American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. American Mathematical Society, 2003.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Q. Guo, Nonscattering solutions to the $ L^{2} $-supercritical NLS equations, preprint,, , ().   Google Scholar

[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.  Google Scholar

[10]

M. Keel and T. Tao, Endpoint Strichartz Estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Ch. Miao and B. Zhang, Harmonic Annlysis Method Apply to Partial Differential Equations,(in Chinese) Beijing, Science Press, 2008. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer, New York, 1999.  Google Scholar

[22]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. American Mathematical Society, 2003.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Q. Guo, Nonscattering solutions to the $ L^{2} $-supercritical NLS equations, preprint,, , ().   Google Scholar

[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.  Google Scholar

[10]

M. Keel and T. Tao, Endpoint Strichartz Estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Ch. Miao and B. Zhang, Harmonic Annlysis Method Apply to Partial Differential Equations,(in Chinese) Beijing, Science Press, 2008. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer, New York, 1999.  Google Scholar

[22]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.   Google Scholar

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