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

Abstract

We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger 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 $s=\frac{d}{2}-\frac{2}{p-1}$ 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 Schrödinger 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.

arXiv comment57 pages, 4 figures and updated references

References in corpus (2)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Cited by in corpus (1)

• Going beyond the threshold: scattering and blow-up in the focusing NLS equation