The 3D energy-critical inhomogeneous nonlinear Schrodinger equation with strong singularity

Abstract

In this paper, we study the Cauchy problem for the 3D energy-critical inhomogeneous nonlinear Schrödinger equation(INLS) $$i{\partial }_{t}u+\Delta u=\pm |x{|}^{-\alpha }|u{|}^{4-2\alpha }u$$ with strong singularity $3/2\le \alpha <2$. The well-posedness problem is well-understood for $0<\alpha <3/2$, but the case $3/2\le \alpha <2$ has remained open so far. We address the local/small data global well-posedness result for $3/2\le \alpha <11/6$ by improving the inhomogeneous Strichartz estimates on the weighted space.

25pages, 3 figures