Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects

Laurent Desvillettes; Michèle Grillot; Philippe Grillot; Simona Mancini

doi:10.3934/dcds.2018205

Article Contents

2018, Volume 38, Issue 9: 4675-4692. Doi: 10.3934/dcds.2018205

This issue Previous Article Lyapunov stability for regular equations and applications to the Liebau phenomenon Next Article Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential

Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects

1.
Université Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, F-75013, Paris, France
2.
Institut Denis Poisson, Université d'Orléans, Université de Tours, UMR 7013, CNRS, Route de Chartres, BP 6759, F-45067 Orléans cedex 2, France

* Corresponding author: S. Mancini
* Corresponding author: S. Mancini

Received: November 30, 2017

Revised: March 31, 2018

Published: August 2018

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Abstract

In this paper we study a degenerate parabolic system of reaction-diffusion equations arising in cellular biology models. Its specificity lies in the fact that one of the concentrations does not diffuse. Under realistic conditions on the reaction term, we prove existence and uniqueness of a nonnegative solution to the considered system, and we study its regularity. Moreover, we discuss the existence and linear stability of the steady solutions (equilibria), and give a sufficient condition on the reaction term for Turing-like instabilities to be triggered. These results are finally illustrated by some numerical simulations.

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Mathematics Subject Classification: Primary: 35K65, 82B21; Secondary: 35B35.

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Figure 1. Time evolutions of $\min_{x\in\Omega} u(t,x)$ and $\max_{x\in\Omega} u(t,x)$. Both curves converge to the value 0.5943519.

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Figure 2. Time evolutions of $\min_{x\in\Omega} v(t,x)$ and $\max_{x\in\Omega} v(t,x)$. Both curves converge to the value 0.4015822.

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Figure 3. Time evolution of $\max_{x\in \Omega}u(t,x) - \min_{x\in\Omega} u(t,x)$.

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Figure 4. Time evolutions of $\min_{x\in\Omega} u(t,x)$ and $\max_{x\in\Omega} u(t,x)$. Both curves converge to the value $U^*\cong 0.4536921$.

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Figure 5. Time evolutions of $\min_{x\in\Omega} v(t,x)$ and $\max_{x\in\Omega} v(t,x)$. The $\min_{x\in\Omega} v(t,x)$ curve converges to the value $V_1^*\cong 0.0073564$, while the $\max_{x\in\Omega} v(t,x)$ one converges to $V_2^*\cong 0.7526096$.

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Figure 6. The bi-dimensional distribution $v(T,x)$ for T very large. Patterns induced by the initial data defined by 35 clearly appear.

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