Divergent solutions to the 5D Hartree Equations

Abstract

We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+\Delta u+(|\cdot|^{-3}\ast|u|^{2})u=0$ 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}$, we prove that if $u_{0}\in H^{1}$ 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 $t_{n}\rightarrow+\infty$ such that $\|\nabla u(t_{n})\|_{2}\rightarrow+\infty.$ A similar result holds for negative time.