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 $\partial_t^2 u^\epsilon +\gamma\partial_t u^\epsilon-{\rm div} \left(a\left( \tfrac{x}{\epsilon} \right)\nabla u^\epsilon \right)+f(u^\epsilon)=g$, 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 ${\rm div}\left(a\left( \tfrac{x}{\epsilon} \right)\nabla \right)$ 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 ${\rm dist}_X(\mathcal{A}^\epsilon, \mathcal{A}^0) \le C \epsilon^\varkappa$ and ${\rm dist}^s_X(\mathcal{M}^\epsilon, \mathcal{M}^0) \le C \epsilon^\varkappa$ in the spaces $X =L^2(\Omega)\times H^{-1}(\Omega)$ and $X =(C^\beta(\overline{\Omega}))^2$. In the natural energy space $\mathcal{E} : = H^1_0(\Omega) \times L^2(\Omega)$, error estimates ${\rm dist}_{\mathcal{E}}(\mathcal{A}^\epsilon, {T}_\epsilon \mathcal{A}^0) \le C \sqrt{\epsilon}^\varkappa$ and ${\rm dist}^s_{\mathcal{E}}(\mathcal{M}^\epsilon, {T}_\epsilon \mathcal{M}^0) \le C \sqrt{\epsilon}^\varkappa$ 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.