On the inhomogeneous biharmonic nonlinear Schr\"odinger equation: local, global and stability results

We consider the inhomogeneous biharmonic nonlinear Schr"odinger equation (IBNLS) $$i{u}_t+{\mathrm{\Delta }}^{2}u+\lambda \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u=0,$$ where $\lambda =\pm 1$ and $\alpha$, $b>0$. We show local and global well-posedness in $H^s({\mathbb{R}}^N)$ in the $H^s$-subcritical case, with $s=0,2$. Moreover, we prove a stability result in $H^2({\mathbb{R}}^N)$, in the mass-supercritical and energy-subcritical case. The fundamental tools to prove these results are the standard Strichartz estimates related to the linear problem.