Divergent solutions to the 5D Hartree Equations

Abstract

We consider the Cauchy problem for the focusing Hartree equation $i{u}_t+\Delta u+(|\cdot {|}^{-3}\ast |u{|}^2)u=0$ 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 $\Vert \nabla u_0{\Vert }_2\Vert u_0{\Vert }_2>\Vert \nabla Q{\Vert }_2\Vert Q{\Vert }_2,$ 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 $t_n\to +\infty$ such that $\Vert \nabla u(t_n){\Vert }_2\to +\infty .$ A similar result holds for negative time.

References in corpus (1)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation