
Abstract
We study the global attractors for the dissipative sineGordon
type wave equation with time dependent external force $g(x,t)$. We
assume that the function $g(x,t)$ is translationary compact in
$L^{l o c}_2(\mathbb R,L_2 (\Omega))$ and the nonlinear function
$f(u)$ is bounded and satisfies a global Lipschitz condition. If
the Lipschitz constant $K$ is smaller than the first eigenvalue of
the Laplacian with homogeneous Dirichlet conditions and the dissipation
coefficient is large, then the global attractor has a simple
structure: it is the closure of all the values of the unique
bounded complete trajectory of the wave equation. Moreover, the
attractor attracts all the solutions of the equation with
exponential rate.
We also consider the wave equation with rapidly oscillating
external force $g^\varepsilon(x,t)=g(x,t,t/\varepsilon)$ having the average
$g^0(x,t)$ as $\varepsilon\to 0+$. We assume that the function
$g(x,t,\zeta)g^0(x,t)$ has a bounded primitive with respect to
$\zeta$. Then we prove that the Hausdorff distance between the global
attractor $\mathcal A_\varepsilon$ of the original equation and the global
attractor $\mathcal A_0$ of the averaged equation is less than
$O(\varepsilon^{1/2})$.
Mathematics Subject Classification: 35B40, 35L70, 34D45, 34C29.
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