A hyperbolic-elliptic-parabolic PDE model describing chemotactic E. coli colonies

TL;DR

This paper analyzes a simplified PDE model of E. coli chemotaxis, establishing local solutions, uniform estimates, and finite-time blow-up behavior, revealing differences from classical Keller-Segel models.

Contribution

It introduces a hyperbolic-elliptic-parabolic PDE system for bacterial chemotaxis and proves existence, uniform bounds, and blow-up phenomena, extending understanding of pattern formation mechanisms.

Findings

Constructed local-in-time solutions for the PDE system.

Proved uniform a priori estimates for solutions.

Identified conditions leading to finite-time blow-up.

Abstract

We study a modified version of an initial-boundary value problem describing the formation of colony patterns of bacteria \textit{Escherichia Coli}. The original system of three parabolic equations was studied numerically and analytically and gave insights into the underlying mechanisms of chemotaxis. We focus here on the parabolic-elliptic-parabolic approximation and the hyperbolic-elliptic-parabolic limiting system which describes the case of pure chemotactic movement without random diffusion. We first construct local-in-time solutions for the parabolic-elliptic-parabolic system. Then we prove uniform \textit{a priori} estimates and we use them along with a compactness argument in order to construct local-in-time solutions for the hyperbolic-elliptic-parabolic limiting system. Finally, we prove that some initial conditions give rise to solutions which blow-up in finite time in the $L^\infty$ norm in all space dimensions. This last result violet is true even in space dimension 1, which is not the case for the full parabolic or parabolic-elliptic Keller-Segel systems.