Threshold solutions for the $3d$ cubic INLS: the energy-subcritical case

TL;DR

This paper extends the classification of solution dynamics at the ground state threshold for the 3D cubic inhomogeneous nonlinear Schrödinger equation to the full energy-subcritical range by modifying analysis techniques.

Contribution

It introduces a modified modulation analysis and employs Strichartz estimates to generalize previous results to a broader parameter range.

Findings

Extended classification to the full energy-subcritical range $b ext{ in }(0,1)$.

Modified analysis techniques applicable to other dispersive equations with singular potentials.

Demonstrated the effectiveness of Strichartz estimates over pointwise bounds.

Abstract

We revisit the work [L. Campos and J. Murphy, SIAM J. Math. Anal., 55 (2023), pp. 3807--3843], which classified the dynamics of $H^1$ solutions at the ground state threshold for cubic inhomogeneous nonlinear Schr\"odinger equations of the form $i\partial_t u + \Delta u + |x|^{-b}|u|^2 u = 0$ in the range $b\in(0,\tfrac12)$. By modifying the modulation analysis and using Strichartz estimates in place of pointwise bounds, we extend the result to the full energy-subcritical range $b\in(0,1)$. This strategy is expected to carry over to other dispersive equations with singular potentials.