Abstract We consider an initial–boundary value problem for the beam vibration equation, which is a fourth-order nonlinear equation with two independent variables. It is shown that under certain conditions on the initial data this problem can be reduced to the Cauchy problem for a countable system of quasilinear ordinary differential equations. Using the method of energy inequalities, we prove that this Cauchy problem has a solution. Based on this, we establish the existence of a local solution of the original initial–boundary value problem and construct it in closed form. A theorem on the uniqueness of a global solution is proved.

中文翻译：

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非线性梁振动方程的初边值问题

摘要 我们考虑梁振动方程的初边值问题，它是一个具有两个自变量的四阶非线性方程。结果表明，在初始数据的一定条件下，该问题可以简化为拟线性常微分方程可数系统的柯西问题。使用能量不等式的方法，我们证明这个柯西问题有解。在此基础上，我们建立了原始初边值问题的局部解的存在性，并以封闭形式构造它。证明了一个关于全局解唯一性的定理。