In this paper, we study a class of nonlinear wave equations with variable exponent sources. By introducing a family of potential wells, we first prove the global existence of solutions with initial data in the potential wells and the finite time blow-up for solutions starting in the unstable sets. The boundedness and asymptotic behavior of global solutions are also concerned. Finally, we obtain the finite time blow-up with high energy initial data.

中文翻译：

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具有可变指数源的阻尼非线性波动方程的稳定集和不稳定集。

在本文中，我们研究了一类具有可变指数源的非线性波动方程。通过引入一系列潜在的井，我们首先用潜在井中的初始数据证明溶液的全局存在性，以及从不稳定集中开始的溶液的有限时间爆炸。整体解的有界性和渐近行为也受到关注。最后，我们获得了具有高能量初始数据的有限时间爆炸。