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

Abstract

Homogenisation of global ${\mathcal{A}}^{\epsilon }$ and exponential ${\mathcal{M}}^{\epsilon }$ attractors for the damped semi-linear anisotropic wave equation ${\partial }_t^2{u}^{\epsilon }+\gamma {\partial }_t{u}^{\epsilon }-\mathrm{div}\left(a\left(\frac{x}{\epsilon }\right)\nabla u^{\epsilon }\right)+f(u^{\epsilon })=g$, on a bounded domain $\Omega \subset {\mathbb{R}}^3$, is performed. Order-sharp estimates between trajectories $u^{\epsilon }(t)$ 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 $\mathrm{div}\left(a\left(\frac{x}{\epsilon }\right)\nabla \right)$ and its homogenised limit $\mathrm{div}\left(a^h\nabla \right)$. 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 ${\mathrm{dist}}_X({\mathcal{A}}^{\epsilon },{\mathcal{A}}^0)\le C{\epsilon }^{\varkappa }$ and ${\mathrm{dist}}_X^s({\mathcal{M}}^{\epsilon },{\mathcal{M}}^0)\le C{\epsilon }^{\varkappa }$ in the spaces $X=L^2(\Omega )\times H^{-1}(\Omega )$ and $X=(C^{\beta }(\overline{\Omega }){)}^2$. In the natural energy space $\mathcal{E}:=H_0^1(\Omega )\times L^2(\Omega )$, error estimates ${\mathrm{dist}}_{\mathcal{E}}({\mathcal{A}}^{\epsilon },T_{\epsilon }{\mathcal{A}}^0)\le C{\sqrt{\epsilon }}^{\varkappa }$ and ${\mathrm{dist}}_{\mathcal{E}}^s({\mathcal{M}}^{\epsilon },T_{\epsilon }{\mathcal{M}}^0)\le C{\sqrt{\epsilon }}^{\varkappa }$ are established where $T_{\epsilon }$ is first-order correction for the homogenised attractors suggested by asymptotic expansions. Our results are applied to Dirchlet, Neumann and periodic boundary conditions.