Threshold solutions for the Hartree equation

TL;DR

This paper studies special threshold solutions for the 5d focusing Hartree equation, establishing their existence, classification, and spectral properties, revealing solutions that either blow up, scatter, or approach a ground state exponentially.

Contribution

It extends the analysis of threshold solutions to the 5d Hartree equation, identifying and classifying special solutions analogous to those known for NLS.

Findings

Existence of special solutions Q+, Q- at the threshold

Classification of all radial threshold solutions

Spectral analysis of the linearized operator

Abstract

We consider the focusing $5$d Hartree equation, which is $L^2$-supercritical, with finite energy initial data, and investigate the solutions at the mass-energy threshold. We establish the existence of special solutions following the work of Duyckaerts-Roudenko [11] for the $3$d focusing cubic nonlinear Schr\"odinger equation (NLS). In particular, apart from the ground state solution $Q$, which is global but non-scattering, there exist special solutions $Q^+$ and $Q^-$, which in one time direction approach $Q$ exponentially, and in the other time direction $Q^+$ blows up in finite time and $Q^-$ exists for all time, exhibiting scattering behavior. We then characterize all radial threshold solutions either as scattering and blow up solutions in both time directions (similar to the solutions under the mass-energy threshold, see Arora-Roudenko [3]), or as the special solutions described above. To obtain the existence and classification result, in this paper we perform a thorough and meticulous investigation of the spectral properties of the linearized operator associated to the Hartree equation.