Flow invariance for nonautonomous nonlinear partial differential delay equations

Seyedeh Marzieh Ghavidel; Wolfgang M. Ruess

doi:10.3934/cpaa.2012.11.2351

Article Contents

2012, Volume 11, Issue 6: 2351-2369. Doi: 10.3934/cpaa.2012.11.2351

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Flow invariance for nonautonomous nonlinear partial differential delay equations

Seyedeh Marzieh Ghavidel^1, and
Wolfgang M. Ruess^2,

1.
Department of Mathematics, Razi University, Kermanshah
2.
Fakultät für Mathematik, Universität Duisburg-Essen, D-45117 Essen

Received: February 28, 2011

Revised: April 30, 2011

Published: October 2012

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Abstract

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.

Keywords:

Nonlinear partial differential delay equations,
accretive operators,
flow-invariance,
population models.

Mathematics Subject Classification: Primary: 47J35, 35R10; Secondary: 47H06, 47H20, 47N60, 92D25.

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