A model of morphogen transport II

TL;DR

This paper analyzes a complex PDE-ODE model of morphogen transport, proving global well-posedness and convergence to a 1D limit as the domain width shrinks, addressing challenges from singular source terms.

Contribution

It extends previous work by establishing well-posedness and convergence results for a coupled PDE-ODE morphogen transport model with singular sources.

Findings

Proved global existence and uniqueness of solutions.

Showed convergence to a 1D system as domain width approaches zero.

Addressed mathematical challenges from singular source terms.

Abstract

A model of morphogen transport consisting of two evolutionary PDEs of reaction-diffusion type and three ODEs posed on a rectangular domain is analysed. We prove that the problem is globally well-posed and that the corresponding solutions converge, as the width of the rectangle tends to zero, to the unique solution of the one dimensional system which was analyzed in the first paper of the series. Main difficulties in the analysis stem from the presence of a singular source term - a Dirac Delta combined with no smoothing effect in the ODE part of the system.