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Flow invariance for nonlinear partial differential delay equations


Author: Wolfgang M. Ruess
Journal: Trans. Amer. Math. Soc. 361 (2009), 4367-4403
MSC (2000): Primary 47J35, 35R10; Secondary 47H06, 47H20, 47N60, 92D25
DOI: https://doi.org/10.1090/S0002-9947-09-04833-8
Published electronically: March 4, 2009
MathSciNet review: 2500891
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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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Additional Information

Wolfgang M. Ruess
Affiliation: Fachbereich Mathematik, Universität Duisburg-Essen, D-45117 Essen, Germany
Email: wolfgang.ruess@uni-due.de

DOI: https://doi.org/10.1090/S0002-9947-09-04833-8
Keywords: Nonlinear partial differential delay equations, accretive operators, flow-invariance, linearized stability, regularity, population models
Received by editor(s): October 2, 2007
Published electronically: March 4, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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