We have the abstract evolution equation $$U^{\prime}=A(t)U+F(U), \quad U(0) = U_0.$$ If the operator $A$ is independent of time, we can get local existence of solutions by proving that $A$ is the infinitesimal generator of a $C_0$-semigroup and apply results from Pazy's book (Section 6, Thm. 1.4 and Thm. 1.5).

In the case where $A$ depends on time, can I apply the same theorems?

If I can, what is the method to prove that $A(\cdot)$ is the infinitesimal generator of a $C_0$-semigroup which depends on time?