Homogenization of attractors of non-linear hyperbolic equations with asymptotically degenerate coefficients
Leonid Pankratov, Igor Dmitrievich Chueshov · Sbornik Mathematics · 1999
A non-linear initial-boundary-value problem for a hyperbolic equation with dissipation is considered in a bounded domain {omega} u{sub tt}{sup {epsilon}} + {delta}u{sub t}{sup {epsilon}} - div(a{sup {epsilon}}(x){nabla}u{sup {epsilon}}) + f(u{sup {epsilon}}) = h{sup {epsilon}}(x) where {delta}>0 and the coefficient a{sup {epsilon}}(x) is of order {epsilon}{sup 3+{gamma}} (0{<=}{gamma}<1) on the union of spherical annuli of thickness d{sub {epsilon}}=d{epsilon}{sup 2+{gamma}}. The annuli are periodically, with period {epsilon}, distributed in a bounded domain {omega}. Outside the union of the annuli a{sup {epsilon}}(x){identical_to}1. The asymptotic behaviour of the solutions and the global attractor of the problem are studied as {epsilon}{yields}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.