Sharp Threshold of Global Existence and Mass Concentration for the Schrödinger–Hartree Equation with Anisotropic Harmonic Confinement

Abstract

This article is concerned with the initial-value problem of a Schrödinger–Hartree equation in the presence of anisotropic partial/whole harmonic confinement. First, we get a sharp threshold for global existence and finite time blow-up on the ground state mass in the $L^2$ -critical case. Then, some new cross-invariant manifolds and variational problems are constructed to study blow-up versus global well-posedness criterion in the $L^2$ -critical and $L^2$ -supercritical cases. Finally, we research the mass concentration phenomenon of blow-up solutions and the dynamics of the $L^2$ -minimal blow-up solutions in the $L^2$ -critical case. The main ingredients of the proofs are the variational characterisation of the ground state, a suitably refined compactness lemma, and scaling techniques. Our conclusions extend and compensate for some previous results.