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On blow-up criterion for the nonlinear Schrödinger equation
1. | School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China |
2. | Center for Applied Mathematics, Tianjin University, Tianjin 300072, China, China |
References:
[1] |
L. Bergé, Wave collapse in physics: Principle and applications to light and plasma waves, Phys. Rep., 303 (1998), 259-370.
doi: 10.1016/S0370-1573(97)00092-6. |
[2] |
D. Cao and Q. Guo, Divergent solutions to the 5D Hartree equations, Colloquium Mathematicum, 125 (2011), 255-287.
doi: 10.4064/cm125-2-10. |
[3] |
T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^ s$, Nonlinear Anal., Theory, Methods & Applications, 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[4] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. American Mathematical Society, 2003. |
[5] |
L. Glangetas and F. Merle, A Geometrical Approach of Existence of Blow up Solutions in $H^1$ for Nonlinear Schrödinger Equation, in Rep. No. R95031, Laboratoire d'Analyse Numérique, Univ. Pierre and Marie Curie, 1995. |
[6] |
J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.
doi: 10.1007/BF02099195. |
[7] |
R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equation, J. Math. Phys., 18 (1977), 1794-1797.
doi: 10.1063/1.523491. |
[8] |
Q. Guo, Nonscattering solutions to the $ L^{2} $-supercritical NLS equations, preprint, arXiv:1101.2271. |
[9] |
J. Holmer and S. Roudenko, Divergence of infinite-variance nonradial solutions to the 3d NLS equation, Comm. Partial Differ. Eqns, 35 (2010), 878-905.
doi: 10.1080/03605301003646713. |
[10] |
M. Keel and T. Tao, Endpoint Strichartz Estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[11] |
C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[12] |
R. Killip and M. Visan, Energy-supercritical NLS: Critical $\dotH^s$-bounds imply scattering, Comm. Partial Differ. Eqns, 35 (2010), 945-987.
doi: 10.1080/03605301003717084. |
[13] |
J. E. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245-263.
doi: 10.1016/0022-1236(78)90073-3. |
[14] |
F. Merle and P. Raphael, The blow-up dynamics and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math., 161 (2005), 157-222.
doi: 10.4007/annals.2005.161.157. |
[15] |
Y. Martel, Blow-up for the nonlinear Schrödinger equation in nonisotropic spaces, Nonlinear Anal., 28 (1997), 1903-1908.
doi: 10.1016/S0362-546X(96)00036-3. |
[16] |
Ch. Miao and B. Zhang, Harmonic Annlysis Method Apply to Partial Differential Equations,(in Chinese) Beijing, Science Press, 2008. |
[17] |
H. Nawa, Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power, Comm. Pure Appl. Math., 52 (1999), 193-270.
doi: 10.1002/(SICI)1097-0312(199902)52:2<193::AID-CPA2>3.0.CO;2-3. |
[18] |
T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[19] |
T. Ogawa and Y. Tsutsumi, Blowup of $H^1$-solution for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.
doi: 10.2307/2048340. |
[20] |
P. Raphael and J. Szeftel, Standing ring blow up solutions to the $N$ dimensional quintic NLS, Comm. Math. Phys., 290 (2009), 973-996.
doi: 10.1007/s00220-009-0796-2. |
[21] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer, New York, 1999. |
[22] |
M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. |
show all references
References:
[1] |
L. Bergé, Wave collapse in physics: Principle and applications to light and plasma waves, Phys. Rep., 303 (1998), 259-370.
doi: 10.1016/S0370-1573(97)00092-6. |
[2] |
D. Cao and Q. Guo, Divergent solutions to the 5D Hartree equations, Colloquium Mathematicum, 125 (2011), 255-287.
doi: 10.4064/cm125-2-10. |
[3] |
T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^ s$, Nonlinear Anal., Theory, Methods & Applications, 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[4] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. American Mathematical Society, 2003. |
[5] |
L. Glangetas and F. Merle, A Geometrical Approach of Existence of Blow up Solutions in $H^1$ for Nonlinear Schrödinger Equation, in Rep. No. R95031, Laboratoire d'Analyse Numérique, Univ. Pierre and Marie Curie, 1995. |
[6] |
J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.
doi: 10.1007/BF02099195. |
[7] |
R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equation, J. Math. Phys., 18 (1977), 1794-1797.
doi: 10.1063/1.523491. |
[8] |
Q. Guo, Nonscattering solutions to the $ L^{2} $-supercritical NLS equations, preprint, arXiv:1101.2271. |
[9] |
J. Holmer and S. Roudenko, Divergence of infinite-variance nonradial solutions to the 3d NLS equation, Comm. Partial Differ. Eqns, 35 (2010), 878-905.
doi: 10.1080/03605301003646713. |
[10] |
M. Keel and T. Tao, Endpoint Strichartz Estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[11] |
C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.
doi: 10.1007/s00222-006-0011-4. |
[12] |
R. Killip and M. Visan, Energy-supercritical NLS: Critical $\dotH^s$-bounds imply scattering, Comm. Partial Differ. Eqns, 35 (2010), 945-987.
doi: 10.1080/03605301003717084. |
[13] |
J. E. Lin and W. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245-263.
doi: 10.1016/0022-1236(78)90073-3. |
[14] |
F. Merle and P. Raphael, The blow-up dynamics and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math., 161 (2005), 157-222.
doi: 10.4007/annals.2005.161.157. |
[15] |
Y. Martel, Blow-up for the nonlinear Schrödinger equation in nonisotropic spaces, Nonlinear Anal., 28 (1997), 1903-1908.
doi: 10.1016/S0362-546X(96)00036-3. |
[16] |
Ch. Miao and B. Zhang, Harmonic Annlysis Method Apply to Partial Differential Equations,(in Chinese) Beijing, Science Press, 2008. |
[17] |
H. Nawa, Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power, Comm. Pure Appl. Math., 52 (1999), 193-270.
doi: 10.1002/(SICI)1097-0312(199902)52:2<193::AID-CPA2>3.0.CO;2-3. |
[18] |
T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[19] |
T. Ogawa and Y. Tsutsumi, Blowup of $H^1$-solution for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.
doi: 10.2307/2048340. |
[20] |
P. Raphael and J. Szeftel, Standing ring blow up solutions to the $N$ dimensional quintic NLS, Comm. Math. Phys., 290 (2009), 973-996.
doi: 10.1007/s00220-009-0796-2. |
[21] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer, New York, 1999. |
[22] |
M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. |
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