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

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.
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,, \emph{J. Math. Pures. Appl.}, 55 (1976), 269.

[2]

H. Bahouri and P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131.

[3]

J. Bourgain, Global well-posedness of defocusing $3D$ critical NLS in the radial case,, \emph{J. Amer. Math. Soc.}, 12 (1999), 145.

[4]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003). doi: 10.1090/cln/010.

[5]

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

[6]

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

[7]

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

[8]

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

[9]

M. Keel and T. Tao, Endpoint Strichartz estimates,, \emph{Amer. J. Math.}, 120 (1998), 955.

[10]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations,, \emph{J. Diff. Eq.}, 175 (2001), 353. doi: 10.1006/jdeq.2000.3951.

[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,, \emph{Invent. Math.}, 166 (2006), 645. doi: 10.1007/s00222-006-0011-4.

[12]

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

[13]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics of the nonradial energy-critical wave equation above the ground state energy,, \emph{Discrete, 33 (2013), 2423.

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

W. Schlag, Spectral theory and nonlinear differential equations: a survey,, \emph{Discrete, 15 (2006), 703. doi: 10.3934/dcds.2006.15.703.

[23]

G. Talenti, Best Constant In Sobolev Inequality,, \emph{Ann. Mat. Pura. Appl.}, 110 (1976), 353.

show all references

References:
[1]

T. Aubin, Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire,, \emph{J. Math. Pures. Appl.}, 55 (1976), 269.

[2]

H. Bahouri and P. Gerard, High frequency approximation of solutions to critical nonlinear wave equations,, \emph{Amer. J. Math.}, 121 (1999), 131.

[3]

J. Bourgain, Global well-posedness of defocusing $3D$ critical NLS in the radial case,, \emph{J. Amer. Math. Soc.}, 12 (1999), 145.

[4]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003). doi: 10.1090/cln/010.

[5]

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

[6]

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

[7]

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

[8]

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

[9]

M. Keel and T. Tao, Endpoint Strichartz estimates,, \emph{Amer. J. Math.}, 120 (1998), 955.

[10]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations,, \emph{J. Diff. Eq.}, 175 (2001), 353. doi: 10.1006/jdeq.2000.3951.

[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,, \emph{Invent. Math.}, 166 (2006), 645. doi: 10.1007/s00222-006-0011-4.

[12]

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

[13]

J. Krieger, K. Nakanishi and W. Schlag, Global dynamics of the nonradial energy-critical wave equation above the ground state energy,, \emph{Discrete, 33 (2013), 2423.

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

W. Schlag, Spectral theory and nonlinear differential equations: a survey,, \emph{Discrete, 15 (2006), 703. doi: 10.3934/dcds.2006.15.703.

[23]

G. Talenti, Best Constant In Sobolev Inequality,, \emph{Ann. Mat. Pura. Appl.}, 110 (1976), 353.

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

[5]

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

[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]

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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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 - A, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637

[15]

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

[16]

Laurent Di Menza, Olivier Goubet. Stabilizing blow up solutions to nonlinear schrÖdinger equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1059-1082. doi: 10.3934/cpaa.2017051

[17]

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

[18]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183

[19]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

[20]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]