On the blow up phenomenon for the $L^2$-critical focusing Hartree equation in $\Bbb R^4$

TL;DR

This paper investigates the finite time blow-up solutions of the focusing mass-critical Hartree equation in four-dimensional space, characterizing their dynamics, mass concentration, and minimal mass solutions using advanced inequalities and profile decomposition techniques.

Contribution

It provides a detailed analysis of blow-up behavior, including mass concentration and minimal mass solutions, for the $L^2$-critical focusing Hartree equation in $br^4$, employing refined inequalities and decomposition methods.

Findings

Characterization of minimal mass blow-up solutions.

Analysis of mass concentration phenomena.

Use of refined Gagliardo-Nirenberg inequality and profile decomposition.

Abstract

We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with $H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we also analyze the mass concentration phenomenon of such blow up solutions.