Strong instability of standing waves for Hartree equation with harmonic potential

This paper is concerned with the standing waves of Hartree equation with harmonic potential. Via construction of a cross-constrained invariant set for Hartree equation with harmonic potential, it is shown that if the initial data is in the invariant set then the corresponding solution blows up in finite time. Under an appropriate assumption on the frequency ω, the strong instability of standing waves is investigated by the above blow-up result. Furthermore, without the assumption we also study the property of the standing waves for Hartree equation with harmonic potential.