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On uniqueness of positive entire solutions and other properties of linear parabolic equations
On non-autonomous sine-Gordon 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, GSP-4, Russian Federation |
2. | Institute for Problems of Information Transmission, Russian Academy of Sciences, 101 447 Moscow GSP-4, 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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