Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential

In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $0<c\le c^{\ast }$ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $0<c<\frac{N^2+4N}{(N+2{)}^2}{c}^{\ast }$. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $L^2$-critical setting for $0<c<c^{\ast }$, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.