Even though too late but I would like to mention further referenes that I find very intersting. Kisyński, J.. "Sur les opérateurs de Green des problèmes de Cauchy abstraits." Studia Mathematica 23.3 (1964): 285-328. <eudml.org/doc/217075>. Contrarily to most of the other results, this paper treats operators with time dependant domains. Gregor Nickel's thesis and papers with Rainer Nagel contain some intersting examples that emphasise the difficulties in the non autonomous case.

@S.Maths Yes, sorry for the delay.

@Guest You are not quoting correctly, in fact, uniform continuous isn't the same as immediately uniform continuous, the same for diffierentiable.

@S.Maths Necessary and/or sufficient conditions on the operator $(A, D(A))$ to be a generator of a semigroup or a strongly continuous semigroup on a Hilbert space.

Even though this is too late, but maybe someone will be interested in the following paper which seems to answer the question in a different way than what @Igor Khavkine answer's suggestion. 1. Liu K, Rao B, Zhang X (2002) Stabilization of the wave equations with potential and indefinite damping. Journal of Mathematical Analysis and Applications 269: 747–769.