Non-linear acoustic oscillations in a strongly inhomogeneous medium
Leonid Pankratov, Igor Dmitrievich Chueshov · 2005
In a bounded domain I?. we study the Dirichlet boundary value problem for a non-linear hyperbolic equation: where 6 is a strictly positive parameter and the coefficient a'(z) equals at3 on the union of sperical annuli of thickness de = ds3. The set of these annuli is periodically, with period E, distributed in R. Outside of this set &(I) = 1. We study the asymptotic behaviour of the solutions of the boundary value problem and the global attractor of the comsponding dynamical system as E - 0. For any finite time interval, the homogenization of the boundary value problem leads to a system consisting of a nonlinear hypekolic equation and an ordinary second order differential equation with respect to the time variable. It is also shown that the global attractor of the initial boundary value problem approaches in a certain sense a weak global attractor of the homogenized problem. +&U: - div (a(z)Vu') + f(u') = h'(z),