Partial differential equations with discrete and distributed state-dependent delays

Abstract

This work is an attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with state-dependent delays). We study local existence and long-time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor.

Dynamics of Quasi-Stable Dissipative Systems

“…[121] and the references therein). Partial differential equations with state-dependent delay have been essentially less investigated; see the discussions in the papers REZOUNENKO [190,191] devoted to the parabolic case. The simplest case of a state-dependent delay is a delay explicitly given by a realvalued function Á W R !…”

Wellposedness of abstract differential equations with state‐dependent delay

“…, Walther and the references in these works. Concerning first order ordinary abstract differential equations and partial differential equations with state‐dependent delay, we refer the reader to the early paper by Hernandez, Prokopczyk & Ladeira , the works by Rezounenko , Rezounenko & Wu , Kosovalic, Magpantay, Chen & Wu and some recent interesting papers by Krisztin & Rezounenko , Yunfei, Yuan & Pei , Kosovalic, Chen & Wu and Hernandez, Pierri & Wu .…”

Existence and regularity of pullback attractors for an incompressible non-Newtonian fluid with delays

“…There are many papers on retarded ODEs, PDEs and functional differential equations (see e.g. [43,44,45,46]). For instance, Navier-Stokes equations with delays were systematically studied in [17,18,19].…”