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:
[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.

[2]

D. Cao and Q. Guo, Divergent solutions to the 5D Hartree equations,, Colloquium Mathematicum, 125 (2011), 255. 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., 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A.

[4]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (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, (1995).

[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.

[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.

[8]

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

[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.

[10]

M. Keel and T. Tao, Endpoint Strichartz Estimates,, Amer. J. Math., 120 (1998), 955. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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, (1999).

[22]

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

show all references

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.

[2]

D. Cao and Q. Guo, Divergent solutions to the 5D Hartree equations,, Colloquium Mathematicum, 125 (2011), 255. 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., 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A.

[4]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (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, (1995).

[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.

[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.

[8]

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

[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.

[10]

M. Keel and T. Tao, Endpoint Strichartz Estimates,, Amer. J. Math., 120 (1998), 955. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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, (1999).

[22]

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

[1]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781

[2]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[3]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

[4]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[5]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[6]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

[7]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[8]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203

[9]

Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683

[10]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501

[11]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493

[12]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042

[13]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11

[14]

Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827

[15]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

[16]

Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066

[17]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[18]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072

[19]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123

[20]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]