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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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations," Textos de Metodos Matematicos, Vol. 22, Rio de Janeiro, 1989. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré Physique Théorique, 46 (1987), 113-129.  Google Scholar [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.  Google Scholar [12] C. Laurey, The Cauchy problem for a generalized Zakharov system, Diffe. Integral Equ., 8 (1995), 105-130.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [16] C. X. Miao, "The Modern Method of Nonlinear Wave Equations," Lectures in Contemporary Mathematics, No. 2, Science Press, Beijing, 2005. Google Scholar [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. Google Scholar [18] L. Nirenberg, On elliptic partial differential equations, Ann. della Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  Google Scholar [19] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Henri. Poincaré Phys. Théor., 62 (1995), 69-80.  Google Scholar [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), 111-117.  Google Scholar [21] T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger equation, J. Diff. Eq., 92 (1991), 317-330.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] I. Segal, Nonlinear semi-groups, Ann. Math., 78 (1963), 339-364. doi: 10.2307/1970347.  Google Scholar [25] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar [26] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (): 567.   Google Scholar [27] V. E. Zakharov, The collapse of Langmuir waves, Soviet Phys. JETP, 35 (1972), 908-914. Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations," Textos de Metodos Matematicos, Vol. 22, Rio de Janeiro, 1989. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré Physique Théorique, 46 (1987), 113-129.  Google Scholar [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.  Google Scholar [12] C. Laurey, The Cauchy problem for a generalized Zakharov system, Diffe. Integral Equ., 8 (1995), 105-130.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [16] C. X. Miao, "The Modern Method of Nonlinear Wave Equations," Lectures in Contemporary Mathematics, No. 2, Science Press, Beijing, 2005. Google Scholar [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. Google Scholar [18] L. Nirenberg, On elliptic partial differential equations, Ann. della Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  Google Scholar [19] M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Henri. Poincaré Phys. Théor., 62 (1995), 69-80.  Google Scholar [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), 111-117.  Google Scholar [21] T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger equation, J. Diff. Eq., 92 (1991), 317-330.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] I. Segal, Nonlinear semi-groups, Ann. Math., 78 (1963), 339-364. doi: 10.2307/1970347.  Google Scholar [25] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.  Google Scholar [26] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Commun. Math. Phys., 87 (): 567.   Google Scholar [27] V. E. Zakharov, The collapse of Langmuir waves, Soviet Phys. JETP, 35 (1972), 908-914. Google Scholar [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.  Google Scholar [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.  Google Scholar
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