# American Institute of Mathematical Sciences

November  2012, 11(6): 2351-2369. doi: 10.3934/cpaa.2012.11.2351

## Flow invariance for nonautonomous nonlinear partial differential delay equations

 1 Department of Mathematics, Razi University, Kermanshah, Iran 2 Fakultät für Mathematik, Universität Duisburg-Essen, D-45117 Essen, Germany

Received  March 2011 Revised  May 2011 Published  April 2012

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.
Citation: Seyedeh Marzieh Ghavidel, Wolfgang M. Ruess. Flow invariance for nonautonomous nonlinear partial differential delay equations. Communications on Pure &amp; Applied Analysis, 2012, 11 (6) : 2351-2369. doi: 10.3934/cpaa.2012.11.2351
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