A one-dimensional logistic like equation with nonlinear and nonlocal diffusion: strong convergence to equilibrium

Ducrot, Arnaud; Manceau, David

doi:10.1090/proc/14971

“…Our model is related to the study of Ducrot et al [5] who introduced a complete model of in-vitro cell dynamics with many different behaviors at the cellular level. Other features of closely related models have been investigated in [4,6,7,11,12].…”

Section: Introductionmentioning

confidence: 99%

Traveling waves with continuous profile for hyperbolic Keller-Segel equation

Griette¹,

Magal²,

Zhao³

2022

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0

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This work describes a hyperbolic model for cell-cell repulsion with population dynamics. We consider the pressure produced by a population of cells to describe their motion. We assume that cells try to avoid crowded areas and prefer locally empty spaces far away from the carrying capacity. Here, our main goal is to prove the existence of traveling waves with continuous profiles. This article complements our previous results about sharp traveling waves. We conclude the paper with numerical simulations of the PDE problem, illustrating such a result.

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“…For Turing and Turing-Hopf bifurcation due to the non-local effect, we refer to Ducrot et al [18] and Song et al [49]. We also refer to Mogliner et al [40], Eftimie et al [23], Ducrot and Magal [21], Ducrot and Manceau [20] for more topics on non-local advection equations. For the derivation of such models, we refer to the work of Bellomo et al [6] and Morale, Capasso and Oelschläger [41].…”

Section: Introductionmentioning

confidence: 99%

Sharp discontinuous traveling waves in a hyperbolic Keller--Segel equation

Fu

¹

,

Griette

²

,

Magal

³

2020

Preprint

0

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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-posedness property of the associated Cauchy problem on the real line. We start from bounded initial conditions and we consider some invariant properties of the initial conditions such as the continuity, smoothness and monotonicity. We also describe in detail the behavior of the level sets near the propagating boundary of the solution and we find that an asymptotic jump is formed on the solution for a natural class of initial conditions. Finally, we prove the existence of sharp traveling waves for this model, which are particular solutions traveling at a constant speed, and argue that sharp traveling waves are necessarily discontinuous. This analysis is confirmed by numerical simulations of the PDE problem.

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“…For Turing and Turing-Hopf bifurcation due to the non-local effect, we refer to Ducrot et al [16] and Song et al [37]. We also refer to Mogliner et al [30], Eftimie et al [20], Ducrot and Magal [17], Ducrot and Manceau [18] for more topics on non-local advection equations. For the derivation of such models, we refer to the work of Bellomo et al [5] and Morale, Capasso and Oelschläger [31].…”

mentioning

confidence: 99%

Existence and uniqueness of solutions for a hyperbolic Keller�CSegel equation

Fu

¹

,

Griette

²

,

Magal

³

2021

DCDSB

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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.

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A one-dimensional logistic like equation with nonlinear and nonlocal diffusion: strong convergence to equilibrium

Cited by 5 publications

References 27 publications

Traveling waves with continuous profile for hyperbolic Keller-Segel equation

Traveling waves with continuous profile for hyperbolic Keller-Segel equation

Sharp discontinuous traveling waves in a hyperbolic Keller--Segel equation

Existence and uniqueness of solutions for a hyperbolic Keller�CSegel equation

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