Some approaches to investigations of partial differential equations with state-dependent delays

Partial differential equations (PDEs) with discrete (concentrated) and distributed state-dependent delays are studied. Investigations of PDEs with discrete state-dependent delays essentially differ from the ones of equations with constant or time-dependent (discrete) delays. The main mathematical difference is that nonlinearities with discrete state-dependent delays (in contrast to constant or time-dependent 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 state-dependent delays, in general, is not well-posed 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 long-time 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.

References:
[1] A.V. Rezounenko, Partial differential equations with discrete and distributed state-dependent delays // Journal of Mathematical Analysis and Applications, Volume 326, Issue 2 (2007), P.1031-1045. (see detailed preprint version: March 22, 2005 - http://arxiv.org/abs/math.DS/0503470).
[2] A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions // Nonlinear Analysis, Series A: Theory, Methods & Applications, Volume 70, Issue 11 (1 June 2009), P.3978-3986.