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January  2005, 12(1): 27-38. doi: 10.3934/dcds.2005.12.27

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

V. V. Chepyzhov 1, , M. I. Vishik 2, and  W. L. Wendland 3,

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

Received  August 2003 Revised  December 2003 Published  December 2004

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 $O(\varepsilon^{1/2})$.
Keywords: time averaging., dissipative wave equation, Sine-Gordon equation, global attractor.
Mathematics Subject Classification: 35B40, 35L70, 34D45, 34C2.
Citation: V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27
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