Quasi-periodic Solutions for a Class of Higher Dimensional Beam Equation with Quasi-periodic Forcing
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Abstract
This work focuses on higher-dimensional quasi-periodically forced nonlinear beam equation. This means studying 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.
$$\begin{aligned} u_{tt} + ( -\Delta +M_\xi )^2u +\varepsilon \phi (t) ( u+u^3 ) =0, \quad x\in \mathbf {R}^d, t\in \mathbf {R} \end{aligned}$$
Keywords
Beam equation Quasi-periodic solution Infinite dimensional KAM theoryMathematics Subject Classification
37K50 58E05Notes
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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