Global Behavior Of Finite Energy Solutions To The $d$-Dimensional Focusing Nonlinear Schr\"odinger Equation

TL;DR

This paper establishes a sharp threshold for global existence, scattering, or blowup of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, extending previous 3D cubic results using concentration compactness methods.

Contribution

It generalizes known results for the 3D cubic NLS to higher dimensions and different nonlinearities, providing a unified threshold criterion based on the renormalized gradient.

Findings

Solutions with initial renormalized gradient less than 1 exist globally and scatter.

Solutions with initial renormalized gradient greater than 1 blow up or diverge.

The results extend previous 3D cubic NLS findings to broader settings.

Abstract

We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation (NLS), $i \partial_t u+\Delta u+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power $p$ and the dimension $d$ are such that the scaling index $s=\frac{d}2-\frac2{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $\ME[u_0]<1$ ($\ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient $\g_u$ of a solution $u$ to NLS is initially less than 1, i.e., $\g_u(0)<1,$ then the solution exists globally in time and scatters in $H^1$ (approaches some linear Schr\"odinger evolution as $t\to\pm\infty$); if the renormalized gradient $\g_u(0)>1,$ then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of $H^1$ 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.