On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging

We study the global attractors for the dissipative sine--Gordon 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 $O(\varepsilon^{1/2})$.