# American Institute of Mathematical Sciences

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

## 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

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.
Citation: Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639
##### References:
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##### References:
 [1] L. Bergé, Wave collapse in physics: Principle and applications to light and plasma waves,, Phys. Rep., 303 (1998), 259. 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. 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., 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A. Google Scholar [4] T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (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, (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. 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. 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. doi: 10.1080/03605301003646713. Google Scholar [10] M. Keel and T. Tao, Endpoint Strichartz Estimates,, Amer. J. Math., 120 (1998), 955. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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, (1999). Google Scholar [22] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567. Google Scholar
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