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On a model of target detection in molecular communication networks
1-7-11, Akabanedai, Kita-Ku, Tokyo 115-0053, Japan |
This paper is concerned with a target-detection model using bio-nanomachines in the human body that is actively being discussed in the field of molecular communication networks. Although the model was originally proposed as spatially one-dimensional, here we extend it to two dimensions and analyze it. After the mathematical formulation, we first verify the solvability of the stationary problem, and then the existence of a strong global-in-time solution of the non-stationary problem in Sobolev–Slobodetskiĭ space. We also show the non-negativeness of the non-stationary solution.
References:
[1] |
K. Ahn and K. Kang,
On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical, Discrete Contin. Dyn. Syst., 34 (2014), 5165-5179.
doi: 10.3934/dcds.2014.34.5165. |
[2] |
J. T. Beale,
Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84 (1983/84), 307-352.
doi: 10.1007/BF00250586. |
[3] |
L. Corrias, B. Perthame and H. Zaag,
Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[4] |
A. Einolghozati, M. Sardari, A. Beirami and F. Fekri, Capacity of discrete molecular diffusion channels, Proc. IEEE International Symposium on Information Theory, (2011).
doi: 10.1109/ISIT.2011.6034228. |
[5] |
A. Einolghozati, M. Sardari and F. Fekri, Capacity of diffusion-based molecular communication with ligand receptors, Proc. IEEE Information Theory Workshop, (2011).
doi: 10.1109/ITW.2011.6089591. |
[6] |
B. D. Ewald and R. Temam,
Maximum principles for the primitive equations of the atmosphere, Discrete Contin. Dynam. Systems, 7 (2001), 343-362.
doi: 10.3934/dcds.2001.7.343. |
[7] |
M. A. Fontelos, A. Friedman and B. Hu,
Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.
doi: 10.1137/S0036141001385046. |
[8] |
A. Friedman and J. I. Tello,
Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-162.
doi: 10.1016/S0022-247X(02)00147-6. |
[9] |
F. R. Guarguaglini and R. Natalini,
Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology, Comm. Pure and Appl. Anal., 6 (2007), 287-309.
doi: 10.3934/cpaa.2007.6.287. |
[10] |
F. R. Guarguaglini and R. Natalini,
Nonlinear transmission problems for quasilinear diffusion systems, Networks and Heterogeneous Media, 2 (2007), 359-381.
doi: 10.3934/nhm.2007.2.359. |
[11] |
H. Honda and A. Tani,
Some boundedness of solutions for the primitive equations of the atmosphere and the ocean, ZAMM Journal of Applied Mathematics and Mechanics, 95 (2015), 38-48.
doi: 10.1002/zamm.201200216. |
[12] |
H. Honda,
Local-in-time solvability of target detection model in molecular communication network, International Journal of Applied Mathematics, 31 (2018), 427-455.
|
[13] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber Dtsch. Math.-Verein., 105 (2003), 103-165.
|
[14] |
S. Iwasaki,
Convergence of solutions to simplified self-organizing target-detection model, Sci. Math. Japnonicae, 81 (2016), 115-129.
|
[15] |
S. Iwasaki, J. Yang and T. Nakano,
A mathematical model of mon-diffusion-based mobile molecular communication networks, IEEE Comm. Lettr., 21 (2017), 1967-1972.
doi: 10.1109/LCOMM.2017.2681061. |
[16] |
K. Kang, T. Kolokolnikov and M. J. Ward,
The stability and dynamics of a spike in the 1D Keller-Segel Model, IMA J. Appl. Math., 72 (2007), 140-162.
doi: 10.1093/imamat/hxl028. |
[17] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
|
[18] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Society, Providence, R.I., 1968. |
[19] |
O. A. Ladyženskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
![]() ![]() |
[20] |
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der Mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. |
[21] |
T. Nakano, A. Eckford and T. Haraguchi, Molecular Communication, Cambridge University Press, Cambridge, 2013.
![]() |
[22] |
T. Nakano and et al., Performance evaluation of leader-follower-based mobile molecular communication networks for target detection applications, IEEE Trans. Comm., 65 (2017), 663–676. |
[23] |
L. Nirenberg,
Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675.
doi: 10.1002/cpa.3160080414. |
[24] |
Y. Okaie and et al., Modeling and performance evaluation of mobile bionanocensor networks for target tracking, Proc. IEEE ICC, (2014), 3969–3974. |
[25] |
Y. Okaie and et al., Cooperative target tracking by a mobile bionanosensor network, IEEE Trans. Nanobioscience, 13 (2014), 267–277. |
[26] |
K. Osaki and A. Yagi,
Atsushi Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469.
|
[27] |
R. Schaaf,
Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[28] |
T. Senba and T. Suzuki,
Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl., 10 (2000), 191-224.
|
[29] |
M. Struwe and G. Tarantello,
On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 109-121.
|
[30] |
Y. Sugiyama, Y. Tsutsui and J. J. L. Velázquez,
Global solutions to a chemotaxis system with non-diffusive memory, J. Math. Anal. Appl., 410 (2014), 908-917.
doi: 10.1016/j.jmaa.2013.08.065. |
[31] |
A. Marciniak-Czochra, G. Karch and K. Suzuki,
Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., 74 (2017), 583-618.
doi: 10.1007/s00285-016-1035-z. |
[32] |
N. Tanaka and A. Tani,
Surface waves for a compressible viscous fluid, J. Math. Fluid Mech., 5 (2003), 303-363.
doi: 10.1007/s00021-003-0078-2. |
[33] |
G. Wang and J. Wei,
Steady state solutions of a rReaction-diffusion systems modeling chemotaxis, Math. Nachr., 233/234 (2002), 221-236.
doi: 10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.3.CO;2-D. |
[34] |
J. Wloka, Partielle Differentialgleichungen, B. G. Teubner, Stuttgart, 1982,500 pp. |
show all references
References:
[1] |
K. Ahn and K. Kang,
On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical, Discrete Contin. Dyn. Syst., 34 (2014), 5165-5179.
doi: 10.3934/dcds.2014.34.5165. |
[2] |
J. T. Beale,
Large-time regularity of viscous surface waves, Arch. Ration. Mech. Anal., 84 (1983/84), 307-352.
doi: 10.1007/BF00250586. |
[3] |
L. Corrias, B. Perthame and H. Zaag,
Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[4] |
A. Einolghozati, M. Sardari, A. Beirami and F. Fekri, Capacity of discrete molecular diffusion channels, Proc. IEEE International Symposium on Information Theory, (2011).
doi: 10.1109/ISIT.2011.6034228. |
[5] |
A. Einolghozati, M. Sardari and F. Fekri, Capacity of diffusion-based molecular communication with ligand receptors, Proc. IEEE Information Theory Workshop, (2011).
doi: 10.1109/ITW.2011.6089591. |
[6] |
B. D. Ewald and R. Temam,
Maximum principles for the primitive equations of the atmosphere, Discrete Contin. Dynam. Systems, 7 (2001), 343-362.
doi: 10.3934/dcds.2001.7.343. |
[7] |
M. A. Fontelos, A. Friedman and B. Hu,
Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.
doi: 10.1137/S0036141001385046. |
[8] |
A. Friedman and J. I. Tello,
Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-162.
doi: 10.1016/S0022-247X(02)00147-6. |
[9] |
F. R. Guarguaglini and R. Natalini,
Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology, Comm. Pure and Appl. Anal., 6 (2007), 287-309.
doi: 10.3934/cpaa.2007.6.287. |
[10] |
F. R. Guarguaglini and R. Natalini,
Nonlinear transmission problems for quasilinear diffusion systems, Networks and Heterogeneous Media, 2 (2007), 359-381.
doi: 10.3934/nhm.2007.2.359. |
[11] |
H. Honda and A. Tani,
Some boundedness of solutions for the primitive equations of the atmosphere and the ocean, ZAMM Journal of Applied Mathematics and Mechanics, 95 (2015), 38-48.
doi: 10.1002/zamm.201200216. |
[12] |
H. Honda,
Local-in-time solvability of target detection model in molecular communication network, International Journal of Applied Mathematics, 31 (2018), 427-455.
|
[13] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber Dtsch. Math.-Verein., 105 (2003), 103-165.
|
[14] |
S. Iwasaki,
Convergence of solutions to simplified self-organizing target-detection model, Sci. Math. Japnonicae, 81 (2016), 115-129.
|
[15] |
S. Iwasaki, J. Yang and T. Nakano,
A mathematical model of mon-diffusion-based mobile molecular communication networks, IEEE Comm. Lettr., 21 (2017), 1967-1972.
doi: 10.1109/LCOMM.2017.2681061. |
[16] |
K. Kang, T. Kolokolnikov and M. J. Ward,
The stability and dynamics of a spike in the 1D Keller-Segel Model, IMA J. Appl. Math., 72 (2007), 140-162.
doi: 10.1093/imamat/hxl028. |
[17] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
|
[18] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Society, Providence, R.I., 1968. |
[19] |
O. A. Ladyženskaja and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.
![]() ![]() |
[20] |
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der Mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972. |
[21] |
T. Nakano, A. Eckford and T. Haraguchi, Molecular Communication, Cambridge University Press, Cambridge, 2013.
![]() |
[22] |
T. Nakano and et al., Performance evaluation of leader-follower-based mobile molecular communication networks for target detection applications, IEEE Trans. Comm., 65 (2017), 663–676. |
[23] |
L. Nirenberg,
Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675.
doi: 10.1002/cpa.3160080414. |
[24] |
Y. Okaie and et al., Modeling and performance evaluation of mobile bionanocensor networks for target tracking, Proc. IEEE ICC, (2014), 3969–3974. |
[25] |
Y. Okaie and et al., Cooperative target tracking by a mobile bionanosensor network, IEEE Trans. Nanobioscience, 13 (2014), 267–277. |
[26] |
K. Osaki and A. Yagi,
Atsushi Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469.
|
[27] |
R. Schaaf,
Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[28] |
T. Senba and T. Suzuki,
Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl., 10 (2000), 191-224.
|
[29] |
M. Struwe and G. Tarantello,
On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 109-121.
|
[30] |
Y. Sugiyama, Y. Tsutsui and J. J. L. Velázquez,
Global solutions to a chemotaxis system with non-diffusive memory, J. Math. Anal. Appl., 410 (2014), 908-917.
doi: 10.1016/j.jmaa.2013.08.065. |
[31] |
A. Marciniak-Czochra, G. Karch and K. Suzuki,
Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., 74 (2017), 583-618.
doi: 10.1007/s00285-016-1035-z. |
[32] |
N. Tanaka and A. Tani,
Surface waves for a compressible viscous fluid, J. Math. Fluid Mech., 5 (2003), 303-363.
doi: 10.1007/s00021-003-0078-2. |
[33] |
G. Wang and J. Wei,
Steady state solutions of a rReaction-diffusion systems modeling chemotaxis, Math. Nachr., 233/234 (2002), 221-236.
doi: 10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.3.CO;2-D. |
[34] |
J. Wloka, Partielle Differentialgleichungen, B. G. Teubner, Stuttgart, 1982,500 pp. |
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