Several fundamental results on existence, flow-invariance, regularity, and linearized stability of solutions to the nonlinear partial differential delay equationu˙(t)+Bu(t)∋F(ut),t≥0,u0=φ,\dot {u}(t) + Bu(t) \ni F(u_t), t\geq 0, u_0 = \varphi ,withB⊂X×XB\subset X\times Xω−\omega -accretive, are developed for a general Banach spaceX.X.In contrast to existing results, with the history-responseFFglobally defined and, at least, Lipschitz on bounded sets, the results are tailored for situations withFFdefined on (possibly) thin subsets of the initial-history spaceEEonly, and are applied to place several classes of population models in their naturalL1−L^1-setting. The main result solves the open problem of a subtangential condition for flow-invariance of solutions in the fully nonlinear case, paralleling those known for the cases of (a) no delay, (b) ordinary delay equations withB≡0,B\equiv 0,and (c) the semilinear case.