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

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

2011-01-12

arxiv.org/abs/1101.2271v2

Mathematics

Analysis of PDEs

Scientific paper

We investigate the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+|u|^{p-1}u=0$ with $1+\frac{4}{N}\|Q\|_{2}^{\frac{1-s_{c}}{s_{c}}}\|\nabla Q\|_{2},$ 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 .

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