Threshold solutions for the focusing 3D cubic Schrödinger equation. (English) Zbl 1195.35276

Duyckaerts, Thomas; Roudenko, Svetlana

Rev. Mat. Iberoam. 26, No. 1, 1-56 (2010).

This paper is devoted to the study of the focusing 3D cubic nonlinear Schrödinger equation (NLS) with \(H^1\) data, \[ i\partial_t u+\Delta u+|u|^2 u=0, u(x,t)=u_0\in H^1, x\in\mathbb{R}^3. \] More precisely, the authors study the long time dynamics (i.e., scattering and blow up in finite time) of the solutions at the mass-energy threshold, namely, when \(M[u_0] E[u_0]=M[Q] E[Q]\). Here \(Q\) is the ground state, which is the unique positive radial solution of the equation \(-Q+\Delta Q+|Q|^2Q=0\).
This NLS is known to be locally well-posed in \(H^1\), and the solutions satisfy the energy and mass conservation laws \[ E[u](t)=\frac{1}{2}\int |\nabla u(x,t)|^2 dx -\frac{1}{4} \int |u(x,t)|^4 dx =E[u_0]\;, \]
\[ M[u](t)=\int |u(x,t)|^2 dx =M[u_0]\;. \] In earlier works of J. Holmer and S. Roudenko [Commun. Math. Phys. 282, No. 2, 435–467 (2008; Zbl 1155.35094)] and T. Duyckaerts, J. Holmer and S. Roudenko [Math. Res. Lett. 15, No. 5–6, 1233–1250 (2008; Zbl 1171.35472)], the behavior of solutions was classified when \(M[u_0]E[u_0]< M[Q]E[Q]\).
In this paper, the authors first exhibit 3 special solutions: \(e^{it}Q\) and \(Q_{\pm}\), where \(Q_{\pm}\) exponentially approach the ground state solution in the positive time direction, \(Q_+\) has finite time blow up and \(Q_-\) scatters in the negative time direction. Secondly, the authors classify solutions at this threshold and obtain that up to \(\dot H^{1/2}\) symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold.
These results are obtained by studying the spectral properties of the linearized Schrödinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.

Reviewer: Chengbo Wang (Baltimore)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35P25	Scattering theory for PDEs
35B40	Asymptotic behavior of solutions to PDEs
35B44	Blow-up in context of PDEs
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords:

nonlinear Schrödinger equation; scattering; profile decomposition; blow-up

Citations:

Zbl 1155.35094; Zbl 1171.35472

References:

[1]	Bahouri, H. and Gérard, P.: Concentration effects in critical nonlinear wave equation and scattering theory. In Geometrical optics and related topics (Cortona, 1996) , 17-30. Progr. Nonlinear Differential Equations Appl. 32 . Birkhäuser Boston, Boston, MA, 1997. · Zbl 0926.35090
[2]	Banica, V.: Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 1, 139-170. · Zbl 1170.35528
[3]	Bourgain, J.: Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Amer. Math. Soc. 12 (1999), no. 1, 145-171. JSTOR: · Zbl 0958.35126 · doi:10.1090/S0894-0347-99-00283-0
[4]	Cazenave, T.: Semilinear Schrödinger equations . Courant Lecture Notes in Mathematics 10 . New York University, Courant Institute of Mathematical Sciences, New York. AMS, Providence, RI, 2003. · Zbl 1055.35003
[5]	Coffman, C.V.: Uniqueness of the ground state solution for \(\Delta u-u+u^3=0\) and a variational characterization of other solutions. Arch. Rational Mech. Anal. 46 (1972), 81-95. · Zbl 0249.35029 · doi:10.1007/BF00250684
[6]	Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T.: Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on \(\mathbbR^3\). Comm. Pure Appl. Math. 57 (2004), no. 8, 987-1014. · Zbl 1060.35131 · doi:10.1002/cpa.20029
[7]	Duyckaerts, T., Holmer, J. and Roudenko, S.: Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett. 15 (2008), no. 6, 1233-1250. · Zbl 1171.35472 · doi:10.4310/MRL.2008.v15.n6.a13
[8]	Duyckaerts, T. and Merle, F.: Dynamic of threshold solutions for energy-critical NLS. To appear in Geom. Funct. Anal., 2007. · Zbl 1232.35150 · doi:10.1007/s00039-009-0707-x
[9]	Duyckaerts, T. and Merle, F.: Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP 2007 , no. 4, Art. ID rpn002, 67 pp. (2008). · Zbl 1159.35043 · doi:10.1093/imrp/rpn002
[10]	Gérard, P.: Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var. 3 (1998), 213-233 (electronic). · Zbl 0907.46027 · doi:10.1051/cocv:1998107
[11]	Gérard, P.: A critical profile decomposition for the three-dimensional cubic wave equation. Manuscript in preparation.
[12]	Ginibre, J. and Velo, G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case; II. Scattering theory, general case. J. Funct. Anal. 32 (1979), no. 1, 1-32, 33-71. · Zbl 0396.35028 · doi:10.1016/0022-1236(79)90077-6
[13]	Grillakis, M.: Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system. Comm. Pure Appl. Math. 43 (1990), no. 3, 299-333. · Zbl 0731.35010 · doi:10.1002/cpa.3160430302
[14]	Holmer, J. and Roudenko, S.: On blow-up solutions to the 3D cubic nonlinear Schrödinger equation. Appl. Math. Res. Express AMRX 2007 , no. 1, Art. ID abm004, 31pp. · Zbl 1130.35361 · doi:10.1093/amrx/abm004
[15]	Holmer, J. and Roudenko, S.: A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Comm. Math. Phys. 282 (2008), no. 2, 435-467. · Zbl 1155.35094 · doi:10.1007/s00220-008-0529-y
[16]	Kenig, C.E. and 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 (2006), no. 3, 645-675. · Zbl 1115.35125 · doi:10.1007/s00222-006-0011-4
[17]	Kenig, C.E. and Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation in the radial case. Acta Math. 201 (2008), no. 2, 147-212. · Zbl 1183.35202 · doi:10.1007/s11511-008-0031-6
[18]	Kenig, C.E. and Merle, F.: Scattering for \(\dotH^1/2\) bounded solutions to the cubic defocusing NLS in 3D. To appear in Trans. Amer. Math. Soc. · Zbl 1188.35180
[19]	Keraani, S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differential Equations 175 (2001), no. 2, 353-392. · Zbl 1038.35119 · doi:10.1006/jdeq.2000.3951
[20]	Killip, R., Tao, T. and Visan, M.: The cubic nonlinear Schrödinger equation in two dimensions with radial data. To appear in J. Eur. Math. Soc. · Zbl 1187.35237 · doi:10.4171/JEMS/180
[21]	Killip, R. and Visan, M.: The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. To appear in Amer. J. Math. · Zbl 1208.35138 · doi:10.1353/ajm.0.0107
[22]	Killip, R., Visan, M. and Zhang, X.: The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Anal. PDE 1 (2008), no. 2, 229-266. · Zbl 1171.35111 · doi:10.2140/apde.2008.1.229
[23]	Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u-u+u^p=0\) in \(\mathbbR^n\). Arch. Rational Mech. Anal. 105 (1989), no. 3, 243-266. · Zbl 0676.35032 · doi:10.1007/BF00251502
[24]	Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223-283. · Zbl 0704.49004
[25]	Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1 (1985), no. 2, 45-121. · Zbl 0704.49006 · doi:10.4171/RMI/12
[26]	Merle, F.: On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Comm. Pure Appl. Math. 45 (1992), no. 2, 203-254. · Zbl 0767.35084 · doi:10.1002/cpa.3160450204
[27]	Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69 (1993), no. 2, 427-454. · Zbl 0808.35141 · doi:10.1215/S0012-7094-93-06919-0
[28]	Merle, F. and Raphaël, P.: Blow up of the critical norm for some radial \(L^2\) super critical nonlinear Schrödinger equations Amer. J. Math. 130 (2008), no. 4, 945-978. · Zbl 1188.35182 · doi:10.1353/ajm.0.0012
[29]	Merle, F. and Vega, L.: Compactness at blow-up time for \(L^2\) solutions of the critical nonlinear Schrödinger equation in 2D. Internat. Math. Res. Notices 1998 , no. 8, 399-425. · Zbl 0913.35126 · doi:10.1155/S1073792898000270
[30]	Strauss, W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), no. 2, 149-162. · Zbl 0356.35028 · doi:10.1007/BF01626517
[31]	Tao, T.: 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. · Zbl 1106.35001
[32]	Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1982/83), no. 4, 567-576. · Zbl 0527.35023 · doi:10.1007/BF01208265
[33]	Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), no. 3, 472-491. · Zbl 0583.35028 · doi:10.1137/0516034
[34]	Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986), no. 1, 51-67. · Zbl 0594.35005 · doi:10.1002/cpa.3160390103