March  2017, 16(2): 557-590. doi: 10.3934/cpaa.2017028

Global dynamics of solutions with group invariance for the nonlinear schrödinger equation

Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan

Received  July 2016 Revised  November 2016 Published  January 2017

Fund Project: The author is supported by JSPS Research Fellow 15J02570

We consider the focusing mass-supercritical and energy-subcritical nonlinear Schrödinger equation (NLS). We are interested in the global behavior of the solutions to (NLS) with group invariance. By the group invariance, we can determine the global behavior of the solutions above the ground state standing waves.

Citation: Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028
References:
[1]

T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672. doi: 10.1215/21562261-2265914.

[2]

V. Banica and N. Visciglia, Scattering for NLS with a delta potential, J. Differential Equations, 260 (2016), 4410-4439. doi: 10.1016/j.jde.2015.11.016.

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${\mathbb{R}^N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149.

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.

[6]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.

[8]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.

[9]

B. Dodson, Global well -posedness and scattering for the focusing, energy -critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, preprint, arXiv: 1409.1950.

[10]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618. doi: 10.1016/j.aim.2015.04.030.

[11]

D. DuY. Wu and K. Zhang, On blow-up criterion for the nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 3639-3650. doi: 10.3934/dcds.2016.36.3639.

[12]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13.

[13]

T. Duyckaerts, H. Jia, C. E. Kenig, and F. Merle, Soliton resolution along a sequence of times for the focusing energy critical wave equation, preprint, arXiv: 1601.01871.

[14]

T. DuyckaertsC. E. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math., 1 (2013), 75-144. doi: 10.4310/CJM.2013.v1.n1.a3.

[15]

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.

[16]

T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam., 26 (2010), 1-56. doi: 10.4171/RMI/592.

[17]

T. Duyckaerts and S. Roudenko, Going beyond the threshold: scattering and blow-up in the focusing NLS equation, Comm. Math. Phys., 334 (2015), 1573-1615. doi: 10.1007/s00220-014-2202-y.

[18]

D. FangJ. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9.

[19]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90077-6.

[20]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.

[21]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y.

[22]

J. Holmer and S. Roudenko, Divergence of infinite-variance nonradial solutions to the 3D NLS equation, Comm. Partial Differential Equations, 35 (2010), 878-905. doi: 10.1080/03605301003646713.

[23]

S. IbrahimN. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460. doi: 10.2140/apde.2011.4.405.

[24]

S. IbrahimN. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation, Trans. Amer. Math. Soc., 366 (2014), 5653-5669. doi: 10.1090/S0002-9947-2014-05852-2.

[25]

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

[26]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.

[27]

R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, preprint, arXiv: 1606.01512.

[28]

R. KillipB. Stovall and M. Visan, Scattering for the cubic Klein-Gordon equation in two space dimensions, Trans. Amer. Math. Soc., 364 (2012), 1571-1631. doi: 10.1090/S0002-9947-2011-05536-4.

[29]

R. KillipT. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc., 11 (2009), 1203-1258. doi: 10.4171/JEMS/180.

[30]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107.

[31]

R. KillipM. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229.

[32]

M. K. Kwong, Uniqueness of positive solutions of ∆u -u + up = 0 in ${\mathbb{R}^N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[33]

D. Li and X. Zhang, Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions, J. Funct. Anal., 256 (2009), 1928-1961. doi: 10.1016/j.jfa.2008.12.007.

[34]

E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. xxii+346 pp. doi: 10.1090/gsm/014.

[35]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Second edition, Universitext. Springer, New York, 2015. xiv+301 pp. doi: 10.1007/978-1-4939-2181-2.

[36]

Y. Martel, Blow-up for the nonlinear Schrödinger equation in nonisotropic spaces, Nonlinear Anal., 28 (1997), 1903-1908. doi: 10.1016/S0362-546X(96)00036-3.

[37]

Y. Martel and P. Raphaël, Strongly interacting blow up bubbles for the mass critical NLS, preprint, arXiv: 1512.00900.

[38]

J. L. Marzuola and M. E. Taylor, Higher dimensional vortex standing waves for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 41 (2016), 398-446. doi: 10.1080/03605302.2015.1127966.

[39]

S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531. doi: 10.3934/cpaa.2015.14.1481.

[40]

S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234.

[41]

K. Nakanishi and T. Roy, Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data, Commun. Pure Appl. Anal., 15 (2016), 2023-2058. doi: 10.3934/cpaa.2016026.

[42]

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

[43]

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

[44]

T. Ogawa and Y. Tsutsumi, Blow-up of H1 solution for the nonlinear Schrödinger equation, J. Diff. Eq., 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.

[45]

A. Walter Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.

[46]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133. doi: 10.1016/0022-1236(81)90063-X.

[47]

W. A. Strauss, Nonlinear scattering theory at low energy: sequel, J. Funct. Anal., 43 (1981), 281-293. doi: 10.1016/0022-1236(81)90019-7.

[48]

T. Tao, 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. xvi+373 pp. doi: 10.1090/cbms/106.

show all references

References:
[1]

T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672. doi: 10.1215/21562261-2265914.

[2]

V. Banica and N. Visciglia, Scattering for NLS with a delta potential, J. Differential Equations, 260 (2016), 4410-4439. doi: 10.1016/j.jde.2015.11.016.

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${\mathbb{R}^N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149.

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.

[6]

H. Brézis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[7]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.

[8]

T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.

[9]

B. Dodson, Global well -posedness and scattering for the focusing, energy -critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, preprint, arXiv: 1409.1950.

[10]

B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618. doi: 10.1016/j.aim.2015.04.030.

[11]

D. DuY. Wu and K. Zhang, On blow-up criterion for the nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 3639-3650. doi: 10.3934/dcds.2016.36.3639.

[12]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13.

[13]

T. Duyckaerts, H. Jia, C. E. Kenig, and F. Merle, Soliton resolution along a sequence of times for the focusing energy critical wave equation, preprint, arXiv: 1601.01871.

[14]

T. DuyckaertsC. E. Kenig and F. Merle, Classification of radial solutions of the focusing, energy-critical wave equation, Camb. J. Math., 1 (2013), 75-144. doi: 10.4310/CJM.2013.v1.n1.a3.

[15]

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.

[16]

T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam., 26 (2010), 1-56. doi: 10.4171/RMI/592.

[17]

T. Duyckaerts and S. Roudenko, Going beyond the threshold: scattering and blow-up in the focusing NLS equation, Comm. Math. Phys., 334 (2015), 1573-1615. doi: 10.1007/s00220-014-2202-y.

[18]

D. FangJ. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9.

[19]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32. doi: 10.1016/0022-1236(79)90077-6.

[20]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.

[21]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y.

[22]

J. Holmer and S. Roudenko, Divergence of infinite-variance nonradial solutions to the 3D NLS equation, Comm. Partial Differential Equations, 35 (2010), 878-905. doi: 10.1080/03605301003646713.

[23]

S. IbrahimN. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Anal. PDE, 4 (2011), 405-460. doi: 10.2140/apde.2011.4.405.

[24]

S. IbrahimN. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation, Trans. Amer. Math. Soc., 366 (2014), 5653-5669. doi: 10.1090/S0002-9947-2014-05852-2.

[25]

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

[26]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.

[27]

R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, preprint, arXiv: 1606.01512.

[28]

R. KillipB. Stovall and M. Visan, Scattering for the cubic Klein-Gordon equation in two space dimensions, Trans. Amer. Math. Soc., 364 (2012), 1571-1631. doi: 10.1090/S0002-9947-2011-05536-4.

[29]

R. KillipT. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc., 11 (2009), 1203-1258. doi: 10.4171/JEMS/180.

[30]

R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107.

[31]

R. KillipM. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229.

[32]

M. K. Kwong, Uniqueness of positive solutions of ∆u -u + up = 0 in ${\mathbb{R}^N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[33]

D. Li and X. Zhang, Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions, J. Funct. Anal., 256 (2009), 1928-1961. doi: 10.1016/j.jfa.2008.12.007.

[34]

E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. xxii+346 pp. doi: 10.1090/gsm/014.

[35]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Second edition, Universitext. Springer, New York, 2015. xiv+301 pp. doi: 10.1007/978-1-4939-2181-2.

[36]

Y. Martel, Blow-up for the nonlinear Schrödinger equation in nonisotropic spaces, Nonlinear Anal., 28 (1997), 1903-1908. doi: 10.1016/S0362-546X(96)00036-3.

[37]

Y. Martel and P. Raphaël, Strongly interacting blow up bubbles for the mass critical NLS, preprint, arXiv: 1512.00900.

[38]

J. L. Marzuola and M. E. Taylor, Higher dimensional vortex standing waves for nonlinear Schrödinger equations, Comm. Partial Differential Equations, 41 (2016), 398-446. doi: 10.1080/03605302.2015.1127966.

[39]

S. Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 14 (2015), 1481-1531. doi: 10.3934/cpaa.2015.14.1481.

[40]

S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234.

[41]

K. Nakanishi and T. Roy, Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data, Commun. Pure Appl. Anal., 15 (2016), 2023-2058. doi: 10.3934/cpaa.2016026.

[42]

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

[43]

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

[44]

T. Ogawa and Y. Tsutsumi, Blow-up of H1 solution for the nonlinear Schrödinger equation, J. Diff. Eq., 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B.

[45]

A. Walter Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.

[46]

W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133. doi: 10.1016/0022-1236(81)90063-X.

[47]

W. A. Strauss, Nonlinear scattering theory at low energy: sequel, J. Funct. Anal., 43 (1981), 281-293. doi: 10.1016/0022-1236(81)90019-7.

[48]

T. Tao, 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. xvi+373 pp. doi: 10.1090/cbms/106.

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