Dispersive estimates, blow-up and failure of Strichartz estimates for the Schr\"odinger equation with slowly decaying initial data

TL;DR

This paper studies the Schrödinger equation with radially symmetric initial data that decays slowly, showing cases where well-posedness holds despite the failure of Strichartz estimates, highlighting limitations of these estimates.

Contribution

It demonstrates the global well-posedness of the Schrödinger equation for initial data with slow decay where Strichartz estimates do not apply, revealing new insights into dispersive behavior.

Findings

Well-posedness for slowly decaying initial data

Failure of Strichartz estimates in certain regimes

Identification of conditions where dispersive estimates break down

Abstract

The initial value problem for the homogeneous Schr\"odinger equation is investigated for radially symmetric initial data with slow decay rates and not too wild oscillations. Our global wellposedness results apply to initial data for which Strichartz estimates fail.