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
\(0,\pi ]\).