Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions

TL;DR

This paper investigates partial differential equations with discrete state-dependent delays, establishing conditions for well-posedness and analyzing long-term behavior, including the existence of a global attractor.

Contribution

It introduces an additional assumption on the delay function to ensure well-posedness and studies the asymptotic dynamics of the resulting system.

Findings

Well-posedness is guaranteed under a new assumption on the delay function.

Existence of a compact global attractor is proven.

Long-time asymptotic behavior of solutions is characterized.

Abstract

Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional assumption on the state-dependent delay function to guarantee the well posedness. For the constructed dynamical system we study the long-time asymptotic behavior and prove the existence of a compact global attractor.