Abstract. We consider the problem of identifying sharp criteria under which radial H 1 (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i∂ t u + ∆u + |u| 2 u = 0 scatter, i.e. approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the, where u 0 denotes the initial data, and M [u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution e it Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if, then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution e it Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if, then the solution blows-up in finite time. The technique employed is parallel to that employed by in their study of the energy-critical NLS.

Abstract. Scattering of radial H 1 solutions to the 3D focusing cubic nonlinear Schrö-dinger equation below a mass-energy thresholdwhere Q is the ground state, was established in Holmer-Roudenko [7]. In this note, we extend the result in [7] to non-radial H 1 data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle [10], we control via momentum conservation the rate of divergence of the spatial translation parameter and by a convexity argument based on a local virial identity deduce scattering. An application to the defocusing case is also mentioned.

In this article we introduce Triebel-Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Using it we derive molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition we prove optimal trace theorem in the variable indices case.

We study the focusing 3d cubic NLS equation with H 1 data at the mass-energy threshold, namely, when. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when. In this paper, we first exhibit 3 special solutions: e it Q and Q ± , where Q is the ground state, Q ± 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, we classify solutions at this threshold and obtain that up toḢ 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.

We consider the 3d cubic focusing nonlinear Schrödinger equation (NLS) i∂ t u + ∆u + |u| 2 u = 0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H 1 axial (R 3 ) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Raphaël's construction [33] of a family of solutions to the 2d quintic focusing NLS, i∂ t u+∆u+|u| 4 u = 0, that blow-up on a circle.By multiplying the above two inequalities, and then integrating against rdrdz, we getFollowing through with Cauchy-Schwarz in each of the two integrals gives (2.1). ∂ s exp 5π 4 d 0 J ∼ J −3/2 (−J s ) exp 5π 4 d 0 J − J s bΓ b ≥ 1.