Commun. Math. Res., 32 (2016), pp. 289-302.
Published online: 2021-05
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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.
}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2016.04.01}, url = {http://global-sci.org/intro/article_detail/cmr/18901.html} }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.