Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS

TL;DR

This paper characterizes minimal-mass blowup solutions to the focusing mass-critical nonlinear Schrödinger equation in dimensions four and higher, showing they are either the ground state or the pseudo-conformal ground state, up to symmetries.

Contribution

It proves that non-scattering solutions with minimal mass are either the ground state or the pseudo-conformal ground state, extending previous results to higher dimensions.

Findings

Minimal-mass blowup solutions are characterized as either the ground state or pseudo-conformal ground state.

In dimensions d ≥ 4, these solutions are unique up to symmetries.

The results unify and extend prior classifications of blowup solutions.

Abstract

Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schr\"odinger equation $iu_t+\Delta u=-|u|^{\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$. We prove that if $u$ does not scatter then, up to phase rotation and scaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that of Merle \cite{merle2}, we obtain that in dimensions $d\geq 4$, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave.