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On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. (English) Zbl 1067.35017
Summary: 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 \(L_2^{\text{loc}}(\mathbb{R}, L_2 (\Omega))\) 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 \(O(\varepsilon^{1/2})\).

35B41 Attractors
35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
34D45 Attractors of solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
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