Abstract
We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity \(s_c\in (\max \{-1,-\frac{d}{2}\},0)\), we prove that any solution satisfying
on its maximal interval of existence must be global and scatter.
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Barab, J.: Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation. J. Math. Phys. 25(11), 3270–3273 (1984)
Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121(1), 131–175 (1999)
Bégout, P.: Convergence to scattering states in the nonlinear Schrödinger equation. Commun. Contemp. Math. 3, 403–418 (2001)
Bégout, P., Vargas, A.: Mass concentration phenomena for the \(L^2\)-critical nonlinear Schrödinger equation. Trans. Am. Math. Soc. 359(11), 5257–5282 (2007)
Bourgain, J.: Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Int. Math. Res. Not. 1998(5), 253–283 (1998)
Bourgain, J.: Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Am. Math. Soc. 12(1), 145–171 (1999)
Carles, R., Keraani, S.: On the role of quadratic oscillations in nonlinear Schrödinger equations. II. The \(L^2\)-critical case. Trans. Am. Math. Soc. 359(1), 33–62 (2007)
Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, New York. American Mathematical Society, Providence, RI (2003)
Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in \(H^s\). Nonlinear Anal. 14(10), 807–836 (1990)
Colliander, J., Keel, M., Staffilani, G., Takaoka, T., Tao, T.: Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in \({\mathbb{R}}^3\). Ann. Math. (2) 167(3), 767–865 (2008)
Cho, Y., Hwang, G., Ozawa, T.: Global well-posedness of critical nonlinear Schrödinger equations below \(L^2\). Discrete Contin. Dyn. Syst. 33(4), 1389–1405 (2013)
Christ, M., Colliander, J., Tao, T.: A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order. J. Funct. Anal. 254(2), 368–395 (2008)
Christ, M., Weinstein, M.: Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation. J. Funct. Anal. 100(1), 87–109 (1991)
Dodson, B.: Global well-posedness and scattering for the defocusing, \(L^2\)-critical nonlinear Schrödinger equation when \(d\ge 3\). J. Am. Math. Soc. 25(2), 429–463 (2012)
Dodson, B.: Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state. Adv. Math. 285, 1589–1618 (2015)
Dodson, B.: Global well-posedness and scattering for the defocusing, \(L^2\)-critical, nonlinear Schrödinger equation when \(d=1\). Am. J. Math. 138(2), 531–569 (2016)
Dodson, B.: Global well-posedness and scattering for the defocusing, \(L^2\)-critical, nonlinear Schrödinger equation when \(d=2\). Preprint arXiv:1006.1375
Dodson, B.: 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
Dodson, B., Miao, C., Murphy, J., Zheng, J.: The defocusing quintic NLS in four space dimensions. Preprint arXiv:1508.07298
Ginibre, J., Velo, G.: Smoothing properties and retarded estimates for some dispersive evolution equations. Commun. Math. Phys. 144(1), 163–188 (1992)
Grillakis, M.: On nonlinear Schrödinger equations. Commun. Partial Differ. Equ. 25(9–10), 1827–1844 (2000)
Hidano, K.: Nonlinear Schrödinger equations with radially symmetric data of critical regularity. Funkcial. Ekvac. 51(1), 135–147 (2008)
Hunt, R.A.: On L(p,q) spaces. Enseign. Math. (2) 12, 249–276 (1966)
Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)
Kenig, C.E., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166, 645–675 (2006)
Kenig, C.E., Merle, F.: Scattering for \(\dot{H}^{1/2}\) bounded solutions to the cubic, defocusing NLS in 3 dimensions. Trans. Am. Math. Soc. 362(4), 1937–1962 (2010)
Keraani, S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differ. Equ. 175(2), 353–392 (2001)
Keraani, S.: On the blow up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235(1), 171–192 (2006)
Killip, R., Tao, T., Visan, M.: The cubic nonlinear Schrödinger equation in two dimensions with radial data. J. Eur. Math. Soc. (JEMS) 11(6), 1203–1258 (2009)
Killip, R., Visan, M.: The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Am. J. Math. 132(2), 361–424 (2010)
Killip, R., Visan, M.: Energy-supercritical NLS: critical \(\dot{H}^s\)-bounds imply scattering. Commun. Partial Differ. Equ. 35(6), 945–987 (2010)
Killip, R., Visan, M.: Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. Anal. PDE 5(4), 855–885 (2012)
Killip, R., Visan, M.: Nonlinear Schrödinger equations at critical regularity. Clay Math. Proc. 17, 325–437 (2013)
Killip, R., Visan, M., Zhang, X.: The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Anal. PDE 1(2), 229–266 (2008)
Masaki, S.: On minimal non-scattering solution for focusing mass-subcritical nonlinear Schrödinger equation. Preprint arXiv:1301.1742
Masaki, S.: A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Commun. Pure Appl. Anal. 14(4), 1481–1531 (2015)
Masaki, S.: Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation. Preprint arXiv:1605.09234
Masaki, S., Segata, J.: On well-posedness of the generalized Korteweg-de Vries equation in scale critical \(\hat{L}^r\) space. Anal. PDE 9(3), 699–725 (2016)
Merle, F., Vega, L.: Compactness at blow-up time for \(L^2\) solutions of the critical nonlinear Schrödinger equation in 2D. Int. Math. Res. Not. 1998(8), 399–425 (1998)
Miao, C., Murphy, J., Zheng, J.: The defocusing energy-supercritical NLS in four space dimensions. J. Funct. Anal. 267(6), 1662–1724 (2014)
Murphy, J.: Intercritical NLS: critical \(\dot{H}^s\)-bounds imply scattering. SIAM J. Math. Anal. 46(1), 939–997 (2014)
Murphy, J.: The defocusing \(\dot{H}^{1/2}\)-critical NLS in high dimensions. Discrete Contin. Dyn. Syst. 34(2), 733–748 (2014)
Murphy, J.: The radial defocusing nonlinear Schrödinger equation in three space dimensions. Commun. Partial Differ. Equ. 40(2), 265–308 (2015)
Nakanishi, K., Ozawa, T.: Remarks on scattering for nonlinear Schrödinger equations. NoDEA Nonlinear Differ. Equ. Appl. 9(1), 45–68 (2002)
O’Neil, R.: Convolution operators and \(L(p, q)\) spaces. Duke Math. J. 30, 129–142 (1963)
Ryckman, E., Visan, M.: Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in 1+4. Am. J. Math. 129(1), 1–60 (2007)
Strauss, W.: Nonlinear scattering theory. In: Lavita, J. A., Marchand, J.-P. (eds.) Scattering Theory in Mathematical Physics, pp. 53–78. Riedel, Dordrecht (1974)
Strichartz, R.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705–714 (1977)
Tao, T.: Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data. N. Y. J. Math. 11, 57–80 (2005)
Tao, T., Visan, M., Zhang, X.: Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions. Duke Math. J. 140(1), 165–202 (2007)
Tsutsumi, Y., Yajima, K.: The asymptotic behavior of nonlinear Schrödinger equations. Bull. Am. Math. Soc. (N.S.) 11(1), 186–188 (1984)
Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Ph.D. Thesis, UCLA (2006)
Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Duke Math. J. 138(2), 281–374 (2007)
Visan, M.: Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions. Int. Math. Res. Not. IMRN 2012(5), 1037–1067 (2012)
Visan, M.: Dispersive equations. In: Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, vol. 45. Springer Basel, Birkhäuser (2014)
Xie, J., Fang, D.: Global well-posedness and scattering for the defocusing \(\dot{H}^s\)-critical NLS. Chin. Ann. Math. Ser. B 34(6), 801–842 (2013)
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Killip, R., Masaki, S., Murphy, J. et al. Large data mass-subcritical NLS: critical weighted bounds imply scattering. Nonlinear Differ. Equ. Appl. 24, 38 (2017). https://doi.org/10.1007/s00030-017-0463-9
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DOI: https://doi.org/10.1007/s00030-017-0463-9