American Institute of Mathematical Sciences

Advanced
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  November 2020

Fund Project: The author is partially supported by MINECO (Spain), Project MTM2014-53406-R, FEDER resources, as well as by Junta de Andalucía Project P12-FQM-954

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 - A, 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

[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

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

[1]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264
[2]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[3]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276
[4]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[5]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[6]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116
[7]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[8]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[9]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[10]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[11]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[12]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[13]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[14]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[15]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[16]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336
[17]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[18]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
[19]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[20]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

2019 Impact Factor: 1.338

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences