Divergent solutions to the 5D Hartree Equations

We consider the Cauchy problem for the focusing Hartree equation $i{u}_t+\mathrm{\Delta }u+(\mathrm{\mid }\cdot {\mathrm{\mid }}^{-3}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u=0$ in ${\mathbb{R}}^5$ with the initial data in $H^1$, and study the divergent property of infinite-variance and nonradial solutions. Letting $Q$ be the ground state solution of $-Q+\Delta Q+(|\cdot|^{-3}\ast|Q|^{2})Q=0 $in$ $\mathbb{R}$^{5}$,weprovethatif$u_{0}\in H^{1}$satisfying$M(u_0) E(u_0)<M(Q) E(Q)$and$|\nabla u_{0}|{2}|u{0}|{2} >|\nabla Q|{2}|Q|{2} ,$ then the corresponding solution $u(t)$ either blows up in finite forward time, or exists globally for positive time and there exists a time sequence $t_n\to +\mathrm{\infty }$ such that $|\nabla u(t{n})|_{2}\rightarrow+\infty.$ A similar result holds for negative time.