  We investigate an abstract wave equation with a time-dependent propagation
speed, and we consider both the non-dissipative case, and the case with a
strong damping that depends on a power of the elastic operator. Previous
results show that, depending on the values of the parameters and on the time
regularity of the propagation speed, this equation exhibits either
well-posedness in Sobolev spaces, or well-posedness in Gevrey spaces, or
ill-posedness with severe derivative loss.
  In this paper we examine some critical cases that were left open by the
previous literature, and we show that they fall into the pathological regime.
The construction of the counterexamples requires a redesign from scratch of the
basic ingredients, and a suitable application of Baire category theorem in
place of the usual iteration scheme.
