Linearly Stable KAM Tori for One Dimensional Forced Kirchhoff Equations under Periodic Boundary Conditions

We prove an abstract infinite dimensional KAM theorem, which could be applied to prove the existence and linear stability of small-amplitude quasi-periodic solutions for one dimensional forced Kirchhoff equations with periodic boundary conditions $$u_{tt}-(1+{\int }_0^{2\pi }\mathrm{\mid }{u}_x{\mathrm{\mid }}^{2}dx)u_{xx}+M_{\xi }u+\epsilon g(\bar{\omega }t,x)=0,u(t,x+2\pi )=u(t,x),$$ where $M_{\xi }$ is a real Fourier multiplier, $g(\bar{\omega }t,x)$ is real analytic with forced Diophantine frequencies $\bar{\omega }$, $\epsilon$ is a small parameter. The paper generalizes the previous results from the simple eigenvalue to the double eigenvalues under the quasi-linear perturbation.