Ukrainian mathematical congress  2009
Alexander V. Rezounenko (V.N.Karazin Kharkiv National University, Ukraine) Some approaches to investigations of partial differential equations with statedependent delays Partial differential equations (PDEs) with discrete (concentrated) and distributed statedependent delays are studied. Investigations of PDEs with discrete statedependent delays essentially differ from the ones of equations with constant or timedependent (discrete) delays. The main mathematical difference is that nonlinearities with discrete statedependent delays (in contrast to constant or timedependent discrete delays) are not Lipschitz continuous on the space of continuous functions  the classical space of initial data, where a deep theory of delay equations is developed. As a result, the corresponding initial value problem with discrete statedependent delays, in general, is not wellposed in the space of continuous functions. Several approaches have been proposed and developed since 2005. Our main goal is to present and compare some of these approaches. In the cases when the studied initial value problem generates a dynamical system, we study its longtime asymptotic behavior and prove the existence of a compact global attractor. Our results can be used to treat many applied problems, particularly, the diffusive Nicholson’s blowflies equation.
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