A differential equation with state-dependent delay from cell population biology

TL;DR

This paper investigates a differential equation with a state-dependent delay in cell population models, establishing conditions for local and global existence of solutions using advanced functional analysis techniques.

Contribution

It provides new theoretical conditions ensuring the existence of solutions for a cell population model with state-dependent delay, extending previous results to the $C^1$-topology.

Findings

Conditions for local semiflow existence established

Adapted Hale and Lunel's theorem for global existence

Proved solutions remain bounded under certain conditions

Abstract

We analyze a differential equation with a state-dependent delay that is implicitly defined via the solution of an ODE. The equation describes an established though little analyzed cell population model. Based on theoretical results of Hartung, Krisztin, Walther and Wu we elaborate conditions for the model ingredients, in particular vital rates, that guarantee the existence of a local semiflow. Here proofs are based on implicit function arguments. To show global existence, we adapt a theorem from a classical book on functional differential equations by Hale and Lunel, which gives conditions under which - if there is no global existence - closed and bounded sets are left for good, to the $C^1$-topology, which is the natural setting when dealing with state-dependent delays. The proof is based on an older result for semiflows on metric spaces.