Sharp blow-up result for the intercritical Inhomogeneous NLS equation

TL;DR

This paper constructs radial blow-up solutions for the intercritical inhomogeneous nonlinear Schrödinger equation, demonstrating the sharpness of previously established upper bounds on blow-up speed.

Contribution

It introduces the first explicit construction of ring blow-up solutions and confirms the optimality of known blow-up speed bounds for this equation.

Findings

Constructed radial blow-up solutions with ring structure.

Proved the blow-up speed reaches the established upper bound.

Validated the sharpness of previous theoretical bounds.

Abstract

In this paper, we consider the intercritical inhomogeneous nonlinear Schr\"odinger equation. For the radial symmetry initial data, we construct the ring blow-up solutions and obtain blow-up speed. This result implies that the upper bound on the blow-up speed given by Cardoso and Farah [J. Funct. Anal.,281(8) No. 109134, (2021)] is sharp.