In this work, we investigate the problem of finite time blow up as well as the upper bound estimates of lifespan for solutions to small-amplitude semilinear wave equations with mixed nonlinearities $a |u_t|^p+b |u|^q$, posed on asymptotically Euclidean manifolds, which is related to both the Strauss conjecture and Glassey conjecture. In some cases, we obtain existence results, where the lower bound of the lifespan agrees with the upper bound in order. In addition, our results apply for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign condition) and space-independent.

中文翻译：

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渐近欧几里德流形上混合非线性的小幅度半线性波动方程的爆炸

在这项工作中，我们研究了有限时间爆炸问题以及具有混合非线性的小幅度半线性波动方程解的寿命上限估计$a |u_t|^p+b |u|^q$，提出渐近欧几里得流形，这与施特劳斯猜想和格拉西猜想都有关。在某些情况下，我们会得到存在结果，其中寿命的下限与上限按顺序一致。此外，我们的结果适用于半线性阻尼波动方程，当耗散项的系数是可积的（无符号条件）且与空间无关时。