Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 957–978 | Cite as

On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

  • Zhenguo LiangEmail author
  • Xuting Wang


In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on \([0,\pi ]\) with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any \(\epsilon \) small enough, there exist a \(0<m_{\epsilon }\le 1\) and one solution \(u_{\epsilon }(t,x)\) with
$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$
where \(u_{\epsilon }(t,x)\) satisfies 1d wave equation
$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$
with Dirichlet boundery conditions on \([0,\pi ]\).



We thank the anonymous referee(s) for invaluable comments and suggestions. During the preparation of this work we benefit of many suggestions and discussions with Professor Geng Jiansheng.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematical Sciences and Key Lab of Mathematics for Nonlinear ScienceFudan UniversityShanghaiChina

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