Pazy's 1983 book on operator semigroups (link to zbMATH)

*Pazy, A.*, **Semigroups of linear operators and applications to partial differential equations**, Applied Mathematical Sciences, 44. New York etc.: Springer-Verlag. VIII, 279 p. (1983). ZBL0516.47023.

has a some sufficient conditions in the non-autonomous case in Chapter 5. And Section VI.9 in the 2000 book by Engel and Nagel (link to zbMATH)

*Engel, Klaus-Jochen; Nagel, Rainer*, **One-parameter semigroups for linear evolution equations**, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.

analyzes the non-autonomous case by rephrasing it as an autonomous problem on a vector-valued function space (but as far as I know, this might often not be overly helpful to tackle concrete PDEs).

It's probably fair to say that, as of today, the non-autonomous situation has still not been understood as well as the autonomous case.