Divergent solutions to the 5D Hartree Equations

TL;DR

This paper investigates the behavior of solutions to the 5D focusing Hartree equation, showing conditions under which solutions either blow up or grow unbounded over time, extending understanding of solution divergence in high dimensions.

Contribution

It establishes new divergence criteria for nonradial, infinite-variance solutions to the 5D focusing Hartree equation, including blow-up and unbounded growth results.

Findings

Solutions blow up in finite time under certain initial conditions.

Existence of solutions with unbounded gradient norm as time approaches infinity.

Results apply to nonradial, infinite-variance initial data.

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.