Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential

In this article, we conduct a comprehensive investigation into the global existence, blow-up and stability of standing waves for a $L^2$-critical Schrödinger-Choquard equation with harmonic potential. First, by taking advantage of the ground-state solutions and scaling techniques, we obtain some criteria for the global existence and blow-up of the solutions. Second, in terms of the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to the Choquard equation with $p_2=1+\frac{2+\alpha }{N}$, we explore the limiting dynamics of blow-up solutions to the $L^2$-critical Choquard equation with $L^2$-subcritical perturbation, including the $L^2$-mass concentration and blow-up rate. Finally, the orbital stability of standing waves is investigated in the presence of $L^2$-subcritical perturbation, focusing $L^2$-critical perturbation and defocusing $L^2$-supercritical perturbation by using variational methods. Our results supplement the conclusions of some known works.