Well-posedness in weighted spaces for the generalized Hartree equation with $p<2$

TL;DR

This paper proves local and global well-posedness results for the generalized Hartree equation with low nonlinearity powers in weighted Sobolev spaces, and also identifies blow-up solutions in the focusing case.

Contribution

It establishes well-posedness in weighted spaces for $p<2$, extending previous results, and analyzes blow-up phenomena in the focusing, supercritical regime.

Findings

Local well-posedness in weighted Sobolev spaces

Global existence and scattering for certain data

Finite-time blow-up in focusing supercritical case

Abstract

We investigate the well-posedness in the generalized Hartree equation $iu_t + \Delta u + (|x|^{-(N-\gamma)} \ast |u|^p)|u|^{p-2}u=0$, $x \in \mathbb{R}^N$, $0<\gamma<N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin [6]. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.