Nonscattering solutions to the $L^{2}$-supercritical NLS Equations

TL;DR

This paper studies solutions to the energy-critical nonlinear Schrödinger equation with supercritical nonlinearity, identifying conditions under which solutions either blow up or become unbounded in gradient norm over time.

Contribution

It extends previous results on blow-up and divergence properties to a broader class of mass-supercritical and energy-subcritical NLS equations.

Findings

Solutions blow up in finite time or diverge in gradient norm under certain conditions.

Results generalize known 3D cubic NLS blow-up criteria to higher dimensions and different nonlinearities.

Provides conditions involving mass and energy thresholds for solution behavior.

Abstract

We investigate the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+|u|^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1, 2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of infinite-variance and nonradial solutions. If $M(u)^{\frac{1-s_{c}}{s_{c}}}E(u)<M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ and $\|u_{0}\|_{2}^{\frac{1-s_{c}}{s_{c}}}\|\nabla u_{0}\|_{2}>\|Q\|_{2}^{\frac{1-s_{c}}{s_{c}}}\|\nabla Q\|_{2},$ then either $u(t)$~blows up in finite forward time, or $u(t)$ exists globally for positive time and there exists a time sequence $t_{n}\rightarrow+\infty$ such that $\|\nabla u(t_{n})\|_{2}\rightarrow+\infty.$ Here $Q$ is the ground state solution of $-Q+\Delta Q+|Q|^{p-1}Q=0.$ A similar result holds for negative time. This extend the result of the 3D cubic Schr\"{o}dinger equation in \cite{holmer10} to the general mass-supercritical and energy-subcritical case .