A condition on delay for differential equations with discrete state-dependent delay

TL;DR

This paper establishes a new condition on delay functions that ensures well-posedness and the existence of a global attractor for parabolic differential equations with discrete state-dependent delay.

Contribution

It introduces a state-dependent analogue of a previous delay condition, proving well-posedness and the existence of a compact global attractor in the space of continuous functions.

Findings

Established a sufficient condition for well-posedness of delay differential equations

Proved existence of a compact global attractor for the dynamical system

Extended previous results to a broader class of delay functions

Abstract

Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (11) (2009), 3978-3986] is developed. We propose and study a state-dependent analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions $C$. The dynamical system is constructed in $C$ and the existence of a compact global attractor is proved.