This chapter is devoted to the study of the existence results of local
and maximal solutions on the one hand and the existence and uniqueness results
of mild solutions on the second hand, for the non-autonomous evolution equation
d
withfinitedelay.dtu(t)=A(t)u(t)+f(t,ut), t∈[0,T],subjectedtotheinitial
datum .u0 = φ, where .T 0 is some positive constant. The unbounded operators associated to the non-autonomous system are assumed to be stable family that generates .C0-semigroups, while the nonlinear part is supposed to be continuous. Using some boundedness assumptions on the delayed nonlinear continuous part, we prove the local existence of solution that blows up at the finite time. Under some Lipschitz condition on the nonlinear term, we establish the existence and uniqueness of mild solution. Finally, an example of reaction–diffusion non-autonomous partial functional differential equations is used to illustrate our theoretical obtained results.