On Reducibility of Beam Equation with Quasi-Periodic Forcing Potential

Abstract

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation $u_{tt} + ∆^2u + αu + ϵϕ(t)(u + u^3 ) = 0, α > 0$ in the dimension one is considered, where $u(t, x)$ and $ϕ(t$) are analytic quasi-periodic functions in $t$, and $ϵ$ is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.