On blow-up criterion for the nonlinear Schrödinger equation

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July 2016, 36(7): 3639-3650. doi: 10.3934/dcds.2016.36.3639

On blow-up criterion for the nonlinear Schrödinger equation

Dapeng Du ^1, , Yifei Wu ^2, and Kaijun Zhang ^2,

1.	School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2.	Center for Applied Mathematics, Tianjin University, Tianjin 300072, China, China

Received March 2015 Revised November 2015 Published March 2016

Abstract
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The blowup is studied for the nonlinear Schrödinger equation $iu_{t}+\Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $p\ge 1+\frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)<0$ blows up in finite or infinite time. A new proof is also presented for the previous result in [9], in which a similar result in a case of energy-subcritical was shown.

Keywords: Localized virial identity, local well-posedness., Glassey's argument, blow-up, Nonlinear Schrödinger equation.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B4.

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

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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