Figure 1.
Example of acyclic network; the highlighted arcs form the path linking the nodes $N_4$ and $N_5$ .
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Stochastic homogenization of maximal monotone relations and applications
March
2018, 13(1): 47-67.
doi: 10.3934/nhm.2018003
Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network
Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Universitá degli Studi di L'Aquila, Via Vetoio, I-67100 Coppito (L'Aquila), Italy |
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 and Heterogeneous Media,
2018, 13
(1)
: 47-67.
doi: 10.3934/nhm.2018003
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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. |
| [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. |
| [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. |
| [4] |
T. Cazenave and A. Haraux,
An Introduction to Semilinear Evolution Equations, Clarendon Press-Oxford, 1998. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [14] |
T. Hillen,
Hyperbolic models for chemosensitive mevement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034.
doi: 10.1142/S0218202502002008. |
| [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. |
| [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. |
| [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.
|
| [18] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
|
| [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. |
| [20] |
B. B. Mandal and S. C. Kundu,
Cell proliferation and migration in silk broin 3D scaffolds, Biomaterials, 30 (2009), 2956-2965.
|
| [21] |
D. Mugnolo,
Simigroup Methods for Evolutions Equations on Networks, Springer, Berlin, 2014. |
| [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. |
| [23] |
B. Perthame,
Transport Equations in Biology, Frontiers in Mathematics, Birkhauser, 2007. |
| [24] |
L. A. Segel,
A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J.Appl.Math., 32 (1977), 653-665.
|
| [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.
|
| [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. |
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. |
| [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. |
| [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. |
| [4] |
T. Cazenave and A. Haraux,
An Introduction to Semilinear Evolution Equations, Clarendon Press-Oxford, 1998. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [14] |
T. Hillen,
Hyperbolic models for chemosensitive mevement, Math. Models Methods Appl. Sci., 12 (2002), 1007-1034.
doi: 10.1142/S0218202502002008. |
| [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. |
| [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. |
| [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.
|
| [18] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
|
| [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. |
| [20] |
B. B. Mandal and S. C. Kundu,
Cell proliferation and migration in silk broin 3D scaffolds, Biomaterials, 30 (2009), 2956-2965.
|
| [21] |
D. Mugnolo,
Simigroup Methods for Evolutions Equations on Networks, Springer, Berlin, 2014. |
| [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. |
| [23] |
B. Perthame,
Transport Equations in Biology, Frontiers in Mathematics, Birkhauser, 2007. |
| [24] |
L. A. Segel,
A theoretical study of receptor mechanisms in bacterial chemotaxis, SIAM J.Appl.Math., 32 (1977), 653-665.
|
| [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.
|
| [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. |
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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