Non-local PDEs with a state-dependent delay term presented by Stieltjes integral

TL;DR

This paper investigates parabolic PDEs with state-dependent delays represented by Stieltjes integrals, establishing well-posedness conditions and proving the existence of a compact global attractor.

Contribution

It introduces a unified framework for discrete and distributed SDDs using Stieltjes integrals and proves key properties like well-posedness and attractor existence.

Findings

Conditions for well-posedness are established.

Existence of a compact global attractor is proved.

The framework includes singular Lebesgue-Stieltjes measures.

Abstract

Parabolic partial differential equations with state-dependent delays (SDDs) are investigated. The delay term presented by Stieltjes integral simultaneously includes discrete and distributed SDDs. The singular Lebesgue-Stieltjes measure is also admissible. The conditions for the corresponding initial value problem to be well-posed are presented. The existence of a compact global attractor is proved.