February  2020, 19(2): 1057-1087. doi: 10.3934/cpaa.2020049

Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions

Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi di L'Aquila, 67100 Coppito (L'Aquila), Italy

Received  March 2019 Revised  June 2019 Published  October 2019

In this paper we study a semilinear hyperbolic-parabolic system as a model for some chemotaxis phenomena evolving on networks; we consider transmission conditions at the inner nodes which preserve the fluxes and nonhomogeneous boundary conditions having in mind phenomena with inflow of cells and food providing at the network exits. We give some conditions on the boundary data which ensure the existence of stationary solutions and we prove that these ones are asymptotic profiles for a class of global solutions.

Citation: Francesca R. Guarguaglini. Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1057-1087. doi: 10.3934/cpaa.2020049
References:
[1]

R. BorsheS. GottlichA. 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

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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

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R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Mathematiques & Applications, 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.  Google Scholar

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M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

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J. M. Greemberg and W. Alt, Stability results for a diffusion equation with functional drift approximating a chemotaxis model, Trans. Amer. Math. Soc., 300 (1987), 235-258.  doi: 10.2307/2000597.  Google Scholar

[9]

F. R. Guarguaglini, Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network, NHM, 13 (2018), 47-67.  doi: 10.3934/nhm.2018003.  Google Scholar

[10]

F. R. GuarguagliniC. MasciaR. 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

[11]

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

[12]

O. Kedem and A. Katchalsky, Thermo dynamic analysis of the permeability of biological membranes to non-electrolytes, Biochim. Biophys. Acta, 27 (1958). Google Scholar

[13]

B. A. C. HarleyH. KimM. H. ZamanI. V. YannasD. 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

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T. HillenC. Rhode and F. Lutscher, Existence of weak solutions for a hyperbolic model of chemosensitive movement, J. Math. Anal. Appl., 26 (2001), 173-199.  doi: 10.1006/jmaa.2001.7447.  Google Scholar

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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

[16]

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

[17]

D. Mugnolo, Simigroup Methods for Evolutions Equations on Networks, Springer (2014). doi: 10.1007/978-3-319-04621-1.  Google Scholar

[18]

T. Nakagaki, H. Yamada and A. Tóth, Maze-solving by an amoeboid organism, Nature, 407 (2000), 470. Google Scholar

[19]

S. Nicaise, Control and stabilization of $2\times 2$ hyperbolic systems on graphs, Math. Control Relat. Fields, 7 (2017), 53-72.  doi: 10.3934/mcrf.2017004.  Google Scholar

[20]

A. QuarteroniA. Veneziani and P. Zunino, Mathematical and numerical modeling of solute dynamics in blood flow and arterial walls, SIAM J. Numer. Anal., 39 (2001), 1488-1511.  doi: 10.1137/S0036142900369714.  Google Scholar

[21]

A. QuarteroniA. Veneziani and P. Zunino, A domain decompositions method for advection-diffusion processes with application to blood solutes, SIAM J. Sci. Comput., 23 (2002), 1959-1980.  doi: 10.1137/S1064827500375722.  Google Scholar

[22]

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

[23]

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). Google Scholar

[24]

C. Zong and G. Q. Xu, Observability and controllability analysis of blood flow network, Math. Control Relat. Fields, 4 (2014), 521-554.  doi: 10.3934/mcrf.2014.4.521.  Google Scholar

[25]

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]

R. BorsheS. GottlichA. 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

[2]

G. Bretti and R. Natalini, On modeling Maze solving ability of slime mold via a hyperbolic model of chemotaxis, J. Comput. Methods Sci. Eng., 18 (2018), 85-115.   Google Scholar

[3]

G. BrettiR. 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]

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

[6]

R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Mathematiques & Applications, 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.  Google Scholar

[7]

M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.  Google Scholar

[8]

J. M. Greemberg and W. Alt, Stability results for a diffusion equation with functional drift approximating a chemotaxis model, Trans. Amer. Math. Soc., 300 (1987), 235-258.  doi: 10.2307/2000597.  Google Scholar

[9]

F. R. Guarguaglini, Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network, NHM, 13 (2018), 47-67.  doi: 10.3934/nhm.2018003.  Google Scholar

[10]

F. R. GuarguagliniC. MasciaR. 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

[11]

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

[12]

O. Kedem and A. Katchalsky, Thermo dynamic analysis of the permeability of biological membranes to non-electrolytes, Biochim. Biophys. Acta, 27 (1958). Google Scholar

[13]

B. A. C. HarleyH. KimM. H. ZamanI. V. YannasD. 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. HillenC. Rhode and F. Lutscher, Existence of weak solutions for a hyperbolic model of chemosensitive movement, J. Math. Anal. Appl., 26 (2001), 173-199.  doi: 10.1006/jmaa.2001.7447.  Google Scholar

[15]

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

[16]

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

[17]

D. Mugnolo, Simigroup Methods for Evolutions Equations on Networks, Springer (2014). doi: 10.1007/978-3-319-04621-1.  Google Scholar

[18]

T. Nakagaki, H. Yamada and A. Tóth, Maze-solving by an amoeboid organism, Nature, 407 (2000), 470. Google Scholar

[19]

S. Nicaise, Control and stabilization of $2\times 2$ hyperbolic systems on graphs, Math. Control Relat. Fields, 7 (2017), 53-72.  doi: 10.3934/mcrf.2017004.  Google Scholar

[20]

A. QuarteroniA. Veneziani and P. Zunino, Mathematical and numerical modeling of solute dynamics in blood flow and arterial walls, SIAM J. Numer. Anal., 39 (2001), 1488-1511.  doi: 10.1137/S0036142900369714.  Google Scholar

[21]

A. QuarteroniA. Veneziani and P. Zunino, A domain decompositions method for advection-diffusion processes with application to blood solutes, SIAM J. Sci. Comput., 23 (2002), 1959-1980.  doi: 10.1137/S1064827500375722.  Google Scholar

[22]

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

[23]

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). Google Scholar

[24]

C. Zong and G. Q. Xu, Observability and controllability analysis of blood flow network, Math. Control Relat. Fields, 4 (2014), 521-554.  doi: 10.3934/mcrf.2014.4.521.  Google Scholar

[25]

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

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