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

Abstract

Homogenisation of global $\mathcal{A}^ε$ and exponential $\mathcal{M}^ε$ attractors for the damped semi-linear anisotropic wave equation $\partial_t^2 u^ε+γ\partial_t u^ε-{\rm div} \left(a\left( \tfrac{x}ε \right)\nabla u^ε\right)+f(u^ε)=g$, on a bounded domain $Ω\subset \mathbb{R}^3$, 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 difference between resolvents of the elliptic operator ${\rm div}\left(a\left( \tfrac{x}ε \right)\nabla \right)$ and its homogenised limit ${\rm div}\left(a^h\nabla \right)$. Consequently, norm-resolvent estimates on the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts $\mathcal{A}^0$ and $\mathcal{M}^0$ are established. These results imply error estimates of the form ${\rm dist}_X(\mathcal{A}^ε, \mathcal{A}^0) \le C ε^\varkappa$ and ${\rm dist}^s_X(\mathcal{M}^ε, \mathcal{M}^0) \le C ε^\varkappa$ in the spaces $X =L^2(Ω)\times H^{-1}(Ω)$ and $X =(C^β(\overlineΩ))^2$. In the natural energy space $\mathcal{E} : = H^1_0(Ω) \times L^2(Ω)$, error estimates ${\rm dist}_{\mathcal{E}}(\mathcal{A}^ε, {T}_ε\mathcal{A}^0) \le C \sqrtε^\varkappa$ and ${\rm dist}^s_{\mathcal{E}}(\mathcal{M}^ε, {T}_ε\mathcal{M}^0) \le C \sqrtε^\varkappa$ are established where ${T}_ε$ is first-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditions.