Two models of partial differential equations with discrete and distributed state-dependent delays

TL;DR

This paper tackles partial differential equations with state-dependent delays, addressing both discrete (concentrated) $state$-dependent delays and their distributed counterparts, a problem not previously treated. The main approach is to approximate the discrete delay term by a sequence of distributed delay terms, all with $state$-dependent delays, and to analyze the resulting systems. The authors establish local existence and study long-time asymptotic behavior, proving that the distributed-delay model admits a $global attractor$ while the discrete-delay model possesses a $trajectory attractor$. These results provide a foundational framework for understanding dynamics of PDEs with $state$-dependent delays and their attractors, with implications for stability and long-term dynamics.

Abstract

This work is the first 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.