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

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$N=1, 2$,$1+\frac{4}{N}<p<\infty$)inenergyspace$H^1$ 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 $|u_{0}|{2}^{\frac{1-s{c}}{s_{c}}}|\nabla u_{0}|{2}>|Q|{2}^{\frac{1-s_{c}}{s_{c}}}|\nabla Q|_{2},$theneither$u(t)$blowsupinfiniteforwardtime,or$u(t)$ exists globally for positive time and there exists a time sequence $t_n\to +\mathrm{\infty }$ such that $\mathrm{\parallel }\mathrm{\nabla }u(t_n){\mathrm{\parallel }}_2\to +\mathrm{\infty }\mathrm{.}$ Here $Q$ is the ground state solution of $-Q+\mathrm{\Delta }Q+\mathrm{\mid }Q{\mathrm{\mid }}^{p-1}Q=0.$ 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 .