Quasi-periodic solutions of forced Kirchhoff equation

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• 作者：Riccardo Montalto
• 关键词：Kirchhoff equation ; Quasi ; linear PDEs ; Quasi ; periodic solutions ; Infinite ; dimensional dynamical systems ; KAM for PDEs ; Nash–Moser theory
• 刊名：Nonlinear Differential Equations and Applications NoDEA
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：24
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-9004
• 卷排序：24

文摘

In this paper we prove the existence and the stability of small-amplitude quasi-periodic solutions with Sobolev regularity, for the 1-dimensional forced Kirchhoff equation with periodic boundary conditions. This is the first KAM result for a quasi-linear wave-type equation. The main difficulties are: (i) the presence of the highest order derivative in the nonlinearity which does not allow to apply the classical KAM scheme, (ii) the presence of double resonances, due to the double multiplicity of the eigenvalues of \(-\partial _{xx}\). The proof is based on a Nash–Moser scheme in Sobolev class. The main point concerns the invertibility of the linearized operator at any approximate solution and the proof of tame estimates for its inverse in high Sobolev norm. To this aim, we conjugate the linearized operator to a \(2 \times 2\), time independent, block-diagonal operator. This is achieved by using changes of variables induced by diffeomorphisms of the torus, pseudo-differential operators and a KAM reducibility scheme in Sobolev class.