Global Behavior Of Finite Energy Solutions To The d-Dimensional Focusing Nonlinear Schrödinger Equation

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We study the global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation (NLS), i ∂ₜ u+Δu+ |u|ᵖ⁻¹u=0, with initial data u₀∈ H¹, x ∈ Rⁿ. The nonlinearity power p and the dimension d are such that the scaling index s=d2-2p-1 is between 0 and 1, thus, the NLS is mass-supercritical (s>0) and energy-subcritical (s<1). For solutions with [u₀]<1 ([u₀] stands for an invariant and conserved quantity in terms of the mass and energy of u₀), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient ᵤ of a solution u to NLS is initially less than 1, i.e., ᵤ(0)<1, then the solution exists globally in time and scatters in H¹ (approaches some linear Schrödinger evolution as t→±∞); if the renormalized gradient ᵤ(0)>1, then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of H¹ norm in infinite time. This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle.