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On uniqueness of positive entire solutions and other properties of linear parabolic equations
On nonautonomous sineGordon type equations with a simple global attractor and some averaging
1.  Institute for Problems of Information Transmission, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 101447, GSP4, Russian Federation 
2.  Institute for Problems of Information Transmission, Russian Academy of Sciences, 101 447 Moscow GSP4, Russian Federation 
3.  Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, 70550 Stuttgart, Germany 
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 $O(\varepsilon^{1/2})$.
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