A simple way to well-posedness in $H^{1}$ of a delay differential equation from cell biology

TL;DR

This paper applies recent well-posedness results in Sobolev spaces to a delay differential equation model in cell biology, enabling broader initial conditions and ensuring the existence of solutions for stem cell maturation models.

Contribution

It introduces a simplified Sobolev-space approach to establish well-posedness for delay differential equations in biological models, expanding initial condition flexibility.

Findings

Broader class of initial prehistories allowed

Simplified verification of well-posedness conditions

Guarantees the non-emptiness of the solution manifold

Abstract

We present an application of recent well-posedness results in the theory of delay differential equations for ordinary differential equations arXiv:2308.04730 to a generalized population model for stem cell maturation. The weak approach using Sobolev-spaces we take allows for a larger class of initial prehistories and makes checking the requirements for well-posedness of such a model considerably easier compared to previous approaches. In fact the present approach is a possible means to guarantee that the solution manifold is not empty, which is a necessary requirement for a $C^{1}$-approach to work.