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Quasi-periodic Solutions for a Class of Higher Dimensional Beam Equation with Quasi-periodic Forcing

  • Yanling Shi
  • Junxiang Xu
  • Xindong Xu
Article
  • 90 Downloads

Abstract

This work focuses on higher-dimensional quasi-periodically forced nonlinear beam equation. This means studying
$$\begin{aligned} u_{tt} + ( -\Delta +M_\xi )^2u +\varepsilon \phi (t) ( u+u^3 ) =0, \quad x\in \mathbf {R}^d, t\in \mathbf {R} \end{aligned}$$
with periodic boundary conditions, where \(\varepsilon \) is a small positive parameter, \(\phi (t)\) is a real analytic quasi-periodic function in t with frequency vector \(\omega =(\omega _1,\omega _2,\ldots ,\omega _m).\) It is proved that there are many quasi-periodic solutions for the above equation via KAM theory.

Keywords

Beam equation Quasi-periodic solution Infinite dimensional KAM theory 

Mathematics Subject Classification

37K50 58E05 

Notes

Acknowledgements

The authors would like to thank the referees for their invaluable comments and suggestions which help to improve the presentation of this paper.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and PhysicsYancheng Institute of TechnologyYanchengPeople’s Republic of China
  2. 2.Department of MathematicsSoutheast UniversityNanjingPeople’s Republic of China

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