doi: 10.3934/cpaa.2020284

Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France, Department of Mathematics, HCMC University of Education, 280 An Duong Vuong, Ho Chi Minh, Vietnam

Received  April 2020 Revised  September 2020 Published  December 2020

Fund Project: This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01)

We consider the cubic nonlinear fourth-order Schrödinger equation
$ i \partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 $
on
$ \mathbb R^N, N\geq 5 $
with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.
Citation: Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020284
References:
[1]

M. Ben-ArtziH. Koch and J. C. Saut, Dispersion estimates for fourth-order Schrödinger equations, C. R. Acad. Sci., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.  Google Scholar

[2]

A. Bényi, T. Oh and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, in Excursions in Harmonic Analysis, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-20188-7_1.  Google Scholar

[3]

A. BényiT. Oh and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb R^d, d\geq 3$, T. Am. Math. Soc. Ser. B, 2 (2015), 1-50.  doi: 10.1090/btran/6.  Google Scholar

[4]

A. BényiT. Oh and O. Pocovnicu, Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb R^3$, T. Am. Math. Soc. B, 6 (2019), 114-160.  doi: 10.1090/btran/29.  Google Scholar

[5]

T. Boulenger and E. Lenzmann, Blowup for biharmonic NL4S, Ann. Sci. Éc. Norm. Supér., 50 (2017), 503-544.  doi: 10.24033/asens.2326.  Google Scholar

[6]

J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., 176 (1996), 421-445.   Google Scholar

[7]

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), 449-475.  doi: 10.1007/s00222-008-0124-z.  Google Scholar

[8]

N. BurqL. Thomann and N. Tzvetkov, Long time dynamics for the one dimensional nonlinear Schrödinger equation, Ann. Inst. Fourier, 63 (2013), 2137-2198.   Google Scholar

[9]

M. Chen and S. Zhang, Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities, Nonlinear Anal., 190 (2020), 111608. doi: 10.1016/j.na.2019.111608.  Google Scholar

[10]

Y. Deng, Two-dimensional nonlinear Schrödinger equation with random radial data, Anal. PDE, 5 (2012), 913-960.  doi: 10.2140/apde.2012.5.913.  Google Scholar

[11]

V. D. Dinh, On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 415-437.   Google Scholar

[12]

V. D. Dinh, Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space, Nonlinear Anal., 172 (2018), 115-140.  doi: 10.1016/j.na.2018.03.003.  Google Scholar

[13]

V. D. Dinh, On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation, J. Dynam. Differ. Equ., 31 (2019), 1793-1823.  doi: 10.1007/s10884-018-9690-y.  Google Scholar

[14]

V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations, arXiv: 2001.03022. Google Scholar

[15]

B. DodsonJ. Lührmann and D. Mendelson, Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation, Adv. Math., 347 (2019), 619-676.  doi: 10.1016/j.aim.2019.02.001.  Google Scholar

[16]

Q. Guo, Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Commun. Partial Differ. Equ., 41 (2016), 185-207.  doi: 10.1080/03605302.2015.1116556.  Google Scholar

[17]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[18]

S. HerrD. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\mathbb{T}^3)$, Duke Math. J., 159 (2011), 329-349.  doi: 10.1215/00127094-1415889.  Google Scholar

[19]

H. Hirayama and M. Okamoto, Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity, Discrete Contin. Dyn. Syst., 36 (2016), 6943-6974.  doi: 10.3934/dcds.2016102.  Google Scholar

[20]

H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity, arXiv: 1505.06497. Google Scholar

[21]

V. I. Karpman, Stabiliztion of solition instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.   Google Scholar

[22]

V. I. Karpman and A. G. Shagalov, Stability of solition described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210.  doi: 10.1016/S0375-9601(97)00063-7.  Google Scholar

[23]

M. Keel and T. Tao, Endpoint Strichartz estimates, Am. J. Math., 120 (1998), 955-980.   Google Scholar

[24]

R. KillipJ. Murphy and M. Visan, Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb R^4)$, Commun. Partial Differ. Equ., 44 (2019), 51-71.  doi: 10.1080/03605302.2018.1541904.  Google Scholar

[25]

H. Koch, D. Tataru and M. Visan, Dispersive equations and nonlinear waves, Birkhäuse 45, Springer Basel, 2014.  Google Scholar

[26]

J. Lührmann and D. Mendelson, Random data Cauchy theory for the nonlinear wave equations of power-type on $ \mathbb R^3$, Commun. Partial Differ. Equ., 39 (2014), 2262-2283.  doi: 10.1080/03605302.2014.933239.  Google Scholar

[27]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the defocusing energy critical nonlinear Schrödinger equation of fourth order in the radial case, J. Differ. Equ., 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[28]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy critical nonlinear Schrödinger equations of fourth order in dimensions $d\geq 9$, J. Differ. Equ., 251 (2011), 3381-3402.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[29]

C. MiaoH. Wu and J. Zhang, Scattering theory below energy for the cubic fourth-order Schrödinger equation, Math. Narchr., 288 (2015), 798-823.  doi: 10.1002/mana.201400012.  Google Scholar

[30]

T. OhM. Okamoto and O. Pocovnicu, On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities, Discrete Contin. Dyn. Syst., 39 (2019), 3479-3520.  doi: 10.3934/dcds.2019144.  Google Scholar

[31]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[32]

B. Pausader, The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 1275-1292.  doi: 10.3934/dcds.2009.24.1275.  Google Scholar

[33]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[34]

B. Pausader and S. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyper. Differ. Equ., 7 (2010), 651-705.  doi: 10.1142/S0219891610002256.  Google Scholar

[35]

B. Pausader and S. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191.  doi: 10.1088/0951-7715/26/8/2175.  Google Scholar

[36]

S. Zhang and S. Xu, The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities, Commun. Pure Appl. Anal., 19 (2020), 3367-3385.  doi: 10.3934/cpaa.2020149.  Google Scholar

[37]

S. ZhuH. Yang and J. Zhang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 7 (2010), 187-205.  doi: 10.4310/DPDE.2010.v7.n2.a4.  Google Scholar

show all references

References:
[1]

M. Ben-ArtziH. Koch and J. C. Saut, Dispersion estimates for fourth-order Schrödinger equations, C. R. Acad. Sci., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.  Google Scholar

[2]

A. Bényi, T. Oh and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, in Excursions in Harmonic Analysis, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-20188-7_1.  Google Scholar

[3]

A. BényiT. Oh and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb R^d, d\geq 3$, T. Am. Math. Soc. Ser. B, 2 (2015), 1-50.  doi: 10.1090/btran/6.  Google Scholar

[4]

A. BényiT. Oh and O. Pocovnicu, Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb R^3$, T. Am. Math. Soc. B, 6 (2019), 114-160.  doi: 10.1090/btran/29.  Google Scholar

[5]

T. Boulenger and E. Lenzmann, Blowup for biharmonic NL4S, Ann. Sci. Éc. Norm. Supér., 50 (2017), 503-544.  doi: 10.24033/asens.2326.  Google Scholar

[6]

J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., 176 (1996), 421-445.   Google Scholar

[7]

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), 449-475.  doi: 10.1007/s00222-008-0124-z.  Google Scholar

[8]

N. BurqL. Thomann and N. Tzvetkov, Long time dynamics for the one dimensional nonlinear Schrödinger equation, Ann. Inst. Fourier, 63 (2013), 2137-2198.   Google Scholar

[9]

M. Chen and S. Zhang, Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities, Nonlinear Anal., 190 (2020), 111608. doi: 10.1016/j.na.2019.111608.  Google Scholar

[10]

Y. Deng, Two-dimensional nonlinear Schrödinger equation with random radial data, Anal. PDE, 5 (2012), 913-960.  doi: 10.2140/apde.2012.5.913.  Google Scholar

[11]

V. D. Dinh, On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 415-437.   Google Scholar

[12]

V. D. Dinh, Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space, Nonlinear Anal., 172 (2018), 115-140.  doi: 10.1016/j.na.2018.03.003.  Google Scholar

[13]

V. D. Dinh, On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation, J. Dynam. Differ. Equ., 31 (2019), 1793-1823.  doi: 10.1007/s10884-018-9690-y.  Google Scholar

[14]

V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations, arXiv: 2001.03022. Google Scholar

[15]

B. DodsonJ. Lührmann and D. Mendelson, Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation, Adv. Math., 347 (2019), 619-676.  doi: 10.1016/j.aim.2019.02.001.  Google Scholar

[16]

Q. Guo, Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Commun. Partial Differ. Equ., 41 (2016), 185-207.  doi: 10.1080/03605302.2015.1116556.  Google Scholar

[17]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[18]

S. HerrD. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\mathbb{T}^3)$, Duke Math. J., 159 (2011), 329-349.  doi: 10.1215/00127094-1415889.  Google Scholar

[19]

H. Hirayama and M. Okamoto, Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity, Discrete Contin. Dyn. Syst., 36 (2016), 6943-6974.  doi: 10.3934/dcds.2016102.  Google Scholar

[20]

H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity, arXiv: 1505.06497. Google Scholar

[21]

V. I. Karpman, Stabiliztion of solition instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.   Google Scholar

[22]

V. I. Karpman and A. G. Shagalov, Stability of solition described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210.  doi: 10.1016/S0375-9601(97)00063-7.  Google Scholar

[23]

M. Keel and T. Tao, Endpoint Strichartz estimates, Am. J. Math., 120 (1998), 955-980.   Google Scholar

[24]

R. KillipJ. Murphy and M. Visan, Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb R^4)$, Commun. Partial Differ. Equ., 44 (2019), 51-71.  doi: 10.1080/03605302.2018.1541904.  Google Scholar

[25]

H. Koch, D. Tataru and M. Visan, Dispersive equations and nonlinear waves, Birkhäuse 45, Springer Basel, 2014.  Google Scholar

[26]

J. Lührmann and D. Mendelson, Random data Cauchy theory for the nonlinear wave equations of power-type on $ \mathbb R^3$, Commun. Partial Differ. Equ., 39 (2014), 2262-2283.  doi: 10.1080/03605302.2014.933239.  Google Scholar

[27]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the defocusing energy critical nonlinear Schrödinger equation of fourth order in the radial case, J. Differ. Equ., 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[28]

C. MiaoG. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy critical nonlinear Schrödinger equations of fourth order in dimensions $d\geq 9$, J. Differ. Equ., 251 (2011), 3381-3402.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[29]

C. MiaoH. Wu and J. Zhang, Scattering theory below energy for the cubic fourth-order Schrödinger equation, Math. Narchr., 288 (2015), 798-823.  doi: 10.1002/mana.201400012.  Google Scholar

[30]

T. OhM. Okamoto and O. Pocovnicu, On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities, Discrete Contin. Dyn. Syst., 39 (2019), 3479-3520.  doi: 10.3934/dcds.2019144.  Google Scholar

[31]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[32]

B. Pausader, The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 1275-1292.  doi: 10.3934/dcds.2009.24.1275.  Google Scholar

[33]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[34]

B. Pausader and S. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyper. Differ. Equ., 7 (2010), 651-705.  doi: 10.1142/S0219891610002256.  Google Scholar

[35]

B. Pausader and S. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191.  doi: 10.1088/0951-7715/26/8/2175.  Google Scholar

[36]

S. Zhang and S. Xu, The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities, Commun. Pure Appl. Anal., 19 (2020), 3367-3385.  doi: 10.3934/cpaa.2020149.  Google Scholar

[37]

S. ZhuH. Yang and J. Zhang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 7 (2010), 187-205.  doi: 10.4310/DPDE.2010.v7.n2.a4.  Google Scholar

[1]

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

[2]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[3]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[4]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[5]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[6]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[7]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[8]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

[9]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[10]

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, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[11]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[12]

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, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[13]

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

[14]

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

[15]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[16]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[17]

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

[18]

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

[19]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[20]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328

2019 Impact Factor: 1.105

Article outline

[Back to Top]