On the inhomogeneous biharmonic nonlinear Schrödinger equation: local, global and stability results

Abstract

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS) $$i{u}_t+{\Delta }^{2}u+\lambda |x{|}^{-b}|u{|}^{\alpha }u=0,$$ 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.