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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, Theorems 1.4 and 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?

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    $\begingroup$ Well, in the second paragraph of Chapter 6, Pazy writes: "Most of the results of this and the following sections [...] can be easily extended to the case where $A$ depends on $t$ in a way that insures the existence of an evolution system [...] for the family $\{A(t)\}_{t\in[0,T]}$". Evolution systems are treated in Chapter 5. Maybe you should narrow your question down somewhat to avoid "work (more) with the book" answers. $\endgroup$
    – Hannes
    Commented Oct 17, 2018 at 12:17

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