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CPAA

Several fundamental results on existence and flow-invariance
of solutions to the nonlinear nonautonomous partial differential delay
equation $ \dot{u}(t) + B(t)u(t) \ni F(t; u_t), 0 \leq s \leq t, u_s = \varphi, $
with $ B(t)\subset X\times X$ $\omega-$accretive,
are developed for a general Banach space $X.$
In contrast to existing results, with
the history-response $F(t;\cdot)$ globally defined and, at least, Lipschitz
on bounded sets,
the results are tailored for situations
with $F(t;\cdot)$ defined on -- possibly --
thin subsets of the initial-history space $E$ only, and are applied to place
several classes of population models in their
natural $L^1-$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 $B(\cdot)\equiv 0,$ and (c) the semilinear case.

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