Existence and uniqueness of solutions for a hyperbolic Keller–Segel equation

Xiaoming Fu; Quentin Griette; Pierre Magal

doi:10.3934/dcdsb.2020326

Article Contents

2021, Volume 26, Issue 4: 1931-1966. Doi: 10.3934/dcdsb.2020326

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Existence and uniqueness of solutions for a hyperbolic Keller–Segel equation

Université de Bordeaux – Institut de Mathématiques de Bordeaux, 351 cours de la Libération, 33400 Talence, France

^* Corresponding author: Pierre Magal, pierre.magal@u-bordeaux.fr
^* Corresponding author: Pierre Magal, pierre.magal@u-bordeaux.fr

Received: August 2020

Early access: November 2020

Published: April 2021

The research of the first author is supported by China Scholarship Council.

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Abstract

In this work we describe a hyperbolic model with cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call "pressure") which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We analyze the well-posed property of the associated Cauchy problem on the real line. Moreover we obtain a convergence result for bounded initial distributions which are positive and stay away from zero uniformly on the real line.

Keywords:

Cell motion,
nonlinear first-order hyperbolic equation,
nonlinear diffusion.

Mathematics Subject Classification: Primary:35L60, 92C17;Secondary:35D30.

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References

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