Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network

Francesca R. Guarguaglini

doi:10.3934/nhm.2018003

Article Contents

2018, Volume 13, Issue 1: 47-67. Doi: 10.3934/nhm.2018003

This issue Previous Article Stochastic homogenization of maximal monotone relations and applications Next Article On a vorticity-based formulation for reaction-diffusion-Brinkman systems

Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network

Francesca R. Guarguaglini

Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Universitá degli Studi di L'Aquila, Via Vetoio, I-67100 Coppito (L'Aquila), Italy

Received: November 2016

Revised: October 2017

Published: March 2018

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Abstract

This paper approaches the question of existence and uniqueness of stationary solutions to a semilinear hyperbolic-parabolic system and the study of the asymptotic behaviour of global solutions. The system is a model for some biological phenomena evolving on a network composed by a finite number of nodes and oriented arcs. The transmission conditions for the unknowns, set at each inner node, are crucial features of the model.

Keywords:

Nonlinear hyperbolic systems,
transmission conditions,
stationary solutions,
asymptotic behaviour of global solutions,
chemotaxis.

Mathematics Subject Classification: Primary: 35R02; Secondary: 35M33, 35L50, 35B40, 35Q92.

Citation:

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Figure 1. Example of acyclic network; the highlighted arcs form the path linking the nodes $N_4$ and $N_5$.

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Figure 2. Example: the highlighted arcs form the path from the outer point $e_1$ to the inner node $N_4$ and $I_5$ is an arc incident with $N_4$, not belonghing to the path.

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References

	W. Alt and J. M. Greemberg , Stability results for a diffusion equation with functional drift approximating a chemotaxis model, Trans. Amer. Math. Soc., 300 (1987) , 235-258. doi: 10.1090/S0002-9947-1987-0871674-4.
	R. Borsche , S. Gottlich , A. Klar and P. Schillen , The scalar Keller-Segel model on networks, Math. Models Methods Appl. Sci., 24 (2014) , 221-247. doi: 10.1142/S0218202513400071.
	G. Bretti , R. Natalini and M. Ribot , A hyperbolic model of chemotaxis on a network: A numerical study, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014) , 231-258. doi: 10.1051/m2an/2013098.
	T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press-Oxford, 1998.
	G. M. Coclite , M. Garavello and B. Piccoli , Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005) , 1862-1886. doi: 10.1137/S0036141004402683.
	L. Corrias and F. Camilli , Parabolic models for chemotaxis on weighted nerworks, J. Math. Pures Appl., 108 (2017) , 459-480. doi: 10.1016/j.matpur.2017.07.003.
	R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Vol. 50 of Mathematiques & Applications [Mathematics & Applications] Springer-Verlag, Berlin, 2006.
	Y. Dolak and T. Hillen , Cattaneo models for chemosensitive movement. Numerical solution and pattern formation, J. Math. Biol., 46 (2003) , 153-170. doi: 10.1007/s00285-002-0173-7.
	F. Filbet , P. Laurencot and B. Pertame , Derivation of hyperbolic model for chemosensitive movement, J. Math. Biol., 50 (2005) , 189-207. doi: 10.1007/s00285-004-0286-2.
	M. Garavello and B. Piccoli, Traffic Flow on Networks -Conservation Laws Models, AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.
	F. R. Guarguaglini , C. Mascia , R. Natalini and M. Ribot , Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009) , 39-76. doi: 10.3934/dcdsb.2009.12.39.
	F. R. Guarguaglini and R. Natalini , Global smooth solutions for a hyperbolic chemotaxis model on a network, SIAM J. Math. Anal., 47 (2015) , 4652-4671. doi: 10.1137/140997099.
	B. A. C. Harley , H. Kim , M. H. Zaman , I. V. Yannas , D. A. Lauffenburger and L. J. Gibson , Microarchitecture of three-dimensional scaffold influences cell migration behavior via junction interaction, Biophysical Journal, 29 (2008) , 4013-4024.
	T. Hillen , Hyperbolic models for chemosensitive mevement, Math. Models Methods Appl. Sci., 12 (2002) , 1007-1034. doi: 10.1142/S0218202502002008.
	T. Hillen , C. Rhode and F. Lutscher , Existence of weak solutions for a hyperbolic model of chemosensitive movement, J. Math. Anal. Appl., 260 (2001) , 173-199. doi: 10.1006/jmaa.2001.7447.
	T. Hillen and A. Stevens , Hyperbolic model for chemotaxis in 1-D, Nonlinear Anal. Real World Appl., 1 (2000) , 409-433. doi: 10.1016/S0362-546X(99)00284-9.
	D. Horstmann , From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I.Jahresber.Deutsch Math-Verein, 105 (2003) , 103-165.
	E. F. Keller and L. A. Segel , Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970) , 399-415.
	M. Kramar and E. Sikolya , Spectral properties and asymptotic periodicity of flows in networks, Mathematische Zeitschrift, 249 (2005) , 139-162. doi: 10.1007/s00209-004-0695-3.
	B. B. Mandal and S. C. Kundu , Cell proliferation and migration in silk broin 3D scaffolds, Biomaterials, 30 (2009) , 2956-2965.
	D. Mugnolo, Simigroup Methods for Evolutions Equations on Networks, Springer, Berlin, 2014.
	J. D. Murray, Mathematical Biology. I An Introduction, Third edition. Interdisciplinary Applied Mathematics, 17 Springer Verlag, New York, 2002; Mathematical Biology. Ⅱ Spatial models and biomedical applications, Third edition. Interdisciplinary Applied Mathematics, 18 Springer Verlag, New York, 2003.
	B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhauser, 2007.
	L. A. Segel , A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J.Appl.Math., 32 (1977) , 653-665.
	C. Spadaccio , A. Rainer , S. De Porcellinis , M. Centola , F. De Marco , M. Chello , M. Trombetta and J. A. Genovese , A G-CSF functionalized PLLA scaffold for wound repair: An in vitro preliminary study, Conf. Proc. IEEE Eng.Med.Biol.Soc., (2010) , 843-846.
	J. Valein and E. Zuazua , Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009) , 2771-2797. doi: 10.1137/080733590.