We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think there are two kinds of blow-up phenomenons when the exponent of the nonlinear term is small. It also means there are two kinds of law to determine the critical exponent. In this paper, we obtain a blow-up result and get the estimate of the upper bound of the lifespan in critical and sub-critical cases. All of the results support such a conjecture, although for now, the existence part is still open.

中文翻译：

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具有阻尼和势能的Strauss型波动方程的爆破

我们研究了一类具有阻尼和势能的非线性波动方程，其系数在缩放意义上至关重要，并且仅取决于空间变量。根据较早的工作，当非线性项的指数小时，可能会出现两种爆炸现象。这也意味着确定临界指数有两种规律。在本文中，我们获得了爆炸结果，并获得了临界和次临界情况下寿命上限的估计。所有结果都支持这种推测，尽管到目前为止，存在部分仍未解决。