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Global dynamics above the ground state energy for the cubic NLS equation in 3D. (English) Zbl 1237.35148

Summary: We extend the result in [the authors, J. Differ. Equations 250, No. 5, 2299–2333 (2011; Zbl 1213.35307)] on the nonlinear Klein-Gordon equation to the nonlinear Schrödinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state. We prove that the initial data set splits into nine nonempty, pairwise disjoint regions which are characterized by the distinct behaviors of the solution for large time: blow-up, scattering to 0, or scattering to the family of ground states generated by the phase and scaling freedom. Solutions of this latter type form a smooth center-stable manifold, which contains the ground states and separates the phase space locally into two connected regions exhibiting blow-up and scattering to 0, respectively. The special solutions found by T. Duyckaerts and S. Roudenko [Rev. Mat. Iberoam. 26, No. 1, 1–56 (2010; Zbl 1195.35276)], following their seminal work on threshold solutions [Geom. Funct. Anal. 18, No. 6, 1787–1840 (2008; Zbl 1232.35150)], appear here as the unique one-dimensional unstable/stable manifolds emanating from the ground states. In analogy with [the authors, loc. cit.], the proof combines the hyperbolic dynamics near the ground states with the variational structure away from them. The main technical ingredient in the proof is a “one-pass” theorem which precludes “almost homoclinic orbits”, i.e., those solutions starting in, then moving away from, and finally returning to, a small neighborhood of the ground states. The main new difficulty compared with the Klein-Gordon case is the lack of finite propagation speed. We need the radial Sobolev inequality for the error estimate in the virial argument. Another major difference between [the authors, loc. cit.] and this paper is the need to control two modulation parameters.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35P15 Estimates of eigenvalues in context of PDEs
37D10 Invariant manifold theory for dynamical systems
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[1] Bahouri H., Gérard P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121(1), 131–175 (1999) · Zbl 0919.35089
[2] Bates, P.W., Jones, C.K.R.T.: Invariant manifolds for semilinear partial differential equations. In: Dynamics Reported, vol.1, pp. 1–38. Dynam. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester (1989) · Zbl 0674.58024
[3] Beceanu M.: A centre-stable manifold for the focussing cubic NLS in $${\(\backslash\)mathbb {R}\^{1+3}}$$ . Commun. Math. Phys. 280(1), 145–205 (2008) · Zbl 1148.35082
[4] Beceanu, M.: New estimates for a time-dependent Schrödinger equation (preprint 2009), to appear in Duke Math. J.
[5] Beceanu, M.: A critical centre-stable manifold for the Schroedinger equation in three dimensions (preprint 2009), to appear in Commun. Pure Appl. Math.
[6] Berestycki H., Cazenave T.: Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires. C. R. Acad. Sci. Paris I Math. 293(9), 489–492 (1981) · Zbl 0492.35010
[7] Berestycki H., Lions P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983) · Zbl 0533.35029
[8] Bourgain J., Wang W.: Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1-2), 197–215 (1997) · Zbl 1043.35137
[9] Buslaev, V.S., Perelman, G.S.: Scattering for the nonlinear Schrödinger equation: states that are close to a soliton. (Russian) Algebra i Analiz. 4(6), 63–102 (1992); translation in St. Petersburg Math. J. 4(6), 1111–1142 (1993) · Zbl 0795.35111
[10] Buslaev, V.S., Perelman, G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. In: Nonlinear Evolution Equations. American Mathematical Society Translations: Series 2, vol. 164, pp. 75–98. American Mathematical Society, Providence, RI (1995) · Zbl 0841.35108
[11] Cazenave, T.: Semilinear Schrödinger equations. In: Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003) · Zbl 1055.35003
[12] Cazenave T., Lions P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85(4), 549–561 (1982) · Zbl 0513.35007
[13] Coffman C.: Uniqueness of the ground state solution for {\(\Delta\)}u u + u 3 = 0 and a variational characterization of other solutions. Arch. Ration. Mech. Anal. 46, 81–95 (1972) · Zbl 0249.35029
[14] Cuccagna S.: Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54(9), 1110–1145 (2001) · Zbl 1031.35129
[15] Cuccagna S., Mizumachi T.: On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations. Commun. Math. Phys. 284(1), 51–77 (2008) · Zbl 1155.35092
[16] Demanet L., Schlag W.: Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation. Nonlinearity 19(4), 829–852 (2006) · Zbl 1106.35044
[17] Duyckaerts, T., Merle, F.: Dynamic of threshold solutions for energy-critical NLS. Geom. Funct. Anal. 18(6), 1787–1840 (2009); Dynamics of threshold solutions for energy-critical wave equation. International Mathematics Research Papers, IMRP (2008) · Zbl 1232.35150
[18] Duykaerts T., Roudenko S.: Thresholdsolutions for the focusing 3D cubic Schrödinger equation. Rev. Mater. Iberoam. 26(1), 1–56 (2010) · Zbl 1195.35276
[19] Duyckaerts T., Holmer J., Roudenko S.: Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett. 15(6), 1233–1250 (2008) · Zbl 1171.35472
[20] Erdogan B., Schlag W.: Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three. II. J. Anal. Math. 99, 199–248 (2006) · Zbl 1146.35324
[21] Fibich G., Merle F., Raphaël P.: Proof of a spectral property related to the singularity formation for the L 2 critical nonlinear Schrödinger equation. Phys. D 220(1), 1–13 (2006) · Zbl 1100.35097
[22] Gesztesy F., Jones C.K.R.T., Latushkin Y., Stanislavova M.: A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations. Indiana Univ. Math. J. 49(1), 221–243 (2000) · Zbl 0969.35123
[23] Ginibre J., Velo G.: On a class of nonlinear Schrödinger equation. I. The Cauchy problems; II. Scattering theory, general case. J. Funct. Anal. 32(1-32), 33–71 (1979) · Zbl 0396.35029
[24] Ginibre J., Velo G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. (9) 64(4), 363–401 (1985) · Zbl 0535.35069
[25] Glassey R.T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equation. J. Math. Phys. 18(9), 1794–1797 (1977) · Zbl 0372.35009
[26] Grillakis M.: Linearized instability for nonlinear Schrödinger and Klein–Gordon equations. Commun. Pure Appl. Math. 41, 747–774 (1988) · Zbl 0632.70015
[27] Grillakis M.: Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system. Commun. Pure Appl. Math. 43, 299–333 (1990) · Zbl 0731.35010
[28] Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74(1), 160–197 (1987) · Zbl 0656.35122
[29] Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94(2), 308–348 (1990) · Zbl 0711.58013
[30] Hirsch M.W., Pugh C.C., Shub M.: Invariant manifolds. In: Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)
[31] Holmer J., Roudenko S.: A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Commun. Math. Phys. 282(2), 435–467 (2008) · Zbl 1155.35094
[32] Hundertmark D., Lee Y.-R.: Exponential decay of eigenfunctions and generalized eigenfunctions of a non-self-adjoint matrix Schrödinger operator related to NLS. Bull. Lond. Math. Soc. 39(5), 709–720 (2007) · Zbl 1155.35065
[33] Ibrahim, S., Masmoudi, N., Nakanishi, K.: Scattering threshold for the focusing nonlinear Klein–Gordon equation. to appear in Anal. PDE · Zbl 1270.35132
[34] Keel M., Tao T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998) · Zbl 0922.35028
[35] Kenig C., Merle F.: Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case. Invent. Math. 166(3), 645–675 (2006) · Zbl 1115.35125
[36] Kenig C., Merle F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201(2), 147–212 (2008) · Zbl 1183.35202
[37] Keraani S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equation. J. Differ. Equ. 175, 353–392 (2001) · Zbl 1038.35119
[38] Krieger J., Schlag W.: Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Am. Math. Soc. 19(4), 815–920 (2006) · Zbl 1281.35077
[39] Krieger J., Schlag W.: Non-generic blow-up solutions for the critical focusing NLS in 1-D. J. Eur. Math. Soc. (JEMS) 11(1), 1–125 (2009) · Zbl 1163.35035
[40] Kwong M.: Uniqueness of positive solutions of {\(\Delta\)}u + u + u p = 0 in $${\(\backslash\)mathbb {R}\^{n}}$$ . Arch. Ration. Mech. Anal. 105(3), 243–266 (1989) · Zbl 0676.35032
[41] Marzuola J., Simpson G.: Spectral analysis for matrix Hamiltonian operators. Nonlinearity 24, 389–429 (2011) · Zbl 1213.35371
[42] Merle F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69(2), 427–454 (1993) · Zbl 0808.35141
[43] Merle, F., Raphael, P.: On a sharp lower bound on the blow-up rate for the L 2L2 critical nonlinear Schrödinger equation. J. Am. Math. Soc. 191, 37–90 (2006). The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. Math. 2 161(1), 157–222 (2005); On universality of blow-up profile for L 2L2 critical nonlinear Schrödinger equation. Invent. Math. 156(3), 565–672 (2004)
[44] Merle F., Vega L.: Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2D. Intern. Math. Res. Notice 8, 399–425 (1998) · Zbl 0913.35126
[45] Merle F., Raphael P., Szeftel J.: Stable self-similar blow-up dynamics for slightly L 2 super-critical NLS equations. Geom. Funct. Anal. 20(4), 1028–1071 (2010) · Zbl 1204.35153
[46] Merle, F., Raphael, P., Szeftel, J.: The instability of Bourgain–Wang solutions for the L 2 critical NLS, preprint, arXiv:1010.5168 · Zbl 1294.35145
[47] Nakanishi K., Schlag W.: Global dynamics above the ground state energy for the focusing nonlinear Klein–Gordon equation. J. Differ. Equ. 250, 2299–2333 (2011) · Zbl 1213.35307
[48] Ogawa T., Tsutsumi Y.: Blow-Up of H 1, solution for the Nonlinear Schrödinger Equation. J. Differ. Equ. 92, 317–330 (1991) · Zbl 0739.35093
[49] Perelman G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2(4), 605–673 (2001) · Zbl 1007.35087
[50] Pillet C.A., Wayne C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Differ. Equ. 141(2), 310–326 (1997) · Zbl 0890.35016
[51] Schlag W.: Stable manifolds for an orbitally unstable nonlinear Schrödinger equation. Ann. Math. (2) 169(1), 139–227 (2009) · Zbl 1180.35490
[52] Soffer A., Weinstein M.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990) · Zbl 0721.35082
[53] Soffer A., Weinstein M.: Multichannel nonlinear scattering, II. The case of anisotropic potentials and data. J. Differ. Equ. 98, 376–390 (1992) · Zbl 0795.35073
[54] Strauss W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149–162 (1977) · Zbl 0356.35028
[55] Strauss, W.A.: Nonlinear wave equations. In: CBMS Regional Conference Series in Mathematics, vol. 73. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI (1989) · Zbl 0714.35003
[56] Sulem C., Sulem P-L.: The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139. Springer, New York (1999) · Zbl 0928.35157
[57] Tao, T.: Nonlinear dispersive equations: local and global analysis. In: CBMS Regional Conference Series in Mathematics, vol. 106. American Mathematical Society, Providence, RI (2006) · Zbl 1106.35001
[58] Tsai T.P., Yau H.T.: Stable directions for excited states of nonlinear Schroedinger equations. Commun. Partial Differ. Equ. 27(11&12), 2363–2402 (2002) · Zbl 1021.35113
[59] Vanderbauwhede, A.: Centre manifolds, normal forms and elementary bifurcations. In: Dynamics Reported, vol. 2, pp. 89–169. Dynam. Report. Ser. Dynam. Systems Appl., 2, Wiley, Chichester (1989) · Zbl 0677.58001
[60] Weinstein M.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985) · Zbl 0583.35028
[61] Weinstein M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39(1), 51–67 (1986) · Zbl 0594.35005
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