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Equivalence of conditions on initial data below the ground state to NLS with a repulsive inverse power potential

Masaru Hamano, Masahiro Ikeda

Abstract

In this paper, we consider the nonlinear Schrödinger equation with a repulsive inverse power potential. First, we show that some global well-posedness results and "blow-up or grow-up" results below the ground state without the potential. Then, we prove equivalence of the conditions on the initial data below the ground state without potential. We note that recently, we established existence of a radial ground state and characterized it by the virial functional for NLS with a general potential in two or higher space dimensions in [8]. Then, we also prove a global well-posedness result and a "blow-up or grow-up" result below the radial ground state with a repulsive inverse power potential obtained in [8].

Equivalence of conditions on initial data below the ground state to NLS with a repulsive inverse power potential

Abstract

In this paper, we consider the nonlinear Schrödinger equation with a repulsive inverse power potential. First, we show that some global well-posedness results and "blow-up or grow-up" results below the ground state without the potential. Then, we prove equivalence of the conditions on the initial data below the ground state without potential. We note that recently, we established existence of a radial ground state and characterized it by the virial functional for NLS with a general potential in two or higher space dimensions in [8]. Then, we also prove a global well-posedness result and a "blow-up or grow-up" result below the radial ground state with a repulsive inverse power potential obtained in [8].

Paper Structure

This paper contains 12 sections, 35 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Organization of the paper
  5. Preliminaries
  6. Notations and definitions
  7. Some tools
  8. Coercivity lemma and global well-posedness
  9. Blow-up or grow-up
  10. Blow-up
  11. Appendix A
  12. Appendix B

Key Result

Theorem \oldthetheorem

Let $d \geq 1$, $2 < p+1 < 2^\ast$, $\gamma > 0$, and $0 < \mu < \min\{2,d\}$. For every $u_0 \in H^1(\mathbb{R}^d)$, there exist $T_\text{max} \in (0,\infty]$, $T_\text{min} \in [-\infty,0)$, and a unique solution to NLS such that Here, the solution $u$ does not exist beyond the interval $(T_\text{min},T_\text{max})$ and the interval $(T_\text{min},T_\text{max})$ is called maximal lifespan of $u

Theorems & Definitions (66)

  • Theorem \oldthetheorem: Local well-posedness, Caz03
  • Definition \oldthetheorem: Scattering, Blow-up, Grow-up, and Standing wave
  • Theorem \oldthetheorem: Holmer--Roudenko, HolRou08
  • Theorem \oldthetheorem: Dinh, Din182
  • Proposition \oldthetheorem: Gagliardo-Nirenberg inequality without a potential, Wei82
  • Proposition \oldthetheorem: Minimization problem, HamIkeISAAC
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem: Boundedness versus unboundedness I
  • Theorem \oldthetheorem: Equivalence of conditions on the initial data below the ground state
  • ...and 56 more