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.
Categories: math.AP math.DS math.FA