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On the vanishing discount problem from the negative direction
A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation
Departamento de Matemática Aplicada and Excellence Research Unit "Modeling Nature" (MNat), Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain |
The parabolic-parabolic Keller-Segel model of chemotaxis is shown to come up as the hydrodynamic system describing the evolution of the modulus square $ n(t,x) $ and the argument $ S(t,x) $ of a wavefunction $ \psi = \sqrt{n} \, e^{iS} $ that solves a cubic Schrödinger equation with focusing interaction, frictional Kostin nonlinearity and Doebner-Goldin dissipation mechanism. This connection is then exploited to construct a family of quasi-stationary solutions to the Keller-Segel system under the influence of no-flux and anti-Fick laws.
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
F. Andreu, V. Caselles, J. M. Mazón and S. Moll,
Finite propagation speed for limited flux diffusion equations, Arch. Rat. Mech. Anal., 182 (2006), 269-297.
doi: 10.1007/s00205-006-0428-3. |
[3] |
M. Arias, J. Campos and J. Soler,
Cross-diffusion and traveling waves in porous-media flux-saturated Keller-Segel models, Math. Models Meth. Appl. Sci., 28 (2018), 2103-2129.
doi: 10.1142/S0218202518400092. |
[4] |
G. Auberson and P. C. Sabatier,
On a class of homogeneous nonlinear Schrödinger equations, J. Math. Phys., 35 (1994), 4028-4040.
doi: 10.1063/1.530840. |
[5] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[6] |
N. Bellomo and M. Winkler,
A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up, Comm. PDE, 42 (2017), 436-473.
doi: 10.1080/03605302.2016.1277237. |
[7] |
A. Bellouquid, J. Nieto and L. Urrutia,
About the kinetic description of fractional diffusion equations modeling chemotaxis, Math. Models Meth. Appl. Sci., 26 (2016), 249-268.
doi: 10.1142/S0218202516400029. |
[8] |
L. Bergé,
Wave collapse in physics: Principles and applications to light and plasma waves, Phys. Rep., 303 (1998), 259-370.
doi: 10.1016/S0370-1573(97)00092-6. |
[9] |
I. Bialynicki–Birula and J. Mycielski,
Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9. |
[10] |
A. Blanchet, On the Parabolic-elliptic Patlak-Keller-Segel System in Dimension $2$ and Higher, Séminaire Laurent Schwartz–EDP et applications, Exposé n. Ⅷ, Palaiseau, 2013. |
[11] |
N. Bournaveas and V. Calvez,
The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923-935.
doi: 10.1088/0951-7715/23/4/009. |
[12] |
A. O. Caldeira and A. J. Leggett,
Path integral approach to quantum Brownian motion, Physica A, 121 (1983), 587-616.
doi: 10.1016/0378-4371(83)90013-4. |
[13] |
J. Calvo, J. Campos, V. Caselles, O. Sánchez and J. Soler, Flux-saturated porous media equation and applications, JEMS Surveys in Mathematical Sciences 2 (2015), 131–218.
doi: 10.4171/EMSS/11. |
[14] |
V. Calvez, L. Corrias and M. A. Ebde,
Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension, Comm. PDE, 37 (2012), 561-584.
doi: 10.1080/03605302.2012.655824. |
[15] |
V. Calvez, B. Perthame and S. Yasuda,
Traveling wave and aggregation in a flux-limited Keller-Segel model, Kinetic & Related Models, 11 (2018), 891-909.
doi: 10.3934/krm.2018035. |
[16] |
M. A. J. Chaplain and J. I. Tello,
On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6.
doi: 10.1016/j.aml.2015.12.001. |
[17] |
W. Chen and J. Dávila,
Resonance phenomenon for a Gelfand-type problem, Nonlinear Anal., 89 (2013), 299-321.
doi: 10.1016/j.na.2013.05.008. |
[18] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinetic & Related Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[19] |
M. del Pino and J. Wei,
Collapsing steady states of the Keller-Segel system, Nonlinearity, 19 (2006), 661-684.
doi: 10.1088/0951-7715/19/3/007. |
[20] |
H. D. Doebner and G. A. Goldin,
On a general nonlinear Schrödinger equation admitting diffusion currents, Phys. Lett. A, 162 (1992), 397-401.
doi: 10.1016/0375-9601(92)90061-P. |
[21] |
S. A. Dyachenko, P. M. Lushnikov and N. Vladimirova,
Logarithmic scaling of the collapse in the critical Keller-Segel equation, Nonlinearity, 26 (2013), 3011-3041.
doi: 10.1088/0951-7715/26/11/3011. |
[22] |
C. Escudero,
The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.
doi: 10.1088/0951-7715/19/12/010. |
[23] |
H. Gajewski and K. Zacharias,
Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[24] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[25] |
P. Guerrero, J. L. López, J. Montejo–Gámez and J. Nieto,
Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner–Goldin type modeling quantum dissipation, J. Nonlinear Sci., 22 (2012), 631-663.
doi: 10.1007/s00332-012-9123-8. |
[26] |
Y. Huang and A. Bertozzi,
Self-similar blowup solutions to an aggregation equation in $\mathbb{R}^N$, SIAM J. Appl. Math., 70 (2010), 2582-2603.
doi: 10.1137/090774495. |
[27] |
Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, S${\bar{u}}$rikaisekikenky${\bar{u}}$sho K${\bar{o}}$ky${\bar{u}}$roku, 1025 (1998), 44–65. Variational problems and related topics (Kyoto, 1997) |
[28] |
J. L. Kazdan and F. W. Warner,
Curvature functions for compact 2-manifolds, Ann. Math., 99 (1974), 14-47.
doi: 10.2307/1971012. |
[29] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90050-6. |
[30] |
M. D. Kostin,
On the Schrödinger–Langevin equation, J. Stat. Phys., 12 (1975), 145-151.
doi: 10.1063/1.1678812. |
[31] |
C.-S. Lin, W.-M. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Diff. Equ., 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[32] |
D. Liu, Global solutions in a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production, J. Math. Phys., 61 (2020), 021503, 4pp.
doi: 10.1063/1.5111650. |
[33] |
D. Liu and Y. Tao,
Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ., 31 (2016), 379-388.
doi: 10.1007/s11766-016-3386-z. |
[34] |
J. L. López, Nonlinear Ginzburg–Landau–type approach to quantum dissipation, Phys. Rev. E., 69 (2004), 026110. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.69.026110. Google Scholar |
[35] |
J. L. López and J. Montejo-Gámez,
A hydrodynamic approach to multidimensional dissipation–based Schrödinger models from quantum Fokker–Planck dynamics, Phys. D, 238 (2009), 622-644.
doi: 10.1016/j.physd.2008.12.006. |
[36] |
J. L. López and J. Montejo-Gámez,
On a rigorous interpretation of the quantum Schrödinger-Langevin operator in bounded domains, J. Math. Anal. Appl., 383 (2011), 365-378.
doi: 10.1016/j.jmaa.2011.05.024. |
[37] |
P. M. Lushnikov,
Critical chemotactic collapse., Phys. Lett. A, 374 (2010), 1678-1685.
doi: 10.1016/j.physleta.2010.01.068. |
[38] |
B. Perthame, Transport Equations in Biology, Springer, 2007. https://www.springer.com/gp/book/9783764378417. |
[39] |
B. Perthame, N. Vauchelet and Z. Wang,
The flux-limited Keller-Segel system; properties and derivation from kinetic equtions, Rev. Mat. Iberoamericana, 36 (2020), 357-386.
doi: 10.4171/rmi/1132. |
[40] |
A. L. Sanin and A. A. Smirnovsky,
Oscillatory motion in confined potential systems with dissipation in the context of the Schrödinger-Langevin-Kostin equation, Phys. Lett. A, 372 (2007), 21-27.
doi: 10.1016/j.physleta.2007.07.019. |
[41] |
R. Schaaf,
Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[42] |
G. Wang and J. Wei,
Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 233/234 (2002), 221-236.
doi: 10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.0.CO;2-M. |
[43] |
M. Zhuang, W. Wang and S. Zheng,
Boundedness in a fully parabolic chemotaxis system with logistic-type source and nonlinear production, Nonlinear Anal. RWA, 47 (2019), 473-483.
doi: 10.1016/j.nonrwa.2018.12.001. |
show all references
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
F. Andreu, V. Caselles, J. M. Mazón and S. Moll,
Finite propagation speed for limited flux diffusion equations, Arch. Rat. Mech. Anal., 182 (2006), 269-297.
doi: 10.1007/s00205-006-0428-3. |
[3] |
M. Arias, J. Campos and J. Soler,
Cross-diffusion and traveling waves in porous-media flux-saturated Keller-Segel models, Math. Models Meth. Appl. Sci., 28 (2018), 2103-2129.
doi: 10.1142/S0218202518400092. |
[4] |
G. Auberson and P. C. Sabatier,
On a class of homogeneous nonlinear Schrödinger equations, J. Math. Phys., 35 (1994), 4028-4040.
doi: 10.1063/1.530840. |
[5] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[6] |
N. Bellomo and M. Winkler,
A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up, Comm. PDE, 42 (2017), 436-473.
doi: 10.1080/03605302.2016.1277237. |
[7] |
A. Bellouquid, J. Nieto and L. Urrutia,
About the kinetic description of fractional diffusion equations modeling chemotaxis, Math. Models Meth. Appl. Sci., 26 (2016), 249-268.
doi: 10.1142/S0218202516400029. |
[8] |
L. Bergé,
Wave collapse in physics: Principles and applications to light and plasma waves, Phys. Rep., 303 (1998), 259-370.
doi: 10.1016/S0370-1573(97)00092-6. |
[9] |
I. Bialynicki–Birula and J. Mycielski,
Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.
doi: 10.1016/0003-4916(76)90057-9. |
[10] |
A. Blanchet, On the Parabolic-elliptic Patlak-Keller-Segel System in Dimension $2$ and Higher, Séminaire Laurent Schwartz–EDP et applications, Exposé n. Ⅷ, Palaiseau, 2013. |
[11] |
N. Bournaveas and V. Calvez,
The one-dimensional Keller-Segel model with fractional diffusion of cells, Nonlinearity, 23 (2010), 923-935.
doi: 10.1088/0951-7715/23/4/009. |
[12] |
A. O. Caldeira and A. J. Leggett,
Path integral approach to quantum Brownian motion, Physica A, 121 (1983), 587-616.
doi: 10.1016/0378-4371(83)90013-4. |
[13] |
J. Calvo, J. Campos, V. Caselles, O. Sánchez and J. Soler, Flux-saturated porous media equation and applications, JEMS Surveys in Mathematical Sciences 2 (2015), 131–218.
doi: 10.4171/EMSS/11. |
[14] |
V. Calvez, L. Corrias and M. A. Ebde,
Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension, Comm. PDE, 37 (2012), 561-584.
doi: 10.1080/03605302.2012.655824. |
[15] |
V. Calvez, B. Perthame and S. Yasuda,
Traveling wave and aggregation in a flux-limited Keller-Segel model, Kinetic & Related Models, 11 (2018), 891-909.
doi: 10.3934/krm.2018035. |
[16] |
M. A. J. Chaplain and J. I. Tello,
On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6.
doi: 10.1016/j.aml.2015.12.001. |
[17] |
W. Chen and J. Dávila,
Resonance phenomenon for a Gelfand-type problem, Nonlinear Anal., 89 (2013), 299-321.
doi: 10.1016/j.na.2013.05.008. |
[18] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu,
On a chemotaxis model with saturated chemotactic flux, Kinetic & Related Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[19] |
M. del Pino and J. Wei,
Collapsing steady states of the Keller-Segel system, Nonlinearity, 19 (2006), 661-684.
doi: 10.1088/0951-7715/19/3/007. |
[20] |
H. D. Doebner and G. A. Goldin,
On a general nonlinear Schrödinger equation admitting diffusion currents, Phys. Lett. A, 162 (1992), 397-401.
doi: 10.1016/0375-9601(92)90061-P. |
[21] |
S. A. Dyachenko, P. M. Lushnikov and N. Vladimirova,
Logarithmic scaling of the collapse in the critical Keller-Segel equation, Nonlinearity, 26 (2013), 3011-3041.
doi: 10.1088/0951-7715/26/11/3011. |
[22] |
C. Escudero,
The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.
doi: 10.1088/0951-7715/19/12/010. |
[23] |
H. Gajewski and K. Zacharias,
Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[24] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[25] |
P. Guerrero, J. L. López, J. Montejo–Gámez and J. Nieto,
Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner–Goldin type modeling quantum dissipation, J. Nonlinear Sci., 22 (2012), 631-663.
doi: 10.1007/s00332-012-9123-8. |
[26] |
Y. Huang and A. Bertozzi,
Self-similar blowup solutions to an aggregation equation in $\mathbb{R}^N$, SIAM J. Appl. Math., 70 (2010), 2582-2603.
doi: 10.1137/090774495. |
[27] |
Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, S${\bar{u}}$rikaisekikenky${\bar{u}}$sho K${\bar{o}}$ky${\bar{u}}$roku, 1025 (1998), 44–65. Variational problems and related topics (Kyoto, 1997) |
[28] |
J. L. Kazdan and F. W. Warner,
Curvature functions for compact 2-manifolds, Ann. Math., 99 (1974), 14-47.
doi: 10.2307/1971012. |
[29] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90050-6. |
[30] |
M. D. Kostin,
On the Schrödinger–Langevin equation, J. Stat. Phys., 12 (1975), 145-151.
doi: 10.1063/1.1678812. |
[31] |
C.-S. Lin, W.-M. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Diff. Equ., 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[32] |
D. Liu, Global solutions in a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production, J. Math. Phys., 61 (2020), 021503, 4pp.
doi: 10.1063/1.5111650. |
[33] |
D. Liu and Y. Tao,
Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ., 31 (2016), 379-388.
doi: 10.1007/s11766-016-3386-z. |
[34] |
J. L. López, Nonlinear Ginzburg–Landau–type approach to quantum dissipation, Phys. Rev. E., 69 (2004), 026110. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.69.026110. Google Scholar |
[35] |
J. L. López and J. Montejo-Gámez,
A hydrodynamic approach to multidimensional dissipation–based Schrödinger models from quantum Fokker–Planck dynamics, Phys. D, 238 (2009), 622-644.
doi: 10.1016/j.physd.2008.12.006. |
[36] |
J. L. López and J. Montejo-Gámez,
On a rigorous interpretation of the quantum Schrödinger-Langevin operator in bounded domains, J. Math. Anal. Appl., 383 (2011), 365-378.
doi: 10.1016/j.jmaa.2011.05.024. |
[37] |
P. M. Lushnikov,
Critical chemotactic collapse., Phys. Lett. A, 374 (2010), 1678-1685.
doi: 10.1016/j.physleta.2010.01.068. |
[38] |
B. Perthame, Transport Equations in Biology, Springer, 2007. https://www.springer.com/gp/book/9783764378417. |
[39] |
B. Perthame, N. Vauchelet and Z. Wang,
The flux-limited Keller-Segel system; properties and derivation from kinetic equtions, Rev. Mat. Iberoamericana, 36 (2020), 357-386.
doi: 10.4171/rmi/1132. |
[40] |
A. L. Sanin and A. A. Smirnovsky,
Oscillatory motion in confined potential systems with dissipation in the context of the Schrödinger-Langevin-Kostin equation, Phys. Lett. A, 372 (2007), 21-27.
doi: 10.1016/j.physleta.2007.07.019. |
[41] |
R. Schaaf,
Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[42] |
G. Wang and J. Wei,
Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 233/234 (2002), 221-236.
doi: 10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.0.CO;2-M. |
[43] |
M. Zhuang, W. Wang and S. Zheng,
Boundedness in a fully parabolic chemotaxis system with logistic-type source and nonlinear production, Nonlinear Anal. RWA, 47 (2019), 473-483.
doi: 10.1016/j.nonrwa.2018.12.001. |
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