## 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

### Citations:

Zbl 1213.35307; Zbl 1195.35276; Zbl 1232.35150
Full Text:

### References:

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