The critical constant $\mu$ (see (1.1)) of time-decaying damping in the scale-invariant case is recently conjectured. It also has been expected that the lifespan estimate is the same as for the associated semilinear heat equations if the constant is in the “heat-like” domain. In this paper, we point out that this is not true if the total integral of the sum of initial position and speed vanishes. In such a case, we have a new type of the lifespan estimates which is closely related to the non-damped case in shifted space dimensions.

中文翻译：

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一维空间中具有尺度不变阻尼的半线性波动方程解的寿命

最近推测了在比例不变情况下的衰减衰减的临界常数（\（1.1））。如果常数在“类热”域中，则还可以预期寿命估计与相关的半线性热方程相同。在本文中，我们指出，如果初始位置和速度之和的总积分消失，则情况并非如此。在这种情况下，我们有一种新型的寿命估算，它与空间尺寸变化的非阻尼情况密切相关。