We consider an abstract wave equation with a propagation speed that depends only on time. We investigate well-posedness results with finite derivative loss in the case where the propagation speed is smooth for positive times, but potentially singular at the initial time.

We prove that solutions exhibit a finite derivative loss under a family of conditions that involve the blow up rate of the first and second derivative of the propagation speed, in the spirit that the weaker is the requirement on the first derivative, the stronger is the requirement on the second derivative. Our family of conditions interpolates between the two limit cases that were already known in the literature.

We also provide the counterexamples that show that, as soon as our conditions fail, solutions can exhibit an infinite derivative loss. The existence of such pathologies was an open problem even in the two extreme cases.

我们考虑传播速度仅取决于时间的抽象波动方程。在正向传播速度平稳但初始时间可能奇异的情况下，我们研究了具有有限导数损失的适定性结果。

我们证明了解在一系列条件下表现出有限的导数损失，这些条件涉及传播速度的一阶和二阶导数的爆破速率，本着这样的精神，对一阶导数的要求越弱，对一阶导数的要求就越强关于二阶导数 我们的条件族插值在文献中已知的两个极限情况之间。

我们还提供了反例，这些反例表明，一旦我们的条件失败，解决方案就会表现出无限的导数损失。即使在两个极端情况下，此类病理的存在也是一个未解决的问题。