Linearly stable KAM tori for one dimensional forced Kirchhoff equations with refined Töplitz-Lipschitz property

Abstract

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 Dirichlet boundary conditionsutt−(1+∫0π|ux|2dx)uxx+Mξu+ϵg(ω¯t,x)=0,u(t,0)=u(t,π)=0, where Mξ is a real Fourier multiplier, g(ω¯t,x)=−g(ω¯t,−x) is real analytic with forced Diophantine frequencies ω¯, ϵ is a small parameter. The proof is based on an improved Kuksin lemma and the off-diagonal decay together with the refined Töplitz-Lipschitz property of the forcing term.