Bounded Non-response Solutions with Liouvillean Forced Frequencies for Nonlinear Wave Equations

In this paper, one dimensional nonlinear wave equations $$\begin{aligned} u_{tt}-u_{xx}+M_{\xi _{2}}u +\varepsilon f(\omega _{1} t, x, u)=0 \end{aligned}$$ with Dirichlet boundery conditions are considered, where $$M_{\xi _{2}}$$ is a real Fourier multiplier, f is a real analytic function with $$f(\omega _{1} t, -x, -u)=-f(\omega _{1} t, x, u)$$ and the forced frequencies $$\omega _1=(1,\alpha )$$ are Liouvillean. We obtain a family of $$C^{\infty }$$ smooth, bounded non-response solutions with Liouvillean forced frequencies. This is based on an infinite dimensional KAM theorem with angle-dependent normal form.