Attractors of dissipative hyperbolic equations with singularly oscillating external forces

M. I. Vishik ; V. V. Chepyzhov

Matematičeskie zametki, Tome 79 (2006) no. 4, pp. 522-545

Résumé

We study a uniform attractor $\mathscr A^\varepsilon$ for a dissipative wave equation in a bounded domain $\Omega\Subset\mathbb R^n$ under the assumption that the external force singularly oscillates in time; more precisely, it is of the form $g_0(x,t)+\varepsilon^{-\alpha}g_1(x,t/\varepsilon)$, $x\in\Omega$, $t\in\mathbb R$, where $\alpha>0$, $0\varepsilon\leqslant1$. In $E=H_0^1\times L_2$, this equation has an absorbing set $B^\varepsilon$ estimated as $\|B^\varepsilon\|_E\leqslant C_1+C_2\varepsilon^{-\alpha}$ and, therefore, can increase without bound in the norm of $E$ as $\varepsilon\to0+$. Under certain additional constraints on the function $g_1(x,z)$, $x\in\Omega$, $z\in\mathbb R$, we prove that, for $0\alpha\leqslant\alpha_0$, the global attractors $\mathscr A^\varepsilon$ of such an equation are bounded in $E$, i.e., $\|\mathscr A^\varepsilon\|_E\leqslant C_3$, $0\varepsilon\leqslant1$. Along with the original equation, we consider a “limiting” wave equation with external force $g_0(x,t)$ that also has a global attractor $\mathscr A^0$. For the case in which $g_0(x,t)=g_0(x)$ and the global attractor $\mathscr A^0$ of the limiting equation is exponential, it is established that, for $0\alpha\leqslant\alpha_0$, the Hausdorff distance satisfies the estimate $\operatorname{dist}_E(\mathscr A^\varepsilon,\mathscr A^0)\leqslant C\varepsilon^{\eta(\alpha)}$, where $\eta(\alpha)>0$. For $\eta(\alpha)$ and $\alpha_0$, explicit formulas are given. We also study the nonautonomous case in which $g_0=g_0(x,t)$. It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors $\mathscr A^\varepsilon$ from $\mathscr A^0$, similar to those given above.