Divergent solutions to the 5D Hartree equations
Volume 125 / 2011
Abstract
We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+\varDelta u+(|\cdot|^{-3}\ast|u|^{2})u=0$ in $\mathbb{R}^{5}$ with initial data in $H^1$, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of $-Q+\varDelta Q+(|\cdot|^{-3}\ast|Q|^{2})Q=0 $ in $ \mathbb{R}^{5}$, we prove that if $u_{0}\in H^{1}$ satisfies $M(u_0) E(u_0) < M(Q) E(Q)$ 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.
