In this paper, we focus on the class of almost-periodically forced higher-dimensional beam equations $$ u_{tt}+(-\Delta +\mu )^{2}u+\psi (\omega t)u=0,\quad \mu >0, t \in \mathbb{R}, x\in \mathbb{R}^{d}, $$ subject to periodic boundary conditions, where $\psi (\omega t)$ is real analytic and almost-periodic in t. We show the existence of almost-periodic solutions for this equation under some suitable hypotheses. In the proof, we improve the KAM iteration to deal with the infinite-dimensional frequency $\omega =(\omega _{1},\omega _{2},\ldots)$.

中文翻译：

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具有几乎周期强迫的高维梁方程的全维不变托里

在本文中，我们关注的是几乎周期性强迫的高维梁方程的类：$$ u_ {tt} +（-\ Delta + \ mu）^ {2} u + \ psi（\ omega t）u = 0， \ quad \ mu> 0，t \ in \ mathbb {R}，x \ in \ mathbb {R} ^ {d}，$$受周期边界条件的约束，其中$ \ psi（\ omega t）$是实数解析并且几乎是周期性的 我们在一些合适的假设下证明了该方程的几乎周期解的存在。在证明中，我们改进了KAM迭代以处理无限维频率$ \ omega =（\ omega _ {1}，\ omega _ {2}，\ ldots）$。