Approximation of trajectories lying on a~global attractor of a~hyperbolic equation with exterior force rapidly oscillating in time

M. I. Vishik ; V. V. Chepyzhov

Sbornik. Mathematics, Tome 194 (2003) no. 9, pp. 1273-1300

Résumé

A quasilinear dissipative wave equation is considered for periodic boundary conditions with exterior force $g(x,t/\varepsilon)$ rapidly oscillating in $t$. It is assumed in addition that, as $\varepsilon\to0+$, the function $g(x,t/\varepsilon)$ converges in the weak sense (in $L_{2,w}^{\mathrm{loc}}(\mathbb R,L_2(\mathbb T^n))$ to a function $\overline g(x)$ and the averaged wave equation (with exterior force $\overline g(x)$ has only finitely many stationary points $\{z_i(x),\,i= 1,\dots,N\}$, each of them hyperbolic. It is proved that the global attractor $\mathscr A_\varepsilon$ of the original equation deviates in the energy norm from the global attractor $\mathscr A_0$ of the averaged equation by a quantity $C\varepsilon^\rho$, where $\rho$ is described by an explicit formula. It is also shown that each piece of a trajectory $u^\varepsilon(t)$ of the original equation lying on $\mathscr A_\varepsilon$ that corresponds to an interval of time-length $C\log(1/\varepsilon)$ can be approximated to within $C_1\varepsilon^{\rho_1}$ by means of finitely many pieces of trajectories lying on unstable manifolds $M^u(z_i)$ of the averaged equation, where an explicit expression for $\rho_1$ is provided.