On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation

Van Duong Dinh

doi:10.3934/cpaa.2019034

Article Contents

2019, Volume 18, Issue 2: 689-708. Doi: 10.3934/cpaa.2019034

This issue Previous Article Infinite energy solutions for the (3+1)-dimensional Yang-Mills equation in Lorenz gauge Next Article Weak solutions to stationary equations of heat transfer in a magnetic fluid

On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation

Van Duong Dinh

Institut de Mathématiques de Toulouse UMR5219, Université Toulouse CNRS, 31062 Toulouse Cedex 9, France

Received: January 2018

Revised: May 2018

Published: October 2018

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Abstract

In this paper we study dynamical properties of blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the $L^2$-concentration and the limiting profile with minimal mass of blow-up solutions.

Keywords:

Nonlinear fractional Schrödinger equation,
local well-posedness,
blow-up,
concentration,
limiting profile.

Mathematics Subject Classification: Primary: 35B44; Secondary: 35Q55.

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Table 1. Local well-posedness (LWP) in $H^s$ for NLFS

	$s$	$\alpha$	LWP
$d=1$	$\frac{1}{3}<s<\frac{1}{2}$	$0<\alpha<\frac{4s}{1-2s}$	$u_0$ non-radial
$d=1$	$\frac{1}{2}<s<1$	$0<\alpha<\infty$	$u_0$ non-radial
$d=2$	$\frac{1}{2}<s<1$	$0<\alpha<\frac{4s}{2-2s}$	$u_0$ non-radial
$d=3$	$\frac{3}{5}\leq s\leq \frac{3}{4}$	$0<\alpha<\frac{4s}{3-2s}$	$u_0$ radial
$d=3$	$\frac{3}{4}<s< 1$	$0<\alpha<\frac{4s}{3-2s}$	$u_0$ non-radial
$d\geq 4$	$\frac{d}{2d-1}\leq s<1$	$0<\alpha<\frac{4s}{d-2s}$	$u_0$ radial

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