Several fundamental results on existence, flow-invariance, regularity, and linearized stability of solutions to the nonlinear partial differential delay equation
u
˙
(
t
)
+
B
u
(
t
)
∋
F
(
u
t
)
,
t
≥
0
,
u
0
=
φ
,
\dot {u}(t) + Bu(t) \ni F(u_t), t\geq 0, u_0 = \varphi ,
with
B
⊂
X
×
X
B\subset X\times X
ω
−
\omega -
accretive, are developed for a general Banach space
X
.
X.
In contrast to existing results, with the history-response
F
F
globally defined and, at least, Lipschitz on bounded sets, the results are tailored for situations with
F
F
defined on (possibly) thin subsets of the initial-history space
E
E
only, and are applied to place several classes of population models in their natural
L
1
−
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 with
B
≡
0
,
B\equiv 0,
and (c) the semilinear case.