Thank you so much! In case anyone else is reading this: The invariance property that remains to be checked can be found, e.g., in Section 4.2.2, Thm. 4 of [L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions].

I ended up using the result by Kato mentioned in the answer that I posted. Nevertheless, thank you for your remarks!

In my situation, I am given operators $A(t)$ and I want to show the well-posedness as well as that the propagators are contractions. I can probably apply the theorem from Pazy's book and then use a renorming rechnique to obtain the contraction property (still have to spell out the details). I was expecting that a general theory exists for the generation of propagators (analogous to the case of semigroups) that would immediately solve my problem, however, it seems that this case is much more involved than the case of semigroups.

Thank you for your answer. If I'm not mistaken, in both books they do not cover propagators of contractions. Do you know of any reference for that case? Or is that usually shown directly in the specific case at hand?

Hi Julie, would you mind elaborating how the doubling of variables method is related to the coupling method in stochastic PDEs?

"But you said an orange! You can't cut the orange peel any thinner than the atoms." - What about subatomic particles?