A KAM Theorem for Higher Dimensional Wave Equations Under Nonlocal Perturbation

Abstract

In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional wave equation under nonlocal perturbation $$\begin{aligned} u_{tt}-\triangle u +M_\xi u +\left( \int _{\mathbb {T}^d} u^2 dx\right) u=0,\quad t\in \mathbb {R},\ x\in \mathbb {T}^d\ \end{aligned}$$where $$M_\xi $$ is a real Fourier multiplier.