• Previous Article
    Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent
  • DCDS Home
  • This Issue
  • Next Article
    Global solutions for a semilinear heat equation in the exterior domain of a compact set
March  2012, 32(3): 827-846. doi: 10.3934/dcds.2012.32.827

Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations

1. 

College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068

Received  March 2010 Revised  July 2011 Published  October 2011

We study blow-up, global existence and standing waves for the nonlinear Schrödinger equations with two-dimensional magnetic field in a cold plasma. Under certain conditions on initial data and initial energy, we derive finite time blow-up phenomena of the solutions to the equations under study. Using compactness and Lagrange multiplier method, we establish the existence of standing waves. Finally, by introducing invariant manifolds and utilizing potential well argument as well as concavity method, we obtain the sharp threshold for global existence and blowup.
Citation: Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827
References:
[1]

H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens nonlinéaires daus de plan,, C. R. Acad. Sci. Paris. Série I Math., 297 (1983), 307.

[2]

H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires,, C. R. Acad. Sci. Paris Sér. I. Math., 293 (1981), 489.

[3]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rat. Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555.

[4]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, Arch. Rat. Mech. Anal., 82 (1983), 347.

[5]

T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations,", Textos de Metodos Matematicos, (1989).

[6]

Z. H. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,, Commun. Math. Phys., 283 (2008), 93. doi: 10.1007/s00220-008-0456-y.

[7]

R. T. Glassey, On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger equations,, J. Math. Phys., 18 (1977), 1794. doi: 10.1063/1.523491.

[8]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I, II. The Cauchy problem, general case. Scattering theory, general case,, J. Funct. Anal., 32 (1979), 1. doi: 10.1016/0022-1236(79)90076-4.

[9]

M. Kono, M. M. Skoric and D. Ter Haar, Spontaneous excitation of magnetic fields and collapse dynamics in a Langmuir plasma,, J. Plasma Phys., 26 (1981), 123. doi: 10.1017/S0022377800010588.

[10]

T. Kato, On nonlinear Schrödinger equations,, Ann. Inst. Henri Poincaré Physique Théorique, 46 (1987), 113.

[11]

T. Kato and G. Ponce, Commutator estimates for the Euler and Navier-Stokes equations,, Commun. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704.

[12]

C. Laurey, The Cauchy problem for a generalized Zakharov system,, Diffe. Integral Equ., 8 (1995), 105.

[13]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{\mathcalt\mathcalt}=-Au-F(u)$,, Transactions of the American Mathematical Society, 192 (1974), 1. doi: 10.2307/1996814.

[14]

C. T. Mckinstrie and D. A. Russell, Nonlinear focusing of coupled waves,, Phys. Rev. Lett., 61 (1988), 2929. doi: 10.1103/PhysRevLett.61.2929.

[15]

C. X. Miao, "Harmonic Analysis and Applications to Partial Differential Equations,", Second edition, (2004).

[16]

C. X. Miao, "The Modern Method of Nonlinear Wave Equations,", Lectures in Contemporary Mathematics, (2005).

[17]

C. X. Miao and B. Zhang, "Harmonic Analysis Method of Partial Differential Equations,", Second edition, (2008).

[18]

L. Nirenberg, On elliptic partial differential equations,, Ann. della Scuola Norm. Sup. Pisa, 13 (1959), 115.

[19]

M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system,, Ann. Inst. Henri. Poincaré Phys. Théor., 62 (1995), 69.

[20]

M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in $\mathbbR^2$,, Ann. Inst. Henri. Poincaré Phys. Théor., 63 (1995), 111.

[21]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger equation,, J. Diff. Eq., 92 (1991), 317.

[22]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solutions for the one-dimension nonlinear Schrödinger equation with critical power nonlinearity,, Proc. Amer. Math. Soc., 111 (1991), 487.

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel Journal of Mathematics, 22 (1975), 273. doi: 10.1007/BF02761595.

[24]

I. Segal, Nonlinear semi-groups,, Ann. Math., 78 (1963), 339. doi: 10.2307/1970347.

[25]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517.

[26]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (): 567.

[27]

V. E. Zakharov, The collapse of Langmuir waves,, Soviet Phys. JETP, 35 (1972), 908.

[28]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations,, Nonlinear Analysis, 48 (2002), 191. doi: 10.1016/S0362-546X(00)00180-2.

[29]

J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential,, Commun. in PDE, 30 (2005), 1429. doi: 10.1080/03605300500299539.

show all references

References:
[1]

H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens nonlinéaires daus de plan,, C. R. Acad. Sci. Paris. Série I Math., 297 (1983), 307.

[2]

H. Berestycki and T. Cazenave, Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires,, C. R. Acad. Sci. Paris Sér. I. Math., 293 (1981), 489.

[3]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rat. Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555.

[4]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, Arch. Rat. Mech. Anal., 82 (1983), 347.

[5]

T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations,", Textos de Metodos Matematicos, (1989).

[6]

Z. H. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,, Commun. Math. Phys., 283 (2008), 93. doi: 10.1007/s00220-008-0456-y.

[7]

R. T. Glassey, On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger equations,, J. Math. Phys., 18 (1977), 1794. doi: 10.1063/1.523491.

[8]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I, II. The Cauchy problem, general case. Scattering theory, general case,, J. Funct. Anal., 32 (1979), 1. doi: 10.1016/0022-1236(79)90076-4.

[9]

M. Kono, M. M. Skoric and D. Ter Haar, Spontaneous excitation of magnetic fields and collapse dynamics in a Langmuir plasma,, J. Plasma Phys., 26 (1981), 123. doi: 10.1017/S0022377800010588.

[10]

T. Kato, On nonlinear Schrödinger equations,, Ann. Inst. Henri Poincaré Physique Théorique, 46 (1987), 113.

[11]

T. Kato and G. Ponce, Commutator estimates for the Euler and Navier-Stokes equations,, Commun. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704.

[12]

C. Laurey, The Cauchy problem for a generalized Zakharov system,, Diffe. Integral Equ., 8 (1995), 105.

[13]

H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{\mathcalt\mathcalt}=-Au-F(u)$,, Transactions of the American Mathematical Society, 192 (1974), 1. doi: 10.2307/1996814.

[14]

C. T. Mckinstrie and D. A. Russell, Nonlinear focusing of coupled waves,, Phys. Rev. Lett., 61 (1988), 2929. doi: 10.1103/PhysRevLett.61.2929.

[15]

C. X. Miao, "Harmonic Analysis and Applications to Partial Differential Equations,", Second edition, (2004).

[16]

C. X. Miao, "The Modern Method of Nonlinear Wave Equations,", Lectures in Contemporary Mathematics, (2005).

[17]

C. X. Miao and B. Zhang, "Harmonic Analysis Method of Partial Differential Equations,", Second edition, (2008).

[18]

L. Nirenberg, On elliptic partial differential equations,, Ann. della Scuola Norm. Sup. Pisa, 13 (1959), 115.

[19]

M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system,, Ann. Inst. Henri. Poincaré Phys. Théor., 62 (1995), 69.

[20]

M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in $\mathbbR^2$,, Ann. Inst. Henri. Poincaré Phys. Théor., 63 (1995), 111.

[21]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger equation,, J. Diff. Eq., 92 (1991), 317.

[22]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solutions for the one-dimension nonlinear Schrödinger equation with critical power nonlinearity,, Proc. Amer. Math. Soc., 111 (1991), 487.

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel Journal of Mathematics, 22 (1975), 273. doi: 10.1007/BF02761595.

[24]

I. Segal, Nonlinear semi-groups,, Ann. Math., 78 (1963), 339. doi: 10.2307/1970347.

[25]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Commun. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517.

[26]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (): 567.

[27]

V. E. Zakharov, The collapse of Langmuir waves,, Soviet Phys. JETP, 35 (1972), 908.

[28]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations,, Nonlinear Analysis, 48 (2002), 191. doi: 10.1016/S0362-546X(00)00180-2.

[29]

J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential,, Commun. in PDE, 30 (2005), 1429. doi: 10.1080/03605300500299539.

[1]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

[2]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639

[3]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11

[4]

Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010

[5]

Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825

[6]

Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669

[7]

Laurent Di Menza, Olivier Goubet. Stabilizing blow up solutions to nonlinear schrÖdinger equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1059-1082. doi: 10.3934/cpaa.2017051

[8]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[9]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[10]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[11]

Jianqing Chen, Boling Guo. Sharp global existence and blowing up results for inhomogeneous Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 357-367. doi: 10.3934/dcdsb.2007.8.357

[12]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399

[13]

Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721

[14]

Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535

[15]

Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077

[16]

Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051

[17]

Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193

[18]

François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229

[19]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[20]

Mingqi Xiang, Patrizia Pucci, Marco Squassina, Binlin Zhang. Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1631-1649. doi: 10.3934/dcds.2017067

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]