Initial value problems for partial differential equations with dependence on the past time derivative are considered; they take the abstract form \[ \begin{aligned} \dot u (t) & = Au(t) + F[t, \dot u(\sigma(t, u(t)))], \; t \in [0, a], \\ u_0 & = \varphi \in C^0([-p, 0]; X), \end{aligned}\] with a Banach space \(X\), and the operator \(A \) is the generator of an analytic semigroup. The consistently bumpy language (singular/plural, past/present etc.), especially in the introduction, did apparently not bother any of the referees whose work is acknowledged.
Three types of solutions are distinguished:

1)

Mild solutions (which solve an associated integral equation),

2)

‘strict solutions’ (which are, roughly speaking, \(C^1 \) w.r. to time on \([0, b]\) for some \( b >0\) (but not necessarily on the past interval \([-p, 0]\), and

3)

\(C^{1 + \alpha}\) solutions (which satisfy \(u(\cdot) \in C^{1+\alpha}(I, X)\) for some interval \(I\) around zero).

An important technique is the use of \(L^q\)-Lipschitz functions, roughly meaning that a time-dependent Lipschitz constant (w.r. to the time argument) is of class \(L^q\) w.r. to time. Existence and uniqueness of mild solutions is proved first, using a contraction argument. This is shown under the alternative assumptions \(T(\cdot) \varphi(0) \in C^0( [0, a], X) \) or \(T(\cdot) \varphi(0) \in C^0( [0, a], X_{\gamma}) \), where \(T(\cdot) \) is the semigroup generated by \(A\) and \(X_{\gamma}\) denotes fractional power spaces.
Additional restrictions on the nonlinearity \(F\) and the delay functional \( \sigma\), together with the condition \(\varphi(0) \in D(A)\), allow to conclude that the mild solutions are also strict. Here the proof is based on earlier Work of Hernandez, Fernandes and Wu.
Under lower smoothness assumptions, but with compactness conditions on the semigroup, a Peano type existence theorem is provided.
Finally, \(C^{1 + \alpha}\) solutions are constructed in Theorem 3.2, using a compatibility condition on the nitial state \( \varphi\). Under a boundedness assumption on the delay functional \( \sigma\), they exist on all of \( [-p, a]\). Several alternative conditions, for example of the type \( \sigma(t,x) < t\) for all \(t>0\), allow to extend these solutions to maximal solutions on \([-p, \infty)\).
Examples with the Dirichlet-Laplace operator and with a logistic-type \(F\), but also more general nonlinearities, conclude the paper.