American Institute of Mathematical Sciences

Advanced
November  2016, 15(6): 2023-2058. doi: 10.3934/cpaa.2016026

Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data

Kenji Nakanishi 1, and  Tristan Roy 2,

1. 

Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology Osaka University, Suita, Osaka 565-0871, Japan
2. 

Graduate School of Mathematics, Nagoya University, Japan

Received  October 2015 Revised  June 2016 Published  September 2016

Consider the focusing energy critical Schrödinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with [19], the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference between [19] and this paper is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.
Keywords: blow-up, solitons, Sobolev critical exponent., scattering theory, nonlinear Schrödinger equation.
Mathematics Subject Classification: Primary: 35Q5.
Citation: Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026
References:
[1]

T. Aubin, Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire, J. Math. Pures. Appl., 55 (1976), 269-296.  Google Scholar

[2]

H. Bahouri and P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  Google Scholar

[3]

J. Bourgain, Global well-posedness of defocusing $3D$ critical NLS in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[6]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x.  Google Scholar

[7]

P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, \emph{S\'eminaire E.D.P} (1996-1997), (): 1996.   Google Scholar

[8]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure. Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.  Google Scholar

[9]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  Google Scholar

[10]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.  Google Scholar

[11]

C. E. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical, focusing, non-Linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.  Google Scholar

[12]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation, Amer. J. Math., 135 (2013), 935-965. doi: 10.1353/ajm.2013.0034.  Google Scholar

[13]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics of the nonradial energy-critical wave equation above the ground state energy, Discrete, Cont, Dyn. Syst., 33 (2013), 2423-2450.  Google Scholar

[14]

P. L. Lions, The concentration-compactness principle in the calculus of variations. (The limit case, Part I.), Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[15]

F. Merle and L. Vega, Compactness at blow-up time for $L^{2}$ solutions of the critical nonlinear Schrödinger equation in $2D$, Internat. Math. Res. Notices, 1998, 399-425. doi: 10.1155/S1073792898000270.  Google Scholar

[16]

K. Nakanishi, Energy scattering for nonlinear Klein-Gordon equation and Schrödinger equation in spatial dimensions $1$ and $2$, J. Funct. Anal., 169 (1999), 201-225. doi: 10.1006/jfan.1999.3503.  Google Scholar

[17]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Diff. Eq., 250 (2011), 2299-2333. doi: 10.1016/j.jde.2010.10.027.  Google Scholar

[18]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation without a radial assumption, Arch. Rational Mech. Analysis, 203 (2012), 809-851. doi: 10.1007/s00205-011-0462-7.  Google Scholar

[19]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. and PDE, 44 (2012), 1-45. doi: 10.1007/s00526-011-0424-9.  Google Scholar

[20]

K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095.  Google Scholar

[21]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger qquation, J. Diff. Eq., 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[22]

W. Schlag, Spectral theory and nonlinear differential equations: a survey, Discrete, Cont, Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.  Google Scholar

[23]

G. Talenti, Best Constant In Sobolev Inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  Google Scholar

show all references

References:
[1]

T. Aubin, Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire, J. Math. Pures. Appl., 55 (1976), 269-296.  Google Scholar

[2]

H. Bahouri and P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  Google Scholar

[3]

J. Bourgain, Global well-posedness of defocusing $3D$ critical NLS in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.  Google Scholar

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[6]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x.  Google Scholar

[7]

P. Gerard, Y. Meyer and F. Oru, Inégalités de Sobolev précisées,, \emph{S\'eminaire E.D.P} (1996-1997), (): 1996.   Google Scholar

[8]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure. Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.  Google Scholar

[9]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  Google Scholar

[10]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Diff. Eq., 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.  Google Scholar

[11]

C. E. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical, focusing, non-Linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.  Google Scholar

[12]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for the energy-critical nonlinear wave equation, Amer. J. Math., 135 (2013), 935-965. doi: 10.1353/ajm.2013.0034.  Google Scholar

[13]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics of the nonradial energy-critical wave equation above the ground state energy, Discrete, Cont, Dyn. Syst., 33 (2013), 2423-2450.  Google Scholar

[14]

P. L. Lions, The concentration-compactness principle in the calculus of variations. (The limit case, Part I.), Rev. Mat. Iberoamericana, 1 (1985), 145-201. doi: 10.4171/RMI/6.  Google Scholar

[15]

F. Merle and L. Vega, Compactness at blow-up time for $L^{2}$ solutions of the critical nonlinear Schrödinger equation in $2D$, Internat. Math. Res. Notices, 1998, 399-425. doi: 10.1155/S1073792898000270.  Google Scholar

[16]

K. Nakanishi, Energy scattering for nonlinear Klein-Gordon equation and Schrödinger equation in spatial dimensions $1$ and $2$, J. Funct. Anal., 169 (1999), 201-225. doi: 10.1006/jfan.1999.3503.  Google Scholar

[17]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Diff. Eq., 250 (2011), 2299-2333. doi: 10.1016/j.jde.2010.10.027.  Google Scholar

[18]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation without a radial assumption, Arch. Rational Mech. Analysis, 203 (2012), 809-851. doi: 10.1007/s00205-011-0462-7.  Google Scholar

[19]

K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. and PDE, 44 (2012), 1-45. doi: 10.1007/s00526-011-0424-9.  Google Scholar

[20]

K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, Zurich Lectures in Advanced Mathematics, EMS, 2011. doi: 10.4171/095.  Google Scholar

[21]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^{1}$ solution for the nonlinear Schrödinger qquation, J. Diff. Eq., 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[22]

W. Schlag, Spectral theory and nonlinear differential equations: a survey, Discrete, Cont, Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.  Google Scholar

[23]

G. Talenti, Best Constant In Sobolev Inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  Google Scholar

[1]

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
[2]

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
[3]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[4]

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
[5]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[6]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[7]

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
[8]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
[9]

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
[10]

Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082
[11]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[12]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[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, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827
[15]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[16]

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
[17]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[18]

Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637
[19]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[20]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences