g,end{array}$ on a bounded domain Ω ⊂ ℝ3, is performed. Order-sharp estimates between trajectories uε(t) and their homogenised trajectories u0(t) are established. These estimates are given in terms of the operator-norm difference between resolvents of the elliptic operator divaxε∇$egin{array}{}displaystyleoperatorname{div}left(aleft( frac{x}{arepsilon} ight)abla ight)end{array}$ and its homogenised limit div (ah∇). Consequently, norm-resolvent estimates on the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts ?0 and ?0 are established. These results imply error estimates of the form distX(?ε, ?0) ≤ Cεϰ and distXs(Mε,M0)≤Cεϰ$egin{array}{}displaystyleoperatorname{dist}^s_X(mathcal M^arepsilon, mathcal M^0) le C arepsilon^arkappaend{array}$ in the spaces X = L2(Ω) × H–1(Ω) and X = (Cβ(Ω))2. In the natural energy space ? := H01$egin{array}{}displaystyleH^1_0end{array}$(Ω) × L2(Ω), error estimates dist?(?ε, Tε ?0) ≤ Cεϰ$egin{array}{}displaystyleC sqrt{arepsilon}^arkappaend{array}$ and distEs(Mε,TεM0)≤Cεϰ$egin{array}{}displaystyleoperatorname{dist}^s_mathcal{E}(mathcal M^arepsilon, ext{T}_arepsilon mathcal M^0) le C sqrt{arepsilon}^arkappaend{array}$ are established where Tε is first-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditions.
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