American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020393

Global attractor for damped forced nonlinear logarithmic Schrödinger equations

Olivier Goubet 1,, and  Ezzeddine Zahrouni 2,

1. 

LAMFA, UMR 7352 CNRS UPJV, 33 rue St Leu, 80039, Amiens Cedex, France
2. 

Equipe de recherche Analyse, Probabilités et Fractals, Av. de l'environnement, 5000 Monastir, Tunisie

* Corresponding author: Olivier Goubet

Received  October 2019 Revised  February 2020 Published  June 2020

Fund Project: This article is dedicated to the memory of Ezzeddine Zahrouni who passed away in december 2018.

We consider here a damped forced nonlinear logarithmic Schrödinger equation in $ \mathbb{R}^N $. We prove the existence of a global attractor in a suitable energy space. We complete this article with some open issues for nonlinear logarithmic Schrödinger equations in the framework of infinite-dimensional dynamical systems.

Keywords: Global attractor, damped forced nonlinear logarithmic Schrodinger equations.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B41.
Citation: Olivier Goubet, Ezzeddine Zahrouni. Global attractor for damped forced nonlinear logarithmic Schrödinger equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020393
References:
[1]

M. Abounouh, Asymptotic behaviour for a weakly damped Schrödinger equation in dimension two, Appl. Math. Lett., 6 (1993), 29-32.  doi: 10.1016/0893-9659(93)90073-V.  Google Scholar

[2]

M. Abounouh, H. Al Moatassime, J. P. Chehab and O. Goubet, Discrete Schrödinger equations and dissipative dynamical systems, Commun. Pure Appl. Anal., 7 (2008), 211-127.  doi: 10.3934/cpaa.2008.7.211.  Google Scholar

[3]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.  Google Scholar

[4]

A. H. Ardila, Logarithmic Schrödinger equation: On the orbital stability of the Gausson, Journal of Differential Equations, 335 (2016), 1-9.   Google Scholar

[5]

A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Analysis, 155 (2017), 52-64.  doi: 10.3934/eect.2017009.  Google Scholar

[6]

A. H. Ardila, Stability of ground states for logarithmic Schrödinger equation with a $\delta'$-interaction, Evol. Equ. Control Theory, 6 (2017), 155-175.  doi: 10.3934/eect.2017009.  Google Scholar

[7]

A. H. Ardila and M. Squassina, Gausson dynamic for logarithmic Schrödinger equations, Asymptot. Anal., 107 (2018), 203-226.  doi: 10.3233/ASY-171458.  Google Scholar

[8]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, 348. Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.  Google Scholar

[9]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[10]

W. Z. Bao, R. Carles, C. M. Su and Q. L. Tang, Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation, SIAM J. Numer. Anal., 57 (2019), 657-680.  doi: 10.1137/18M1177445.  Google Scholar

[11]

W. Z. Bao, R. Carles, C. M. Su and Q. L. Tang, Regularized numerical methods for the logarithmic Schrödinger equation, Numer. Math., 143 (2019), 461-487.  doi: 10.1007/s00211-019-01058-2.  Google Scholar

[12]

W. Z. Bao and D. Jacksch, An explicit unconditionaly stable numerical method for solving damped nonlinear Schrödinger equation with focusing nonlinearity, SIAM J. Numer. Anal., 41 (2003), 1406-1426.  doi: 10.1137/S0036142902413391.  Google Scholar

[13]

P. Benilan, M. G. Crandall and A. Pazy, "Bonnes solutions" d'un problème d'évolution semi-linéaire, C. R. Acad. Sci. Paris sér I Math., 306 (1988), 527-530.   Google Scholar

[14]

C. Besse, A relaxation scheme for nonlinear Schrödinger equations, SIAM J. Num. Anal., 42 (2004), 934-952.  doi: 10.1137/S0036142901396521.  Google Scholar

[15]

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

[16]

H. R. Brezis, Les opérateurs monotones, Secrétariat Mathématique, Paris, Séminaire Choquet: 1965/66, Initiation à l' Analyse, Fasc. 2, Exp. 10, (1968), 33 pp.  Google Scholar

[17]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.  Google Scholar

[18]

C. Calgaro, O. Goubet and E. Zahrouni, Finite-dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation, Math. Methods Appl. Sci., 40 (2017), 5563-5574.  doi: 10.1002/mma.4409.  Google Scholar

[19]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.  Google Scholar

[20]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[21]

T. Cazenave and A. Haraux Équation d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 21–51. doi: 10.5802/afst.543.  Google Scholar

[22]

P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15 pp. doi: 10.1142/S0219199713500326.  Google Scholar

[23]

E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schrödinger equation, Differential Integral Equations, 23 (2010), 237-252.   Google Scholar

[24]

G. Ferriere, The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition, arXiv: 1910.09436v1. Google Scholar

[25]

C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differential Equations, 9 (2004), 509-538.   Google Scholar

[26]

P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 765-779.  doi: 10.1016/j.anihpc.2005.09.004.  Google Scholar

[27]

J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405.  doi: 10.1016/S0294-1449(16)30343-2.  Google Scholar

[28]

J.-M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schr${{\rm{\ddot d}}}$inger equations. Attractors, inertial manifolds and their approximation, RAIRO Modél. Math. Anal. Numér., 23 (1989), 433-443.  doi: 10.1051/m2an/1989230304331.  Google Scholar

[29]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119.  doi: 10.1080/00036819608840420.  Google Scholar

[30]

O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schrödinger equations in $L^2(\mathbb{R})$, Nonlinear Anal., 71 (2009), 317-320.  doi: 10.1016/j.na.2008.10.078.  Google Scholar

[31]

O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 7 (2008), 1429-1442.  doi: 10.3934/cpaa.2008.7.1429.  Google Scholar

[32]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.  Google Scholar

[33]

K. N. Lu and B. X. Wang, Global attractor for the Klein-Gordon-Schrödinger equations in unbounded domains, Journal of Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.  Google Scholar

[34]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 103-200.  doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[35]

G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885-982.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[36]

J. M. Sanz-Serna and J. G. Verwer, Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. of Num. Anal., 6 (1986), 25-42.  doi: 10.1093/imanum/6.1.25.  Google Scholar

[37]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[38]

K. Tsugawa, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318.  doi: 10.3934/cpaa.2004.3.301.  Google Scholar

show all references

References:
[1]

M. Abounouh, Asymptotic behaviour for a weakly damped Schrödinger equation in dimension two, Appl. Math. Lett., 6 (1993), 29-32.  doi: 10.1016/0893-9659(93)90073-V.  Google Scholar

[2]

M. Abounouh, H. Al Moatassime, J. P. Chehab and O. Goubet, Discrete Schrödinger equations and dissipative dynamical systems, Commun. Pure Appl. Anal., 7 (2008), 211-127.  doi: 10.3934/cpaa.2008.7.211.  Google Scholar

[3]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.  Google Scholar

[4]

A. H. Ardila, Logarithmic Schrödinger equation: On the orbital stability of the Gausson, Journal of Differential Equations, 335 (2016), 1-9.   Google Scholar

[5]

A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Analysis, 155 (2017), 52-64.  doi: 10.3934/eect.2017009.  Google Scholar

[6]

A. H. Ardila, Stability of ground states for logarithmic Schrödinger equation with a $\delta'$-interaction, Evol. Equ. Control Theory, 6 (2017), 155-175.  doi: 10.3934/eect.2017009.  Google Scholar

[7]

A. H. Ardila and M. Squassina, Gausson dynamic for logarithmic Schrödinger equations, Asymptot. Anal., 107 (2018), 203-226.  doi: 10.3233/ASY-171458.  Google Scholar

[8]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, 348. Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.  Google Scholar

[9]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[10]

W. Z. Bao, R. Carles, C. M. Su and Q. L. Tang, Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation, SIAM J. Numer. Anal., 57 (2019), 657-680.  doi: 10.1137/18M1177445.  Google Scholar

[11]

W. Z. Bao, R. Carles, C. M. Su and Q. L. Tang, Regularized numerical methods for the logarithmic Schrödinger equation, Numer. Math., 143 (2019), 461-487.  doi: 10.1007/s00211-019-01058-2.  Google Scholar

[12]

W. Z. Bao and D. Jacksch, An explicit unconditionaly stable numerical method for solving damped nonlinear Schrödinger equation with focusing nonlinearity, SIAM J. Numer. Anal., 41 (2003), 1406-1426.  doi: 10.1137/S0036142902413391.  Google Scholar

[13]

P. Benilan, M. G. Crandall and A. Pazy, "Bonnes solutions" d'un problème d'évolution semi-linéaire, C. R. Acad. Sci. Paris sér I Math., 306 (1988), 527-530.   Google Scholar

[14]

C. Besse, A relaxation scheme for nonlinear Schrödinger equations, SIAM J. Num. Anal., 42 (2004), 934-952.  doi: 10.1137/S0036142901396521.  Google Scholar

[15]

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

[16]

H. R. Brezis, Les opérateurs monotones, Secrétariat Mathématique, Paris, Séminaire Choquet: 1965/66, Initiation à l' Analyse, Fasc. 2, Exp. 10, (1968), 33 pp.  Google Scholar

[17]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.  Google Scholar

[18]

C. Calgaro, O. Goubet and E. Zahrouni, Finite-dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation, Math. Methods Appl. Sci., 40 (2017), 5563-5574.  doi: 10.1002/mma.4409.  Google Scholar

[19]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.  Google Scholar

[20]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[21]

T. Cazenave and A. Haraux Équation d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 21–51. doi: 10.5802/afst.543.  Google Scholar

[22]

P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15 pp. doi: 10.1142/S0219199713500326.  Google Scholar

[23]

E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schrödinger equation, Differential Integral Equations, 23 (2010), 237-252.   Google Scholar

[24]

G. Ferriere, The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition, arXiv: 1910.09436v1. Google Scholar

[25]

C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differential Equations, 9 (2004), 509-538.   Google Scholar

[26]

P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 765-779.  doi: 10.1016/j.anihpc.2005.09.004.  Google Scholar

[27]

J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405.  doi: 10.1016/S0294-1449(16)30343-2.  Google Scholar

[28]

J.-M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schr${{\rm{\ddot d}}}$inger equations. Attractors, inertial manifolds and their approximation, RAIRO Modél. Math. Anal. Numér., 23 (1989), 433-443.  doi: 10.1051/m2an/1989230304331.  Google Scholar

[29]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119.  doi: 10.1080/00036819608840420.  Google Scholar

[30]

O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schrödinger equations in $L^2(\mathbb{R})$, Nonlinear Anal., 71 (2009), 317-320.  doi: 10.1016/j.na.2008.10.078.  Google Scholar

[31]

O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 7 (2008), 1429-1442.  doi: 10.3934/cpaa.2008.7.1429.  Google Scholar

[32]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.  Google Scholar

[33]

K. N. Lu and B. X. Wang, Global attractor for the Klein-Gordon-Schrödinger equations in unbounded domains, Journal of Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.  Google Scholar

[34]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 103-200.  doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[35]

G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885-982.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[36]

J. M. Sanz-Serna and J. G. Verwer, Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. of Num. Anal., 6 (1986), 25-42.  doi: 10.1093/imanum/6.1.25.  Google Scholar

[37]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[38]

K. Tsugawa, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318.  doi: 10.3934/cpaa.2004.3.301.  Google Scholar

[1]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[2]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147
[3]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270
[4]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[5]

Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364
[6]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[7]

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
[8]

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
[9]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033
[10]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297
[11]

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

Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046
[13]

Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160
[14]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
[15]

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
[16]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229
[17]

Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053
[18]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[19]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267
[20]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

2019 Impact Factor: 1.233

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences