# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.  Google Scholar [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. Google Scholar [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.  Google Scholar [11] D. Du, Y. 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.  Google Scholar [12] T. Duyckaerts, J. 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.  Google Scholar [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. Google Scholar [14] T. Duyckaerts, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] D. Fang, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] S. Ibrahim, N. 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.  Google Scholar [24] S. Ibrahim, N. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, preprint, arXiv: 1606.01512. Google Scholar [28] R. Killip, B. 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.  Google Scholar [29] R. Killip, T. 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.  Google Scholar [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.  Google Scholar [31] R. Killip, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [37] Y. Martel and P. Raphaël, Strongly interacting blow up bubbles for the mass critical NLS, preprint, arXiv: 1512.00900. Google Scholar [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.  Google Scholar [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.  Google Scholar [40] S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [45] Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.  Google Scholar [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. Google Scholar [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.  Google Scholar [11] D. Du, Y. 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.  Google Scholar [12] T. Duyckaerts, J. 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.  Google Scholar [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. Google Scholar [14] T. Duyckaerts, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] D. Fang, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] S. Ibrahim, N. 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.  Google Scholar [24] S. Ibrahim, N. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] R. Killip, S. Masaki, J. Murphy and M. Visan, Large data mass-subcritical NLS: critical weighted bounds imply scattering, preprint, arXiv: 1606.01512. Google Scholar [28] R. Killip, B. 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.  Google Scholar [29] R. Killip, T. 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.  Google Scholar [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.  Google Scholar [31] R. Killip, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [37] Y. Martel and P. Raphaël, Strongly interacting blow up bubbles for the mass critical NLS, preprint, arXiv: 1512.00900. Google Scholar [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.  Google Scholar [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.  Google Scholar [40] S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv: 1605.09234. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [45] Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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