I want to learn about evolution equations, that is equations: $$ u'(t)=A(t)u(t)+f(t)$$ where $A(t)$ are unbounded operators on a Banach Space $X$. I'm interesting also in the case of $A(t)$ being bounded and $t\mapsto A(t)$ continuos with the norm topology and the Dyson's and Magnus' expansions.
I'm following the book of Pazy, "Semigroups and linear operators", chapter 5, there's only theory without any example. The book also distinguish two cases, he named them parabolic and hyperbolic, depending on two different conditions on $\{A(t)\}$ to generate an evolution operator $U(t,s)$. I guess hyperbolic is for a kind of wave equation and parabolic for the heat equation, but I haven't seen how to prove the conditions in this two particular cases. Also, Evans has a section about the Heat equation with coefficient depending on time, I don't know how to relate it with the theory of evolution operators.
Are there any general necessary and sufficient conditions on $\{A(t)\}_{t\in[0,T]}$ to generate a strongly continuos evolution operator $U(t,s)$? And viceversa, under what conditions a strongly continuos operator $U(t,s)$ generates unbounded operators $\{A(t)\}_{t\in [0,T]}$ such that $u(t)=U(t,s)x$ is the solution of the PDE above?
Is there any good book on evolution equations?
I recommend the book by Engel and Nagel , Chapter VI, Section 9. Semigroups for Nonautonomous Cauchy Problems. It contains some basic examples.