Under the acoustic boundary conditions, the initial boundary value problem of a wave equation with multiple nonlinear source terms is considered. This paper gives the energy functional of regular solutions for the wave equation and proves the decreasing property of the energy functional. Firstly, the existence of a global solution for the wave equation is proved by the Faedo–Galerkin method. Then, in order to obtain the nonexistence of global solutions for the wave equation, a new functional is defined. When the initial energy is less than zero, the special properties of the new functional are proved by the method of contraction. Finally, the conditions for the nonexistence of global solutions of the wave equation with acoustic boundary conditions are analyzed by using these special properties.

中文翻译：

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声学边界条件下多非线性源项波动方程的全局解

在声学边界条件下，考虑了具有多个非线性源项的波动方程的初边值问题。本文给出了波动方程正则解的能量泛函，并证明了能量泛函的递减性质。首先，通过 Faedo-Galerkin 方法证明了波动方程的全局解的存在性。然后，为了获得波动方程全局解的不存在性，定义了一个新的函数。当初始能量小于零时，新泛函的特殊性质由收缩法证明。最后，利用这些特殊性质，分析了具有声学边界条件的波动方程不存在全局解的条件。