• Previous Article
    Second order estimates for complex Hessian equations on Hermitian manifolds
  • DCDS Home
  • This Issue
  • Next Article
    Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems
June  2021, 41(6): 2601-2617. doi: 10.3934/dcds.2020376

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

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.

Citation: José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete and 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.

[2]

F. AndreuV. CasellesJ. 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. AriasJ. 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. BellomoA. BellouquidY. 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. BellouquidJ. 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. CalvezL. 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. CalvezB. 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. ChertockA. KurganovX. 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. DyachenkoP. 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. GuerreroJ. L. LópezJ. 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. LinW.-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.

[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. PerthameN. 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. ZhuangW. 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. AndreuV. CasellesJ. 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. AriasJ. 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. BellomoA. BellouquidY. 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. BellouquidJ. 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. CalvezL. 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. CalvezB. 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. ChertockA. KurganovX. 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. DyachenkoP. 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. GuerreroJ. L. LópezJ. 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. LinW.-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.

[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. PerthameN. 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. ZhuangW. 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.

[1]

José Luis López, Jesús Montejo-Gámez. On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models. Kinetic and Related Models, 2012, 5 (3) : 517-536. doi: 10.3934/krm.2012.5.517

[2]

Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038

[3]

Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078

[4]

Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317

[5]

Kei Nakamura, Tohru Ozawa. Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 789-801. doi: 10.3934/dcds.2013.33.789

[6]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613

[7]

Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023

[8]

Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231

[9]

J. Ignacio Tello. Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022045

[10]

J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445

[11]

Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102

[12]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031

[13]

Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks and Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181

[14]

Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure and Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831

[15]

Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503

[16]

Pascal Bégout, Jesús Ildefonso Díaz. A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3371-3382. doi: 10.3934/dcds.2014.34.3371

[17]

Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations and Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016

[18]

Marco Di Francesco, Donatella Donatelli. Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 79-100. doi: 10.3934/dcdsb.2010.13.79

[19]

Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure and Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339

[20]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure and Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (279)
  • HTML views (199)
  • Cited by (0)

Other articles
by authors

[Back to Top]