Finite vs infinite derivative loss for abstract wave equations with singular time-dependent propagation speed

We consider an abstract wave equation with a propagation speed that depends only on time. investigate well-posedness results finite derivative loss in the case where is smooth for positive times, but potentially singular at initial prove solutions exhibit under family of conditions involve blow up rate first and second speed, spirit weaker requirement derivative, stronger derivative. Our interpolates between two limit cases were already known literature. also provide counterexamples show that, as soon our fail, can infinite loss. The existence such pathologies was open problem even extreme cases.