Homogenization of attractors of non-linear hyperbolic equations with asymptotically degenerate coefficients

A non-linear initial-boundary-value problem for a hyperbolic equation with dissipation is considered in a bounded domain $\Omega$
$$ u^\varepsilon _{tt}+\delta u^\varepsilon _t -\operatorname {div}\bigl (a^\varepsilon (x)\nabla u^\varepsilon\bigr ) +f(u^\varepsilon)=h^\varepsilon (x), $$
where $\delta>0$ and the coefficient $a^\varepsilon (x)$ is of order $\varepsilon ^{3+\gamma}$ $(0\leqslant \gamma<1)$ on the union of spherical annuli of thickness $d_\varepsilon=d\varepsilon^{2+\gamma}$. The annuli are periodically, with period $\varepsilon$, distributed in a bounded domain $\Omega$. Outside the union of the annuli $a^\varepsilon (x)\equiv 1$. The asymptotic behaviour of the solutions and the global attractor of the problem are studied as $\varepsilon \to 0$. It is shown that the homogenization of the problem on each finite time interval leads to a system consisting of a non-linear hyperbolic equation and an ordinary second-order differential equation (with respect to $t$). It is also shown that the global attractor of the initial problem approaches in a certain sense a weak global attractor of the homogenized problem.