Abstract
We extend the result in Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011) 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 Duyckaerts and Roudenko (Rev Mater Iberoam 26(1):1–56, 2010), following the seminal work on threshold solutions by Duyckaerts and Merle (Funct Anal 18(6):1787–1840, 2009), appear here as the unique one-dimensional unstable/stable manifolds emanating from the ground states. In analogy with Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011), 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 Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011) and this paper is the need to control two modulation parameters.
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References
Bahouri H., Gérard P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121(1), 131–175 (1999)
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)
Beceanu M.: A centre-stable manifold for the focussing cubic NLS in \({\mathbb {R}^{1+3}}\) . Commun. Math. Phys. 280(1), 145–205 (2008)
Beceanu, M.: New estimates for a time-dependent Schrödinger equation (preprint 2009), to appear in Duke Math. J.
Beceanu, M.: A critical centre-stable manifold for the Schroedinger equation in three dimensions (preprint 2009), to appear in Commun. Pure Appl. Math.
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)
Berestycki H., Lions P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983)
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)
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)
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)
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)
Cazenave T., Lions P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85(4), 549–561 (1982)
Coffman C.: Uniqueness of the ground state solution for Δu − u + u 3 = 0 and a variational characterization of other solutions. Arch. Ration. Mech. Anal. 46, 81–95 (1972)
Cuccagna S.: Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54(9), 1110–1145 (2001)
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)
Demanet L., Schlag W.: Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation. Nonlinearity 19(4), 829–852 (2006)
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)
Duykaerts T., Roudenko S.: Thresholdsolutions for the focusing 3D cubic Schrödinger equation. Rev. Mater. Iberoam. 26(1), 1–56 (2010)
Duyckaerts T., Holmer J., Roudenko S.: Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett. 15(6), 1233–1250 (2008)
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)
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)
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)
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)
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)
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)
Grillakis M.: Linearized instability for nonlinear Schrödinger and Klein–Gordon equations. Commun. Pure Appl. Math. 41, 747–774 (1988)
Grillakis M.: Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system. Commun. Pure Appl. Math. 43, 299–333 (1990)
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)
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)
Hirsch M.W., Pugh C.C., Shub M.: Invariant manifolds. In: Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)
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)
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)
Ibrahim, S., Masmoudi, N., Nakanishi, K.: Scattering threshold for the focusing nonlinear Klein–Gordon equation. to appear in Anal. PDE
Keel M., Tao T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998)
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)
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)
Keraani S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equation. J. Differ. Equ. 175, 353–392 (2001)
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)
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)
Kwong M.: Uniqueness of positive solutions of Δu + u + u p = 0 in \({\mathbb {R}^{n}}\) . Arch. Ration. Mech. Anal. 105(3), 243–266 (1989)
Marzuola J., Simpson G.: Spectral analysis for matrix Hamiltonian operators. Nonlinearity 24, 389–429 (2011)
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)
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)
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)
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)
Merle, F., Raphael, P., Szeftel, J.: The instability of Bourgain–Wang solutions for the L 2 critical NLS, preprint, arXiv:1010.5168
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)
Ogawa T., Tsutsumi Y.: Blow-Up of H 1, solution for the Nonlinear Schrödinger Equation. J. Differ. Equ. 92, 317–330 (1991)
Perelman G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. Henri Poincaré 2(4), 605–673 (2001)
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)
Schlag W.: Stable manifolds for an orbitally unstable nonlinear Schrödinger equation. Ann. Math. (2) 169(1), 139–227 (2009)
Soffer A., Weinstein M.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990)
Soffer A., Weinstein M.: Multichannel nonlinear scattering, II. The case of anisotropic potentials and data. J. Differ. Equ. 98, 376–390 (1992)
Strauss W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149–162 (1977)
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)
Sulem C., Sulem P-L.: The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse. Applied Mathematical Sciences, vol. 139. Springer, New York (1999)
Tao, T.: Nonlinear dispersive equations: local and global analysis. In: CBMS Regional Conference Series in Mathematics, vol. 106. American Mathematical Society, Providence, RI (2006)
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)
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)
Weinstein M.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985)
Weinstein M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39(1), 51–67 (1986)
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Nakanishi, K., Schlag, W. Global dynamics above the ground state energy for the cubic NLS equation in 3D. Calc. Var. 44, 1–45 (2012). https://doi.org/10.1007/s00526-011-0424-9
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DOI: https://doi.org/10.1007/s00526-011-0424-9