Attractors for semilinear wave equations with localized damping and external forces

To Fu Ma; Paulo Nicanor Seminario-Huertas

doi:10.3934/cpaa.2020097

Article Contents

2020, Volume 19, Issue 4: 2219-2233. Doi: 10.3934/cpaa.2020097

This issue Previous Article Chain recurrence and structure of

$ \omega $

-limit sets of multivalued semiflows Next Article Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations

Attractors for semilinear wave equations with localized damping and external forces

To Fu Ma^1, , and
Paulo Nicanor Seminario-Huertas^2,

1.
Institute of Mathematical and Computer Sciences, University of São Paulo, São Carlos 13566-590, SP, Brazil
2.
Faculty of Engineering, University of Ricardo Palma, Lima, Peru

^*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

Received: April 30, 2019

Revised: August 31, 2019

Published: January 2020

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Abstract

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.

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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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