On Reducibility of Beam Equation with Quasi-Periodic Forcing Potential (Communications in Mathematical Research, 2021)
In this paper, the Dirichlet boundary value problems of the nonlinearbeam equation $u_{tt} + ∆^2u + αu + ϵϕ(t)(u + u^3) = 0, α > 0$ in the dimension oneis 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 equationadmits a small-amplitude quasi-periodic solution. The proof is based on an infinitedimensional KAM iteration procedure.