Quasi-periodic solutions for a class of beam equation system

Yanling Shi; Junxiang Xu

doi:10.3934/dcdsb.2019171

Article Contents

2020, Volume 25, Issue 1: 31-53. Doi: 10.3934/dcdsb.2019171

This issue Previous Article Singular perturbations and scaling Next Article Pullback exponential attractors for the three dimensional non-autonomous Navier-Stokes equations with nonlinear damping

Quasi-periodic solutions for a class of beam equation system

Yanling Shi^1, , and
Junxiang Xu^2,

1.
College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China
2.
Department of Mathematics, Southeast University, Nanjing 211189, China

^* Corresponding author: shiyanling96998@163.com
^* Corresponding author: shiyanling96998@163.com

Received: October 2018

Revised: March 2019

Early access: July 2019

Published: January 2020

The first author is partially supported by NSFC Grant(11801492, 61877052), NSFJS Grant (BK 20170472) and NSF of Jiangsu Higher education Institute of China Grant(18KJB110030). The second author is supported by the NSFC Grant(11871146).

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Abstract

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional beam equation system

$ \left\{ \begin{array}{lll} u_{1tt}+ \Delta^2 u_1 +\sigma u_1 +u_1u_2^2 & = & 0 \\ &&\\ u_{2tt}+ \Delta^2 u_2 +\mu u_2 +u_1^2 u_2 & = & 0 \end{array} \right. $

under periodic boundary conditions, where $ 0<\sigma \in [ \sigma_1,\sigma_2 ], $ $ 0<\mu\in [ \mu_1,\mu_2 ] $ are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system.

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Mathematics Subject Classification: Primary: 37K55; Secondary: 35G30.

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