In this paper, we study the well-posedness of the Cauchy problem for a class of partial differential equations with nonautonomous past $$\begin{aligned} {\left\{ \begin{array}{ll} u'(t)=Bu(t)+\Phi \left( \tilde{u}_t\right) , \quad t\ge 0,&{}\\ u(0)=u^0\in E,\quad u_0=g\in L^{p}(I,E), &{} \end{array}\right. } \end{aligned}$$ (0.1) where \(I=[-r,0]\) (finite delay) or \(I=(-\infty ,0]\) (infinite delay), \(1\le p<\infty \), B is the generator of a \(C_0\)-semigroup on a Banach space E, and \(\Phi \) is a bounded linear operator from \(W^{1,p}(I,E)\) to some Banach space \(Z_B\), an intermediate space related to the operator B, for instance the Favard space \(F_B\) of B. This work generalizes (Maniar and Voigt, in: Goldstein, Nagel, Romanelli (eds) Lecture notes in pure and applied mathematics, vol 234, Marcel Dekker, New York, pp 319–330, 2003) and continues the works (Brendle and Nagel in Discrete Contin Dyn Syst 8:1–24, 2002; Fragnelli in Bull Belg Math Soc 11:133–148, 2004; Fragnelli and Nickel in Differ Integral Equ 16:327–348, 2003; Fragnelli and Tonetto in J Math Anal Appl 289:90–99, 2004; Fragnelli, Delay equations with non autonomous past, Ph.D. thesis, Tubingen, 2002; Nickel and Rhandi in Math Nachr 278:1–13, 2005; Nguyen in J Math Anal Appl 289:301–316, 2004).