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

TL;DR

This paper investigates the well-posedness of the 3D energy-critical inhomogeneous nonlinear Schrödinger equation with strong singularity, extending understanding to cases where the singularity parameter is between 3/2 and 11/6.

Contribution

It establishes local and small data global well-posedness results for the INLS with strong singularity, improving inhomogeneous Strichartz estimates in weighted spaces.

Findings

Proved well-posedness for $3/2 \,\leq\, \alpha < 11/6$

Enhanced inhomogeneous Strichartz estimates for weighted spaces

Extended the understanding of INLS with strong singularity

Abstract

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