Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations

Abstract

Abstract Homogenisation of global 𝓐ε and exponential 𝓜ε attractors for the damped semi-linear anisotropic wave equation $\begin{array}{} \displaystyle \partial_t ^2u^\varepsilon + y \partial_t u^\varepsilon-\operatorname{div} \left(a\left( \tfrac{x}{\varepsilon} \right)\nabla u^\varepsilon \right)+f(u^\varepsilon)=g, \end{array}$ 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 $\begin{array}{} \displaystyle \operatorname{div}\left(a\left( \tfrac{x}{\varepsilon} \right)\nabla \right) \end{array}$ and its homogenised limit div (ah∇). Consequently, norm-resolvent estimates on the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts 𝓐0 and 𝓜0 are established. These results imply error estimates of the form distX(𝓐ε, 𝓐0) ≤ Cεϰ and $\begin{array}{} \displaystyle \operatorname{dist}^s_X(\mathcal M^\varepsilon, \mathcal M^0) \le C \varepsilon^\varkappa \end{array}$ in the spaces X = L2(Ω) × H–1(Ω) and X = (Cβ(Ω))2. In the natural energy space 𝓔 := $\begin{array}{} \displaystyle H^1_0 \end{array}$(Ω) × L2(Ω), error estimates dist𝓔(𝓐ε, Tε 𝓐0) ≤ $\begin{array}{} \displaystyle C \sqrt{\varepsilon}^\varkappa \end{array}$ and $\begin{array}{} \displaystyle \operatorname{dist}^s_\mathcal{E}(\mathcal M^\varepsilon, \text{T}_\varepsilon \mathcal M^0) \le C \sqrt{\varepsilon}^\varkappa \end{array}$ 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.