Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on minimal mass blow up for inhomogeneous NLS

Chenjie Fan, Shumao Wang

Abstract

We revisit the work of \cite{raphael2011existence} on the minimal mass blow up solution for $iu_{t}+Δu=-k(x)|u|^{2}u$, and extend the construction of such a solution to the $k\in C^{2}$ case.

A note on minimal mass blow up for inhomogeneous NLS

Abstract

We revisit the work of \cite{raphael2011existence} on the minimal mass blow up solution for , and extend the construction of such a solution to the case.
Paper Structure (34 sections, 12 theorems, 116 equations)

This paper contains 34 sections, 12 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Statement of main results and motivations
  3. Background for NLS and constructive blow up
  4. Structure of the article
  5. Notation
  6. Acknowledgement
  7. Preliminary
  8. The spectral structure
  9. Basics
  10. Coersiveness
  11. The elliptic object
  12. The modulation equations
  13. Motivation for the construction of $Q_{\mathcal{P}}$
  14. Setting up and main ansatz
  15. Overview
  16. ...and 19 more sections

Key Result

Theorem 1.2

Under Assumption as: main, there exists initial data $u_{0}\in H^{1}$ with $\|u_{0}\|_{L_{x}^{2}}=\|Q\|_{L_{x}^{2}}$, such that the associated solution blows up in finite time.

Theorems & Definitions (25)

  • Theorem 1.2
  • Remark 1.3
  • Lemma 3.1: Lemma 3.2 in raphael2011existence
  • Lemma 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Remark 4.1
  • ...and 15 more