A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation

Michiel Bertsch; Danielle Hilhorst; Hirofumi Izuhara; Masayasu Mimura; Tohru Wakasa

doi:10.3934/dcds.2019226

Article Contents

2020, Volume 40, Issue 6: 3117-3142. Doi: 10.3934/dcds.2019226 open access

This issue Previous Article Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions Next Article Signed Radon measure-valued solutions of flux saturated scalar conservation laws

A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation

1.
Dipartimento di Matematica, University of Rome Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy
2.
Istituto per le Applicazioni del Calcolo Mauro Picone, CNR, Rome, Italy
3.
CNRS, Laboratoire de Mathématique, Analyse Numérique et EDP, Université de Paris-Sud, F-91405 Orsay Cedex, France
4.
Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai-nishi, Miyazaki, 889-2192, Japan
5.
Faculty of Engineering, Musashino University, 3-3-3 Ariake, Koto-ku, Tokyo, 135-8181, Japan
6.
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525, Japan
7.
Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, 1-1, Sensui-cho, Tobata, Kitakyushu, 804-8550, Japan

^* Corresponding author: Hirofumi Izuhara
^* Corresponding author: Hirofumi Izuhara

Received: April 2018

Revised: October 2018

Published: March 2020

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Abstract

We consider a mathematical model describing population dynamics of normal and abnormal cell densities with contact inhibition of cell growth from a theoretical point of view. In the first part of this paper, we discuss the global existence of a solution satisfying the segregation property in one space dimension for general initial data. Here, the term segregation property means that the different types of cells keep spatially segregated when the initial densities are segregated. The second part is devoted to singular limit problems for solutions of the PDE system and traveling wave solutions, respectively. Actually, the contact inhibition model considered in this paper possesses quite similar properties to those of the Fisher-KPP equation. In particular, the limit problems reveal a relation between the contact inhibition model and the Fisher-KPP equation.

Keywords:

Mathematics Subject Classification: Primary: 35M31, 35A01, 35R35, 35C07, 35K65; Secondary: 35Q92, 92C17.

Citation:

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Figure 1. Snapshots of dynamics in (1) with compactly supported initial data. The parameter values are $ \alpha = 4 $, $ \beta = 3 $, $ \gamma = 1 $ and $ k = 0.5 $. The solid and dashed curves indicate $ u $ and $ v $, respectively

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Figure 2. The relation between the parameter $ k $ and the wave velocity $ c_k^* $, where the horizontal and vertical axes indicate $ k $ and $ c_k^* $, respectively. The other parameter values are $ \alpha = 4 $, $ \beta = 3 $ and $ \gamma = 1 $

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Figure 3. $ (\varphi, \psi) $-phase planes for (40) with $ k = 2 $ and $ \gamma = 1 $ : (0) $ c = 0 $, (i) $ c = 0.8 $, (ii) $ c = 1 $, (iii) $ c = 1.2 $

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