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March  2018, 13(1): 47-67. doi: 10.3934/nhm.2018003

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

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: Francesca R. Guarguaglini. Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks & Heterogeneous Media, 2018, 13 (1) : 47-67. doi: 10.3934/nhm.2018003
References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press-Oxford, 1998.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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.   Google Scholar

[14]

T. Hillen, Hyperbolic models for chemosensitive mevement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034.  doi: 10.1142/S0218202502002008.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I.Jahresber.Deutsch Math-Verein, 105 (2003), 103-165.   Google Scholar

[18]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.   Google Scholar

[19]

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.  Google Scholar

[20]

B. B. Mandal and S. C. Kundu, Cell proliferation and migration in silk broin 3D scaffolds, Biomaterials, 30 (2009), 2956-2965.   Google Scholar

[21]

D. Mugnolo, Simigroup Methods for Evolutions Equations on Networks, Springer, Berlin, 2014.  Google Scholar

[22]

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.  Google Scholar

[23]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhauser, 2007.  Google Scholar

[24]

L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J.Appl.Math., 32 (1977), 653-665.   Google Scholar

[25]

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.   Google Scholar

[26]

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.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press-Oxford, 1998.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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.   Google Scholar

[14]

T. Hillen, Hyperbolic models for chemosensitive mevement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034.  doi: 10.1142/S0218202502002008.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I.Jahresber.Deutsch Math-Verein, 105 (2003), 103-165.   Google Scholar

[18]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.   Google Scholar

[19]

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.  Google Scholar

[20]

B. B. Mandal and S. C. Kundu, Cell proliferation and migration in silk broin 3D scaffolds, Biomaterials, 30 (2009), 2956-2965.   Google Scholar

[21]

D. Mugnolo, Simigroup Methods for Evolutions Equations on Networks, Springer, Berlin, 2014.  Google Scholar

[22]

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.  Google Scholar

[23]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhauser, 2007.  Google Scholar

[24]

L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J.Appl.Math., 32 (1977), 653-665.   Google Scholar

[25]

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.   Google Scholar

[26]

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.  Google Scholar

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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