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Attractors for semilinear wave equations with localized damping and external forces

  • *Corresponding author. Current affiliation: Department of Mathematics, University of Brasília, Brasília 70910-900, DF, Brazil

    *Corresponding author. Current affiliation: Department of Mathematics, University of Brasília, Brasília 70910-900, DF, Brazil 

Dedicated to Professor Tomás Caraballo on occasion of his Sixtieth Birthday

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  • This paper is concerned with long-time dynamics of semilinear wave equations defined on bounded domains of $ \mathbb{R}^3 $ with cubic nonlinear terms and locally distributed damping. The existence of regular finite-dimensional global attractors established by Chueshov, Lasiecka and Toundykov (2008) reflects a good deal of the current state of the art on this matter. Our contribution is threefold. First, we prove uniform boundedness of attractors with respect to a forcing parameter. Then, we study the continuity of attractors with respect to the parameter in a residual dense set. Finally, we show the existence of generalized exponential attractors. These aspects were not previously considered for wave equations with localized damping.

    Mathematics Subject Classification: Primary: 35B41, 35L71, 35B33; Secondary: 35B40.


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  • Figure 1.  The control region $ \omega $ satisfies (GCC). Any ray of geometric optics inside $ \Omega $ hits $ \omega $

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