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Longtime Dynamics of a Semilinear Lamé System

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Abstract

This paper is concerned with longtime dynamics of semilinear Lamé systems

$$\begin{aligned} \partial ^2_t u - \mu \Delta u - (\lambda + \mu ) \nabla \mathrm{div} u + \alpha \partial _t u + f(u) = b, \end{aligned}$$

defined in bounded domains of \({\mathbb {R}}^3\) with Dirichlet boundary condition. Firstly, we establish the existence of finite dimensional global attractors subjected to a critical forcing f(u). Writing \(\lambda + \mu \) as a positive parameter \(\varepsilon \), we discuss some physical aspects of the limit case \(\varepsilon \rightarrow 0\). Then, we show the upper-semicontinuity of attractors with respect to the parameter when \(\varepsilon \rightarrow 0\). To our best knowledge, the analysis of attractors for dynamics of Lamé systems has not been studied before.

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Acknowledgements

The authors thank the anonymous referee for his/her comments and remarks that improved the previous version of this paper.

Funding

L. E. Bocanegra-Rodríguez was supported by CAPES, finance code 001 (Ph.D. Scholarship). M. A. Jorge Silva was partially supported by Fundação Araucária Grant 066/2019 and CNPq Grant 301116/2019-9. T. F. Ma was partially supported by CNPq Grant 312529/2018-0 and FAPESP Grant 2019/11824-0. P. N. Seminario-Huertas was fully supported by INCTMat-CAPES Grant 88887.507829/2020-00.

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Authors and Affiliations

  1. Institute of Mathematical and Computer Sciences, University of São Paulo, São Carlos, SP, 13566-560, Brazil

    Lito Edinson Bocanegra-Rodríguez

  2. Department of Mathematics, State University of Londrina, Londrina, PR, 86057-970, Brazil

    Marcio Antonio Jorge Silva

  3. Department of Mathematics, University of Brasília, Brasília, DF, 70910-900, Brazil

    To Fu Ma & Paulo Nicanor Seminario-Huertas

  4. Academic Department of Mathematics, National University of Callao, Bellavista, 07011, Callao, Peru

    Paulo Nicanor Seminario-Huertas

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  1. Lito Edinson Bocanegra-Rodríguez
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  2. Marcio Antonio Jorge Silva
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  3. To Fu Ma
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  4. Paulo Nicanor Seminario-Huertas
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Correspondence to Paulo Nicanor Seminario-Huertas.

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Bocanegra-Rodríguez, L.E., Silva, M.A.J., Ma, T.F. et al. Longtime Dynamics of a Semilinear Lamé System. J Dyn Diff Equat 35, 1435–1456 (2023). https://doi.org/10.1007/s10884-021-09955-7

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