# American Institute of Mathematical Sciences

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

 CNRS, Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, 91405 Orsay cedex, France

* Corresponding author: Pierre Roux

Cet article est dédié á la mémoire du Professeur Ezzedine Zahrouni

Received  November 2020 Revised  January 2021 Published  March 2021

We study a modified version of an initial-boundary value problem describing the formation of colony patterns of bacteria 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 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 is true even in space dimension 1, which is not the case for the full parabolic or parabolic-elliptic Keller-Segel systems.

Citation: Danielle Hilhorst, Pierre Roux. A hyperbolic-elliptic-parabolic PDE model describing chemotactic E. Coli colonies. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021033
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