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June  2021, 41(6): 2601-2617. doi: 10.3934/dcds.2020376

A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation

José Luis López

Departamento de Matemática Aplicada and Excellence Research Unit "Modeling Nature" (MNat), Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Received  March 2020 Revised  September 2020 Published  June 2021 Early access  November 2020

Fund Project: The author is partially supported by the MINECO-Feder (Spain) research grant number RTI2018-098850-B-I00, as well as by the Junta de Andalucía (Spain) Project PY18-RT-2422 & A-FQM-311-UGR18

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.

Keywords: Nonlinear Schrödinger equation, cubic nonlinearity, Kostin nonlinearity, quantum dissipation, Doebner-Goldin equations, chemotaxis, Keller-Segel system, quasi-stationary solutions.
Mathematics Subject Classification: Primary: 35-XX; Secondary: 35Q40, 35K57, 35J60.
Citation: José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Bialynicki–Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.  doi: 10.1088/0951-7715/19/12/010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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)  Google Scholar

[28]

J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. Math., 99 (1974), 14-47.  doi: 10.2307/1971012.  Google Scholar

[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.  Google Scholar

[30]

M. D. Kostin, On the Schrödinger–Langevin equation, J. Stat. Phys., 12 (1975), 145-151.  doi: 10.1063/1.1678812.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

P. M. Lushnikov, Critical chemotactic collapse., Phys. Lett. A, 374 (2010), 1678-1685.  doi: 10.1016/j.physleta.2010.01.068.  Google Scholar

[38]

B. Perthame, Transport Equations in Biology, Springer, 2007. https://www.springer.com/gp/book/9783764378417.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

I. Bialynicki–Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.  doi: 10.1088/0951-7715/19/12/010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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)  Google Scholar

[28]

J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. Math., 99 (1974), 14-47.  doi: 10.2307/1971012.  Google Scholar

[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.  Google Scholar

[30]

M. D. Kostin, On the Schrödinger–Langevin equation, J. Stat. Phys., 12 (1975), 145-151.  doi: 10.1063/1.1678812.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

P. M. Lushnikov, Critical chemotactic collapse., Phys. Lett. A, 374 (2010), 1678-1685.  doi: 10.1016/j.physleta.2010.01.068.  Google Scholar

[38]

B. Perthame, Transport Equations in Biology, Springer, 2007. https://www.springer.com/gp/book/9783764378417.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.  doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475

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