Well-posedness and asymptotic behavior of a multidimensional model of morphogen transport

TL;DR

This paper rigorously analyzes a multidimensional model of morphogen transport, proving well-posedness and exponential convergence to equilibrium, significantly advancing understanding beyond previous one-dimensional results.

Contribution

It establishes existence, uniqueness, and exponential stability of solutions for a multidimensional morphogen transport model, extending prior one-dimensional analyses.

Findings

Solutions exist and are unique for stationary and evolution problems.

Solutions converge exponentially to equilibrium in $C^1\times C^0$ topology.

Results hold for arbitrary domain dimensions.

Abstract

Morphogen transport is a biological process, occurring in the tissue of living organisms, which is a determining step in cell differentiation. We present rigorous analysis of a simple model of this process, which is a system coupling parabolic PDE with ODE. We prove existence and uniqueness of solutions for both stationary and evolution problems. Moreover we show that the solution converges exponentially to the equilibrium in $C^1\times C^0$ topology. We prove all results for arbitrary dimension of the domain. Our results improve significantly previously known results for the same model in the case of one dimensional domain.