On a Parabolic-ODE system of chemotaxis

Mihaela Negreanu; J. Ignacio Tello

doi:10.3934/dcdss.2020016

Article Contents

2020, Volume 13, Issue 2: 279-292. Doi: 10.3934/dcdss.2020016

This issue Previous Article Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity Next Article Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

On a Parabolic-ODE system of chemotaxis

Mihaela Negreanu^1, and
J. Ignacio Tello^2,3, ,

1.
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2.
Departamento de Matemática Aplicada, E.T.S.I. Sistemas Informáticos, Universidad Politécnica de Madrid 28031, Spain
3.
Center for Computation and Simulation, Universidad Politécnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain

^† Corresponding author
^† Corresponding author

Received: April 30, 2017

Revised: March 31, 2018

Published: February 2020

This work has been supported by the Project MTM2017-83391-P from MICINN (Spain).

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Abstract

In this article we consider a coupled system of differential equations to describe the evolution of a biological species. The system consists of two equations, a second order parabolic PDE of nonlinear type coupled to an ODE. The system contains chemotactic terms with constant chemotaxis coefficient describing the evolution of a biological species "$u$" which moves towards a higher concentration of a chemical species "$v$" in a bounded domain of $ \mathbb{R}^n$. The chemical "$v$" is assumed to be a non-diffusive substance or with neglectable diffusion properties, satisfying the equation

$v_t = h(u, v).$

We obtain results concerning the bifurcation of constant steady states under the assumption

$ h_v+χ u h_u>0 $

with growth terms $g$. The Parabolic-ODE problem is also considered for the case $h_v+χ u h_u = 0$ without growth terms, i.e. $g \equiv 0$. Global existence of solutions is obtained for a range of initial data.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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