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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 and Continuous Dynamical Systems, 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), 307-310.

[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), 489-492.

[3]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. 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), 347-375.

[5]

T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations," Textos de Metodos Matematicos, Vol. 22, Rio de Janeiro, 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), 93-125. 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), 1794-1797. 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), 1-71. 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), 123-146. doi: 10.1017/S0022377800010588.

[10]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré Physique Théorique, 46 (1987), 113-129.

[11]

T. Kato and G. Ponce, Commutator estimates for the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[12]

C. Laurey, The Cauchy problem for a generalized Zakharov system, Diffe. Integral Equ., 8 (1995), 105-130.

[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), 1-21. doi: 10.2307/1996814.

[14]

C. T. Mckinstrie and D. A. Russell, Nonlinear focusing of coupled waves, Phys. Rev. Lett., 61 (1988), 2929-2932. doi: 10.1103/PhysRevLett.61.2929.

[15]

C. X. Miao, "Harmonic Analysis and Applications to Partial Differential Equations," Second edition, Monographs on Modern Pure Mathematics, No. 89, Science Press, Beijing, 2004.

[16]

C. X. Miao, "The Modern Method of Nonlinear Wave Equations," Lectures in Contemporary Mathematics, No. 2, Science Press, Beijing, 2005.

[17]

C. X. Miao and B. Zhang, "Harmonic Analysis Method of Partial Differential Equations," Second edition, Monographs on Modern Pure Mathematics, No. 117, Science Press, Beijing, 2008.

[18]

L. Nirenberg, On elliptic partial differential equations, Ann. della Scuola Norm. Sup. Pisa, 13 (1959), 115-162.

[19]

M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Henri. Poincaré Phys. Théor., 62 (1995), 69-80.

[20]

M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in $\mathbb{R}^2$, Ann. Inst. Henri. Poincaré Phys. Théor., 63 (1995), 111-117.

[21]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger equation, J. Diff. Eq., 92 (1991), 317-330.

[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), 487-496.

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303. doi: 10.1007/BF02761595.

[24]

I. Segal, Nonlinear semi-groups, Ann. Math., 78 (1963), 339-364. doi: 10.2307/1970347.

[25]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.

[26]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983/83), 567-576.

[27]

V. E. Zakharov, The collapse of Langmuir waves, Soviet Phys. JETP, 35 (1972), 908-914.

[28]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Analysis, 48 (2002), 191-207. 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), 1429-1443. 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), 307-310.

[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), 489-492.

[3]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. 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), 347-375.

[5]

T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations," Textos de Metodos Matematicos, Vol. 22, Rio de Janeiro, 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), 93-125. 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), 1794-1797. 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), 1-71. 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), 123-146. doi: 10.1017/S0022377800010588.

[10]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré Physique Théorique, 46 (1987), 113-129.

[11]

T. Kato and G. Ponce, Commutator estimates for the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.

[12]

C. Laurey, The Cauchy problem for a generalized Zakharov system, Diffe. Integral Equ., 8 (1995), 105-130.

[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), 1-21. doi: 10.2307/1996814.

[14]

C. T. Mckinstrie and D. A. Russell, Nonlinear focusing of coupled waves, Phys. Rev. Lett., 61 (1988), 2929-2932. doi: 10.1103/PhysRevLett.61.2929.

[15]

C. X. Miao, "Harmonic Analysis and Applications to Partial Differential Equations," Second edition, Monographs on Modern Pure Mathematics, No. 89, Science Press, Beijing, 2004.

[16]

C. X. Miao, "The Modern Method of Nonlinear Wave Equations," Lectures in Contemporary Mathematics, No. 2, Science Press, Beijing, 2005.

[17]

C. X. Miao and B. Zhang, "Harmonic Analysis Method of Partial Differential Equations," Second edition, Monographs on Modern Pure Mathematics, No. 117, Science Press, Beijing, 2008.

[18]

L. Nirenberg, On elliptic partial differential equations, Ann. della Scuola Norm. Sup. Pisa, 13 (1959), 115-162.

[19]

M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Henri. Poincaré Phys. Théor., 62 (1995), 69-80.

[20]

M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in $\mathbb{R}^2$, Ann. Inst. Henri. Poincaré Phys. Théor., 63 (1995), 111-117.

[21]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger equation, J. Diff. Eq., 92 (1991), 317-330.

[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), 487-496.

[23]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22 (1975), 273-303. doi: 10.1007/BF02761595.

[24]

I. Segal, Nonlinear semi-groups, Ann. Math., 78 (1963), 339-364. doi: 10.2307/1970347.

[25]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.

[26]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983/83), 567-576.

[27]

V. E. Zakharov, The collapse of Langmuir waves, Soviet Phys. JETP, 35 (1972), 908-914.

[28]

J. Zhang, Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations, Nonlinear Analysis, 48 (2002), 191-207. 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), 1429-1443. doi: 10.1080/03605300500299539.

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