Flow invariance for nonlinear partial differential delay equations

Ruess, Wolfgang

doi:10.1090/S0002-9947-09-04833-8

Flow invariance for nonlinear partial differential delay equations
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Trans. Amer. Math. Soc. 361 (2009), 4367-4403 Request permission

Abstract:

Several fundamental results on existence, flow-invariance, regularity, and linearized stability of solutions to the nonlinear partial differential delay equation $\dot {u}(t) + Bu(t) \ni F(u_t), t\geq 0, u_0 = \varphi ,$ with $B\subset X\times X$ $\omega -$accretive, are developed for a general Banach space $X.$ In contrast to existing results, with the history-response $F$ globally defined and, at least, Lipschitz on bounded sets, the results are tailored for situations with $F$ 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 $B\equiv 0,$ and (c) the semilinear case.

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