American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Convergence speed of a Ruelle operator associated with a non-uniformly expanding conformal dynamical system and a Dini potential
  • DCDS Home
  • This Issue
  • Next Article
    Lyapunov stability for regular equations and applications to the Liebau phenomenon
September  2018, 38(9): 4675-4692. doi: 10.3934/dcds.2018205

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

Laurent Desvillettes 1, , Michèle Grillot 2, , Philippe Grillot 2, and  Simona Mancini 2,,

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

Received  December 2017 Revised  April 2018 Published  June 2018

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.

Keywords: Degenerate parabolic system, existence of solutions, stability, pattern formation, aggregation.
Mathematics Subject Classification: Primary: 35K65, 82B21; Secondary: 35B35.
Citation: Laurent Desvillettes, Michèle Grillot, Philippe Grillot, Simona Mancini. Study of a degenerate reaction-diffusion system arising in particle dynamics with aggregation effects. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4675-4692. doi: 10.3934/dcds.2018205
References:
[1]

H. Amann, Highly degenerate quasilinear parabolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1991), 135-166.   Google Scholar

[2]

H. Amann, Global existence for a class of highly degenerate parabolic systems, Japan J. Indust. Appl. Math., 8 (1991), 143-151.  doi: 10.1007/BF03167189.  Google Scholar

[3]

K. Anguige and C. Schmeiser, A one-dimensional model for cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion, J. Math. Biol., 58 (2009), 395-427.  doi: 10.1007/s00285-008-0197-8.  Google Scholar

[4]

N. Bedjaoui and P. Souplet, Critical blowup exponents for a system of reaction-diffusion equations with absorption, Zeitschrift für angewandte Mathematik und Physik ZAMP, 53 (2002), 197-210.  doi: 10.1007/s00033-002-8152-9.  Google Scholar

[5]

Cell Mechanics: From Single Scale-Based Models to Multi-Scale Modeling, Editors: A. Chauviere, L. Preziosi and C. Verdier, Publisher: Taylor & Francis Group, Chapman & Hall/CRC Mathematical and Computational Biology Series, 2010. doi: 10.1201/9781420094558.  Google Scholar

[6]

T. Colin, M.-C. Durrieu, J. Joie, Y. Lei, Y. Mammeri, C. Poignard and O. Saut, Modeling of the migration of endothelial cells on bioactive micro-patterned polymers, Mathematical Biosciences and Engineering, 10 (2013), 997-1015.  doi: 10.3934/mbe.2013.10.997.  Google Scholar

[7]

O. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Transl. Math. AMS, Providence R. I., 1968.  Google Scholar

[8]

M. Lambert, O. Thoumine, J. Brevier, D. Choquet, D. Riveline and R.-M. Mège, Nucleation and growth of cadherin adhesions, Exp. Cell Res., 313 (2007), 4025-4040.  doi: 10.1016/j.yexcr.2007.07.035.  Google Scholar

[9]

S. Mancini, R. M. Mège, B. Sarels and P. O. Strale, A phenomenological model of cell-cell adhesion mediated by cadherins, J. Math. Biol., 74 (2017), 1657-1678.  doi: 10.1007/s00285-016-1072-7.  Google Scholar

[10]

M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.  doi: 10.1137/0520057.  Google Scholar

[11]

R. M. Mège, J. Gavard and M. Lambert, Regulation of cell-cell junctions by the cytoskeleton, Current Opinion in Cell Biology, 18 (2006), 541-548.   Google Scholar

[12]

J. D. Murray, Mathematical Biology I. An Introduction, vol. 17, Springer-Verlag New York, 2002.  Google Scholar

[13]

J. D. Murray, Mathematical Biology II. Spatial Models and Biomedical Applications, vol. 18, Springer-Verlag New York, 2003.  Google Scholar

[14]

P. Quittner and P. Souplet, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts Basler Lehrbücher, XI, 2007.  Google Scholar

[15]

A. Rodriguez-Bernal and B. Wang, Attractors for partly dissipative reaction diffusion systems in $ {\mathbb R}^n$, J. Math. Anal. Appl., 252 (2000), 790-803.  doi: 10.1006/jmaa.2000.7122.  Google Scholar

[16]

Y. Wu, J. Vendome, L. Shapiro, A. Ben-Shaul and B. Honig, Transforming binding affinities from three dimensions to two with application to cadherin clustering, Nature, 475 (2011), 510-513.  doi: 10.1038/nature10183.  Google Scholar

show all references

References:
[1]

H. Amann, Highly degenerate quasilinear parabolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1991), 135-166.   Google Scholar

[2]

H. Amann, Global existence for a class of highly degenerate parabolic systems, Japan J. Indust. Appl. Math., 8 (1991), 143-151.  doi: 10.1007/BF03167189.  Google Scholar

[3]

K. Anguige and C. Schmeiser, A one-dimensional model for cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion, J. Math. Biol., 58 (2009), 395-427.  doi: 10.1007/s00285-008-0197-8.  Google Scholar

[4]

N. Bedjaoui and P. Souplet, Critical blowup exponents for a system of reaction-diffusion equations with absorption, Zeitschrift für angewandte Mathematik und Physik ZAMP, 53 (2002), 197-210.  doi: 10.1007/s00033-002-8152-9.  Google Scholar

[5]

Cell Mechanics: From Single Scale-Based Models to Multi-Scale Modeling, Editors: A. Chauviere, L. Preziosi and C. Verdier, Publisher: Taylor & Francis Group, Chapman & Hall/CRC Mathematical and Computational Biology Series, 2010. doi: 10.1201/9781420094558.  Google Scholar

[6]

T. Colin, M.-C. Durrieu, J. Joie, Y. Lei, Y. Mammeri, C. Poignard and O. Saut, Modeling of the migration of endothelial cells on bioactive micro-patterned polymers, Mathematical Biosciences and Engineering, 10 (2013), 997-1015.  doi: 10.3934/mbe.2013.10.997.  Google Scholar

[7]

O. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Transl. Math. AMS, Providence R. I., 1968.  Google Scholar

[8]

M. Lambert, O. Thoumine, J. Brevier, D. Choquet, D. Riveline and R.-M. Mège, Nucleation and growth of cadherin adhesions, Exp. Cell Res., 313 (2007), 4025-4040.  doi: 10.1016/j.yexcr.2007.07.035.  Google Scholar

[9]

S. Mancini, R. M. Mège, B. Sarels and P. O. Strale, A phenomenological model of cell-cell adhesion mediated by cadherins, J. Math. Biol., 74 (2017), 1657-1678.  doi: 10.1007/s00285-016-1072-7.  Google Scholar

[10]

M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM J. Math. Anal., 20 (1989), 816-844.  doi: 10.1137/0520057.  Google Scholar

[11]

R. M. Mège, J. Gavard and M. Lambert, Regulation of cell-cell junctions by the cytoskeleton, Current Opinion in Cell Biology, 18 (2006), 541-548.   Google Scholar

[12]

J. D. Murray, Mathematical Biology I. An Introduction, vol. 17, Springer-Verlag New York, 2002.  Google Scholar

[13]

J. D. Murray, Mathematical Biology II. Spatial Models and Biomedical Applications, vol. 18, Springer-Verlag New York, 2003.  Google Scholar

[14]

P. Quittner and P. Souplet, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts Basler Lehrbücher, XI, 2007.  Google Scholar

[15]

A. Rodriguez-Bernal and B. Wang, Attractors for partly dissipative reaction diffusion systems in $ {\mathbb R}^n$, J. Math. Anal. Appl., 252 (2000), 790-803.  doi: 10.1006/jmaa.2000.7122.  Google Scholar

[16]

Y. Wu, J. Vendome, L. Shapiro, A. Ben-Shaul and B. Honig, Transforming binding affinities from three dimensions to two with application to cadherin clustering, Nature, 475 (2011), 510-513.  doi: 10.1038/nature10183.  Google Scholar

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.
Figure Options

Download full-size image

Download as PowerPoint slide

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.
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 $U^*\cong 0.4536921$.
Figure Options

Download full-size image

Download as PowerPoint slide

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 $V_2^*\cong 0.7526096$.
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Andrea L. Bertozzi, Dejan Slepcev. Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1617-1637. doi: 10.3934/cpaa.2010.9.1617
[2]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[3]

Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108
[4]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
[5]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[6]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081
[7]

Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111
[8]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597
[9]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[10]

H. Malchow, F.M. Hilker, S.V. Petrovskii. Noise and productivity dependence of spatiotemporal pattern formation in a prey-predator system. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 705-711. doi: 10.3934/dcdsb.2004.4.705
[11]

Teemu Lukkari, Mikko Parviainen. Stability of degenerate parabolic Cauchy problems. Communications on Pure & Applied Analysis, 2015, 14 (1) : 201-216. doi: 10.3934/cpaa.2015.14.201
[12]

R. Kowalczyk, A. Gamba, L. Preziosi. On the stability of homogeneous solutions to some aggregation models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 203-220. doi: 10.3934/dcdsb.2004.4.203
[13]

Rui Huang, Yifu Wang, Yuanyuan Ke. Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1005-1014. doi: 10.3934/dcdsb.2005.5.1005
[14]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051
[15]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[16]

Lucio Boccardo, Maria Michaela Porzio. Some degenerate parabolic problems: Existence and decay properties. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 617-629. doi: 10.3934/dcdss.2014.7.617
[17]

Charles A. Stuart. Stability analysis for a family of degenerate semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5297-5337. doi: 10.3934/dcds.2018234
[18]

Genni Fragnelli, Paolo Nistri, Duccio Papini. Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3831-3834. doi: 10.3934/dcds.2013.33.3831
[19]

Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35
[20]

Tariel Sanikidze, A.F. Tedeev. On the temporal decay estimates for the degenerate parabolic system. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1755-1768. doi: 10.3934/cpaa.2013.12.1755

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (87)
  • HTML views (135)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences