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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

 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.

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

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##### 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
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.
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.
Time evolution of $\max_{x\in \Omega}u(t,x) - \min_{x\in\Omega} u(t,x)$.
Time evolutions of $\min_{x\in\Omega} u(t,x)$ and $\max_{x\in\Omega} u(t,x)$. Both curves converge to the value $U^*\cong 0.4536921$.
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 $V_2^*\cong 0.7526096$.
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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