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Volumn 26, Issue 1, 2010, Pages 1-56

Threshold solutions for the focusing 3D cubic Schrödinger equation

Author keywords

Blow up; Nonlinear schr dinger equation; Profile decomposition; Scattering

Indexed keywords


EID: 77951143619     PISSN: 02132230     EISSN: 22350616     Source Type: Journal    
DOI: 10.4171/RMI/592     Document Type: Article
Times cited : (93)

References (34)
  • 1
    • 0007115744 scopus 로고    scopus 로고
    • Concentration effects in critical nonlinear wave equation and scattering theory
    • Cortona, Progr. Nonlinear Differential Equations Appl, Birkhäuser Boston, Boston, MA
    • BAHOURI, H. AND GERARD, P. : Concentration effects in critical nonlinear wave equation and scattering theory. In Geometrical optics and related topics (Cortona, 1996), 17-30. Progr. Nonlinear Differential Equations Appl. 32. Birkhäuser Boston, Boston, MA, 1997.
    • (1996) Geometrical Optics and Related Topics , vol.32 , pp. 17-30
    • Bahouri, H.1    Gerard, P.2
  • 2
    • 28444487069 scopus 로고    scopus 로고
    • Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain
    • BANICA, V. : Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 1, 139-170.
    • (2004) Ann. Sc. Norm. Super. Pisa Cl. Sci. , vol.3 , Issue.1-5 , pp. 139-170
    • Banica, V.1
  • 3
    • 0040248890 scopus 로고    scopus 로고
    • Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case
    • BOURGAIN, J. : Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Amer. Math. Soc. 12 (1999), no. 1, 145-171.
    • (1999) J. Amer. Math. Soc. , vol.12 , Issue.1 , pp. 145-171
    • Bourgain, J.1
  • 4
    • 5444256656 scopus 로고    scopus 로고
    • Semilinear Schrödinger equations
    • New York University, Courant Institute of Mathematical Sciences, New York. AMS, Providence, RJ
    • CAZENAVE, T. : Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics 10. New York University, Courant Institute of Mathematical Sciences, New York. AMS, Providence, RJ, 2003.
    • (2003) Courant Lecture Notes in Mathematics , vol.10
    • Cazenave, T.1
  • 5
    • 0015249887 scopus 로고
    • 3= 0 and a variational characterization of other solutions
    • 3= 0 and a variational characterization of other solutions. Arch. Rational Mech. Anal. 46 (1972), 81-95.
    • (1972) Arch. Rational Mech. Anal. , vol.46 , pp. 81-95
    • Coffman, C.V.1
  • 7
    • 59349111935 scopus 로고    scopus 로고
    • Scattering for the non-radial 3D cubic nonlinear Schrödinger equation
    • DUYCKAERTS, T., HOLMER, J. AND ROUDENKO, S. : Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett. 15 (2008), no. 6, 1233-1250.
    • (2008) Math. Res. Lett. , vol.15 , Issue.6 , pp. 1233-1250
    • Duyckaerts, T.1    Holmer, J.2    Roudenko, S.3
  • 8
    • 85157059298 scopus 로고    scopus 로고
    • Dynamic of threshold solutions for energy-critical NLS
    • To appear in
    • DUYCKAERTS, T. AND MERLE, F. : Dynamic of threshold solutions for energy-critical NLS. To appear in Geom. Funct. Anal., 2007.
    • (2007) Geom. Funct. Anal.
    • Duyckaerts, T.1    Merle, F.2
  • 9
    • 69549133355 scopus 로고    scopus 로고
    • Dynamics of threshold solutions for energy-critical wave equation
    • Art. ID rpn002
    • DUYCKAERTS, T. AND MERLE, F. : Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP 2007, no. 4, Art. ID rpn002, 67 pp. (2008).
    • (2007) Int. Math. Res. Pap. IMRP , Issue.4 , pp. 67
    • Duyckaerts, T.1    Merle, F.2
  • 10
    • 84996132207 scopus 로고    scopus 로고
    • Description du défaut de compacité de l'injection de Sobolev
    • electronic
    • GERARD, P. : Description du défaut de compacité de l'injection de Sobolev. ESAIM Control Optim. Calc. Var. 3 (1998), 213-233 (electronic).
    • (1998) ESAIM Control Optim. Calc. Var. , vol.3 , pp. 213-233
    • Gerard, P.1
  • 12
    • 0002172263 scopus 로고
    • On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case; II. Scattering theory, general case
    • 33-71
    • GINIBRE, J. AND VELO, G. : On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case; II. Scattering theory, general case. J. Funct. Anal. 32 (1979), no. 1, 1-32, 33-71.
    • (1979) J. Funct. Anal. , vol.32 , Issue.1 , pp. 1-32
    • Ginibre, J.1    Velo, G.2
  • 13
    • 84990563336 scopus 로고
    • Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system
    • GRILLAKIS, M. : Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system. Comm. Pure Appl. Math. 43 (1990), no. 3, 299-333.
    • (1990) Comm. Pure Appl. Math. , vol.43 , Issue.3 , pp. 299-333
    • Grillakis, M.1
  • 14
    • 77955700830 scopus 로고    scopus 로고
    • On blow-up solutions to the 3D cubic nonlinear Schrödinger equation
    • Art. ID abm004
    • HOLMER, J. AND ROUDENKO, S. : On blow-up solutions to the 3D cubic nonlinear Schrödinger equation. Appl. Math. Res. Express AMRX 2007, no. 1, Art. ID abm004, 31pp.
    • (2007) Appl. Math. Res. Express AMRX , Issue.1
    • Holmer, J.1    Roudenko, S.2
  • 15
    • 47749109045 scopus 로고    scopus 로고
    • A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation
    • HOLMER, J. AND ROUDENKO, S. : A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Comm. Math. Phys. 282 (2008), no. 2, 435-467.
    • (2008) Comm. Math. Phys. , vol.282 , Issue.2 , pp. 435-467
    • Holmer, J.1    Roudenko, S.2
  • 16
    • 33750526878 scopus 로고    scopus 로고
    • Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case
    • DOI 10.1007/s00222-006-0011-4
    • KENIG, C. E. AND MERLE, F. : Global well-posedness, scattering and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166 (2006), no. 3, 645-675. (Pubitemid 44663366)
    • (2006) Inventiones Mathematicae , vol.166 , Issue.3 , pp. 645-675
    • Kenig, C.E.1    Merle, F.2
  • 17
    • 57049094376 scopus 로고    scopus 로고
    • Global well-posedness, scattering and blowup for the energy-critical focusing non-linear wave equation in the radial case
    • KENIG, C. E. AND MERLE, F. : Global well-posedness, scattering and blowup for the energy-critical focusing non-linear wave equation in the radial case. Acta Math. 201 (2008), no. 2, 147-212.
    • (2008) Acta Math. , vol.201 , Issue.2 , pp. 147-212
    • Kenig, C.E.1    Merle, F.2
  • 18
    • 77950914727 scopus 로고    scopus 로고
    • 1/2 bounded solutions to the cubic defocusing NLS in 3D
    • To appear in
    • 1/2 bounded solutions to the cubic defocusing NLS in 3D. To appear in Trans. Amer. Math. Soc.
    • Trans. Amer. Math. Soc.
    • Kenig, C.E.1    Merle, F.2
  • 19
    • 0035922239 scopus 로고    scopus 로고
    • On the defect of compactness for the Strichartz estimates of the Schrödinger equations
    • DOI 10.1006/jdeq.2000.3951
    • KERAANI, S. : On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differential Equations 175 (2001), no. 2, 353-392. (Pubitemid 33082426)
    • (2001) Journal of Differential Equations , vol.175 , Issue.2 , pp. 353-392
    • Keraani, S.1
  • 20
    • 85036846864 scopus 로고    scopus 로고
    • The cubic nonlinear Schrödinger equation in two dimensions with radial data
    • To appear in
    • KILLIP, R., TAO, T. AND VISAN, M. : The cubic nonlinear Schrödinger equation in two dimensions with radial data. To appear in J. Eur. Math. Soc.
    • J. Eur. Math. Soc.
    • Killip, R.1    Tao, T.2    Visan, M.3
  • 21
    • 77951129591 scopus 로고    scopus 로고
    • The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
    • To appear in
    • KILLIP, R. AND VISAN, M. : The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. To appear in Amer. J. Math.
    • Amer. J. Math.
    • Killip, R.1    Visan, M.2
  • 22
    • 68349155323 scopus 로고    scopus 로고
    • The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
    • KILLIP, R., VISAN, M. AND ZHANG, X. : The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Ana J. PDE 1 (2008), no. 2, 229-266.
    • (2008) Ana J. PDE , vol.1 , Issue.2 , pp. 229-266
    • Killip, R.1    Visan, M.2    Zhang, X.3
  • 25
    • 0001294182 scopus 로고
    • The concentration-compactness principle in the calculus of variations. The limit case. II
    • LIONS, P.-L. : The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1 (1985), no. 2, 45-121.
    • (1985) Rev. Mat. Iberoamericana , vol.1 , Issue.2 , pp. 45-121
    • Lions, P.-L.1
  • 26
    • 84990553632 scopus 로고
    • On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass
    • MERLE, F. : On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Comm. Pure Appl. Math. 45 (1992), no. 2, 203-254.
    • (1992) Comm. Pure Appl. Math. , vol.45 , Issue.2 , pp. 203-254
    • Merle, F.1
  • 27
    • 84974001368 scopus 로고
    • Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power
    • MERLE, F. : Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69 (1993), no. 2, 427-454.
    • (1993) Duke Math. J. , vol.69 , Issue.2 , pp. 427-454
    • Merle, F.1
  • 28
    • 50049127248 scopus 로고    scopus 로고
    • 2 super critical nonlinear Schrödinger equations Amer
    • 2 super critical nonlinear Schrödinger equations Amer. J. Math. 130 (2008), no. 4, 945-978.
    • (2008) J. Math. , vol.130 , Issue.4 , pp. 945-978
    • Merle, F.1    Raphaël, P.2
  • 29
    • 1842709141 scopus 로고    scopus 로고
    • 2 solutions of the critical nonlinear Schrödinger equation in 2D
    • 2 solutions of the critical nonlinear Schrödinger equation in 2D. Internat. Math. Res. Notices 1998, no. 8, 399-425.
    • (1998) Internat. Math. Res. Notices , Issue.8 , pp. 399-425
    • Merle, F.1    Vega, L.2
  • 30
    • 0000540347 scopus 로고
    • Existence of solitary waves in higher dimensions
    • STRAUSS, W. A. : Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), no. 2, 149-162.
    • (1977) Comm. Math. Phys. , vol.55 , Issue.2 , pp. 149-162
    • Strauss, W.A.1
  • 31
    • 34249655802 scopus 로고    scopus 로고
    • Nonlinear dispersive equations. Local and global analysis
    • Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI
    • TAO, T. : Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
    • (2006) CBMS Regional Conference Series in Mathematics , pp. 106
    • Tao, T.1
  • 32
    • 0041473959 scopus 로고
    • Nonlinear Schrödinger equations and sharp interpolation estimates
    • WEINSTEIN, M. I. : Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1982/83), no. 4, 567-576.
    • (1982) Comm. Math. Phys. , vol.87 , Issue.4 , pp. 567-576
    • Weinstein, M.I.1
  • 33
    • 0000686130 scopus 로고
    • Modulational stability of ground states of nonlinear Schrödinger equations
    • WEINSTEIN, M. I. : Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), no. 3, 472-491.
    • (1985) SIAM J. Math. Anal. , vol.16 , Issue.3 , pp. 472-491
    • Weinstein, M.I.1
  • 34
    • 84990553584 scopus 로고
    • Lyapunov stability of ground states of nonlinear dispersive evolution equations
    • WEINSTEIN, M. I. : Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986), no. 1, 51-67.
    • (1986) Comm. Pure Appl. Math. , vol.39 , Issue.1 , pp. 51-67
    • Weinstein, M.I.1


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